(CONTENTS)(UNIT)
3TOTRI-1
$<t>i^ 1 - HiRi<i>ai (Probability)
1.1 v3t?^r (Objectives)1.2 y'Wid'i! (Introduction)
1.3 3Ts| rfSTT 'ctR’W (Meaning and Definition)
1.4 wiRlcp'di W\ (Measurement of Probability)
MO
1
1
2
2
51.5
1.6 ^ ^ (Bayes’s Theorem) •
1.7 . TlRRl (Summary)
3P^JT^-y^ (Exercise Questions) .• TfsT (Reference Books)
2 - Jrrf^raHfT (Probability Distribution)
2.1 (Objectives)
2.2 y'Widdi (Introduction)
2.3 'dMd'lRini (Utility of Theoretical Distribution)
2.4 HiRtd>ai ^ (Normal Probability Curve)
2.5 TfPTHl MiRichdl ^ ^ ^Tc^Rl ^ 'JcpcfPT (Use of the Concept of Norma; P'-obability Curve)
2.6 Hlkllcbl ^ Rb'eTl (Normalizing a Distribution ct Scores)
2.7 'tiixi?! (Summary)
(Exercise Questions)• 179T (Reference Books)
3 - 'tiC'CiVtT HcflnM*!-! (Correlation and Regression)- , #»
3.1 (Objectives)
3.2 wwidii (Introduction)
3.3 3uRicp (Partial Correlation Coefficient)
7
10
1010
11-54
111112
2834
4653
5454
55-83
55
5558653.4
3.5 (Multiple Regression Equation)
3.7 • IIchi' ^ 'tii^cpni (Significance of Beta Coefficient)
3.8 xiixRi (Summary)
(Exercise Questions)• 7121 (Reference Books)
69
3.6 ^ ^ 3Tf^ 72
80
83
83
83
iii
1 > ftcH't**! (Sampling Distribution)
1.1 (Objectives)
1.2 wwRni (Introduction)
• 1.3 'tiMpc (Finite Population)V-'
1.4 ^t-Ro'cwi (t-Distribution)1.5 X2-f^d'e«i aff? §<i<T>i (X2-Distribution and their Uses)
1.6 fitter*! (Degrees of Freedom in Chi-Square Test)
1.7 ' 'HHR «J>\ mRcpcm*ii (Hypothesis of Equal Distribution)
1.8 'HIHRI fcTcR^ ^ hRcpch'Ii (Hypothesis of Normal Distribution)
1.9 ^ qRqxyq'ii (Hypothesis of Independent Distribution)
1.10 (Summary)
3FiTRT-y^ (Exercise Questions)• ^ (Reference Books)
84-1038484
89
9091 ,94
969799
102
103103
5^1^ 2- ^ uIT^ (Test of Hypothesis)
2.1 (Objectives)
2.2 H'cojq'ii (Introduction)
104-163
104
105
2.3 105
2.4 MTT (Standard Error)
2.5 RRN ^ ^qlRtdl (Utility of The Standard Error Concept)
.2.6 (t-Test)2.7 ^ 3RR ^ Wsf^irlT (Si^ificance of Difference between Related Means)
2.8 3pzi ^ 3T^ ^ 'tii4q>oi (Significance ofDifference between other Statistics )
2.9 (Assumptions'of t-Test)
108
108115
124.
131
139
2.10 (Analysis of Covariance)2.11
147^FRT (Computational Process of Analysis of Covariance)
^-yRRUf ^ 3RTPf%cT RRIcTiq (Assumptions underlying Analysis of Covariance)
2.13 (Summary)
(Exercise Questions)• TIST (Reference Books)
• 1512.12 160
162
• " 162163
iv
$<pi^ 1- ^ RKia (Theory of Sampling)
1.1 (Objectives)1.2 Hwiq*!! .(Introduction)
1.3 cTsn JITcfcT (Statistics and Parameters)
1.4 ^ (Population and Sample)
1.5 ^TRRT (Summary)
(Exercise Questions)• (Reference Books)
164-17616416416S167176
176176
(Design and Analysis of Variance)
2.1 (Objectives)
2.2 M’Wiq'ii (Introduction),
2.3 tjTiP-tenr (F-Disdibution)
177-202
177
178
178
^ ^«siPciq> ^ (Theoretical Aspect of Analysis of Variance)
viQbili (Process of Analysis of Variance) fcl?^Tftr|tq^T^ ^TiafgJcTT (Post ANOVA Test of Significance)
^ HWIHhIi’ ^ ^cHT. ^ (ANOVA for Comparison of Two Sample Means)
♦ii-qQiv' (Assumptions of ANOVA)
^TRRT (Summary)
1802.41832.51902.6
(ANOVA by Subtracting a Constant from the Row Scores) 194198
2.7
2.82002.92012.10
202(Exercise Questions) • Tisi (Reference Books) 202
\\
r
\
V
3TSZIFT-I
i^if^chai
(Probability)
(Structuro)
1.1 (Objectives)1.2 Mwuq-ii (Introduction)1.3 (Meaning and Definition)1.4 MiP^qiai ^ ^ (Measurement of Probability)1.5 (Probability Theorems)/1.6 cfn (Bayes’s Theorem)1.7 (Summary)
• (Exercise Questions)• (Reference Books)
1.1 (Objectives)
• lilRl4>dl 3T^I
• ylf4«l>dl ^ iinT
• Mifq(T>ai tWT ^
1.2 UWIcW! (Introduction)
^ MfedI ftrsra (Theory of-ProbabiUty) % ^ ^ ^fhcft «lti ^ (Goddess of Fortune) 3T«i^ fq^tqw (super-stitions) ^
^101^ % % 'SiqtlO ^*qR«T5T epT % fci*<IjftrSi % ■’ini 'Sn^ "3^ VRTf^ R ^ ^ ^3Tf
(Chevalier de Mere) % ^ ^ WITiq«q<lU
■snm t. ^ ■331^1 tter -qi^ ^ 24 4 ^-■^:■^Fq^qi "3^?% T^i^T ^ ■q? n^ ^tnf^ifT# "5^ 'nPnw(Blaise Pascal) % 33Rn, 'qM ^ t^^ldTI ^ ^ 3eTT
3T^ q^fcl^ 33131 83, qftr^ 'EPTfRRft 'qf^TcI^ (Pierre de Fermat) %■m^ f^i jB q^JR qrfq^Rn ftraHi ^"% q^i 3h1r'^' % qi^ ■4' qqq % qfti^qfq35T cTF^ (Laplace) 3^ q^fq ^Pwla^l RTr (Gauss) fR f^^i'rt ^33 RFW'jyf 3fll X3^tq 3^«f-o4^^ ■% fqKiK ■% 3T^ HSi*! 3^33^, 3T^ (De Moivre-1718),qqfer^5ft 33^ q^ ■% ^^ft3 "^l ■^' PtiJfK (R. A. Fisher), (Pearson), '^.(J. Neyman) 37Tfq qrfq3i3T fRSRT ‘'TC 3TRITft3 qlqq^f-'ftrSRT (Sampling Theory) 33 fq«hi'H %3TI '
as
1.3 3Tsf TOT TTfr^TiRT (Meaning and Definition)
qrfqqRTT ^ l^ehW 33 r3) q^<a 3nrq "qtqq % crrqq qq^ ^ ^-3c3^(Random phenomenon) 3)t 34lV>!lf3 T?3T "tl f?s7f3 ^ "f qfq 3R% ^Ruiih 3r3?R RT fq^h: 3R^ t‘i RRt ^13=^ qRuim '^^■533^ R3=r^ f, qc^ q^ ■^' 3:3: q33\ 33 33\ qR^im . ^Iqr, ■'^-fqqfftq q fqjqr "qr qrqiqThR'^jih yiPqchdi 3T?T 133T 33 R3)3T "f I '^3 3f3 3fl ■shR^Pi 33 3eJieT% qpq qfRJimT ^ -qr WRn ti qsrfq qftqrq ^ ^ f % 33^ Ri3% ^3yr^ qr 3? 3T qi fqtqr'qr q^ q^^ 3313^ ■4' 3f 3t q^ fq^ 3?a73?R-3nT3t ^ fqqffct 3R^ ti 3?! q^ qj^ ^ fq3f?qiT3T3J P-iqpMddi q#' ?qt q3TR qq: qi^ 37t ^3T^ qi: ?q 3? ij^-lqqfqq q^' qix 33?^ % qi^ 33 3?N-rt 3nR 33^. qtffq qqt Rwqf^ qft’qrq w 'fl ^“3?3 13^ «HiH<t) R3 fq^^n '^' ¥131 ^ 13Pq^ii'il % 3l1 qii33>3T ■ftf^[F3 33 Wq 3TT3733r q|t3T ^1
TT^ 33^ 5, q^3> q^qq■RTciqq 33T5^ 33) R=t4) %
■^iRs^qTl "4 yiR«t)di 33 13^3 qF?3 33l1% 343) ftfSIRT R3 ttf331 '531 f33R qr 333l1^ ■4 qif33Kn '?q4 qftqq 4 "fq^q <.<adl 4 qqi 5333 qqlq l^-qfqf^ "3)1 '3t3f3T3t 4 "^qi
ti ^ qR; 53 qqqr 4r qrqq t?4 f : "aRST 4 % ^ m(i 4 41, 33lf^ qiql 3^ 34 qraqqqr t”, “argqr 33I33 % 4134 34 ■Rpqrqqr 3341 33t t'\ '*3f qpqTqqr f 14 533 qmr 33^
oijf^ 4> q4^ 4 qRT 414 34 ‘^Rqrqqr qqrq-qqiq t*'', ■qq^ “q^qql 4 arfqf^qqqi 34 qiqqr 34 ^?33 .f4>qi 331 "ti '414 (Goethe) 4 qjqr t" 14,
fi 3k33
qrft 3t4”, “q«R3: g*3Rl 3T3 q4141” arrf^i ft
‘*aT5tT33T 4 334 3^4 4 a4v3T ^ 34 3m 4k 34f’q41 ti q^iTciqi^i 4 4q 4 34 3)t4 4 arfqttqqqr arqqr arqiq 34 qrq 3r4 33 qqrT 1431 ■qrqr ti qTfeT34 l35Tiq 4 ‘q»333T’ -3T33t qTt33)3T 33 374 37f33T 4l qffel34 4 31133)31 14^4 R3> 3331 4 "334 34 141^3331 3134 33 1^3) W<2>HlcH«t) "qrq 4
1.4 Mif^ch^n ^ tm (Measurement, of Probability)
qTl33)3T 34 "RR "^l ^<SHa: 3lq 14331 "tl 4 43143) '^feq^q 33 yiaPif43 eb<a1 t"!4f-
2 3^3rR
"^IT 4^41^ofi (Priori Probability)—sJif^'^di "%Ti?n ■% 3TfT: Mli^=hai % HlH"^ ^ fsrfv ^ ^ '^?IT3Tf % fq^hf^a ^ "tl■jn^T^vriT TTN^ ^ fgfil (Classical) (Priori) yilq«nai "5^^ t'I^ T^EF ■Efz^n % ytr^+di % 3T3^ mRuiwT ^ WTr ^mRijiihT ^ 315'M ■f^. 'SRf^ '3^' hRuiW ^ ^?R5tTT "t (The probability ofan event is simply the ratio of number of favourable observes of an event to the number of possible observes, where each outcome is.equally likely to occur) I 37^ 7T^'
3T^ tT2 i\ eft ^ ■ST2=TT 'Sflf^T^ -3^
% '^Z ^ '^I33T'3ft' ^ Wssn % sr^’TRT ■^' '?Wt!■srCTT ‘A’ ^ 3^3^ ■ET3 ■5n^ % ‘o' w.3T5^ 3 -HTE -sn^ % 'b’w^ yRtJiiH thR'^U^ "t ct®IT ^ ai^q-jH ■!■' ^ A ■^T33T % Hlfq^ell {P (A)} bW—
P(A) =0+0
nif^<f)ai
Number of outcomes favourable to occurence of A .Total number of possible outcomes
38T] ■^1331 A 3 ^ y!fqq>cil {q (A)}
9(A) =0 + 0
Number of outcomes not favourable to occurence of ATotal number of possible outcomes
’3i9Kni (toss) ■^, cit 1^ (Head) fbr^ ^
^ — , «pftt^ yf^^' '^1331 ^ W<sm 1 "t cf^TT W-'>^ Tft’JTWf W<sqi 2 "t I Riq<^ 2 *'
% (Tail) q =
HP =H + T
T 1= - ■^l
___ H + T 2tT^=n‘% ■si3^ (P) alh: R ■«t3^ (q) ^ 1 ^ ti
airarr^ mkuii^(i) M^tu (Random Experiment)—Mifq«hai ■^I3313ft % hR'^iihT 'SPlbT'
3T«T5fT «ftf^ ^ ■4't. W ^ ^ tl ‘y^’ ^ 3Ta^37f^ tl
TRFmTRqWm^ TTiThT tl 3pM % ^ Tpr ttt t-
(3f) y^l ■% mRwiih ttt t, n»>qi ^ ■R^icTT t,. (^) yr<)«ti xqVi% AiriRqci ttt tl Riq«hl t, PT^ f^RTT ^ t 1V Riq«hl■^n RrtTTT, Rpg ■?? m f3f^ ^ ^ "3^ t ■qF Mnr■qr FR ^^iftny, <iHqVl t t hR'JIIH ^ '^tBT t ‘^’-■^TF PT^ ^RFT t % ^ 37f3f^ F^ tl
' (ii) PPPT PW ^ P3r% PTcfl y<£^ri| (Equally Likely Events).—yifqehdi Ri«&m ^ fqq<,»Ji t *PPFT qiqqRi ^ 3PfW 1%^ ^TIFT tl ^331^ t
^I33T P^ tl 13^ 1p^ PI ^ PT P^, 3TcT: PPB PP ^ PIvft P3Pnf^ tl ‘
(Hi) yffW[?f-pJJ^ (Sample Space)-f^ pfipnpf PP\P^ (set or collection) fIft tl IPPPiT <j?9id^ ^ pt Ft PftPHP Ft PPit t—PT P33,
aiF: = (Head, Tail), RPT PTPI •^PR.PP Pft^ FIFT (1,2, 3. 4, 5. 6)1P*^ t ^PT PPt Pr^ pfippp 3TPPP PT yPasj'f (Element of Sample Point) qieciiai tl
pfF ^ f^qqjl Pn ■RPT PIP T^ PR 3?5TPI "PIP ^ yqVi % P^PP PfWP W PPFT Ff^t—
3ePaT
Coin 2
Coin 1 H T
H HH HT
T TH . TT.
^ -gf^ ^ = (HH, HT, TH, TT).^ ^ ^ gRuiiH 36 ^-
Outcome of First die
Outcome of Second die
1 2 3 4 '5 6
(1.1) (1,2). (1.3) (1,4) (1,5) (1,6)(2, 1) (2, 2) (2, 3) (2; 4) (2, 5) (2.6)
(3, 1) (3, 2) (3. 3) (3.4) (3, 5) (3,6)(4.1) (4.2) (4.3) (4,4) (4,5) (4.6)(5.1) (5,2) (5,3) (5,4)- (5, .5) (5.6)(6.1) (6,2) (6,3) (6,4) (6,5) (6,6)
1 The sample space of the
2345 experi
ment6
(iv) yc-ii (Event)—(sample space) (element) ftclT'll (simple event) ®f>fcTRTt f 'g^cTT 't'l
3Tf^of> 3iqqq ^ "3^ ‘^TT ’Mii'ii (compound 01’ complex event) '=fr^ 'll3T^ ■^fg^RT ^ ^ STftTSF 'ETCTTaif tl
(u) MK<*tRct> 3mc»'iiT yc»tn| (Mutually Exclusive Events)-^K'4*<f<sb 3TgcT^ ^ ' 1^R% ■gr 31^ ^ "f cT?TT fqHOn 37^
't’l 3PT ‘4’, 37gg^• ' ’ ■ '
, ■^rf^ ■^fl' ■5fr^ Tc ■g^W7T i\ ^ ti wt ■gf^ ^ •sTTg
■g^gT % -gz -Tc ■g^gT % ^ gft^«T ^
37«TgT ■g^t^^-g^f^f^^gft w-'m^^ grtf 11^ '?TTi^.'3jg7 sir ‘^fT^ gr 37^ ^rn^’ % "STgr 371^ gfi ■^grTT ^ gg# ft gMgfl ■??: 'Hi^'S eldl ^ ■g|gi' "gTr ■3ig7 Tf <Hehd1 ^ 3757: w HKt^Reh sggg^ ‘Hd'liy,
fi
(vi) Wd-S (Independent Events)-‘gfg ■gzgi % "^2 ^ 37T^ felTt ■gzgr% ■geg gr ggrg g^ eft ^ g^-TR ■?grfg ^ f i ■gg ^ ge=iT3^' grr ggra ggr ^ gt g^,
’gn«t, gR Rtggii ■% gR*Jiig gg ggra ^gro ■ERggrreft ^ gdnR % hRwiih gr g^’ gpfii
. {vii) 37TftTeT (Dependent Events)-gfg Tjgr gzgi % gz ^ gg ggig ^R7t g^=ITg7 M-sai 5 eft '^g 'g3gT37t g;^ 377f^ g^gR (dependent events) fi ggiWTRFsf gT?Tg?^ qieTT
g^t ggr g^ if. gr<w fggg^ gjt Rrairggr ^ "A ^ ^ ^ lTg>rd4 ■4'fggr^ 3TierT t gt gR gigw tggr^ ^ gigmi #fti m: wit gr# g^ gg ggjg
gegt gi g^, -4 agfgg gzgR f igtfgggn gft gRgrgr ^ w twz ^ t gsgi % gg^ ^ grfgro ‘0’ ^
■^gR * 1' % gtg '4 ^4fti g^" 5'hRh< 1V gilggrciT fg^r 4 3t?t (numerator) g tn "^gt gr^ieggr ft
gigw
4 ^[fwf
(denominator) V^ f-TJSTi?, ■ET^^^ ^■ ^ ^ #ft WT ^ 4 inf^T^ ‘ 1 ’ ^1 %
yif^ehdi ‘o’ '?Vft Mif^chcii -ST?! 3f?T- (numerator) ¥tTTTI -
p (3wo^^ "^R^n) = 0«<cw ^?T% % % ■ETcT^ ^ 5-rf^'^dl ai^'M (ratio), 3T?I
(fraction) ^ )lfrRi<T (percentage) % ^ ^«tini 'tl ^ie<,J|l^,
n <pa% 1^ ^ ■^Twrn^ , o.s 50%*'^ ■^' ti
a^: P (W^ = 1,vufychdl
(Illustration)There are 50 balls, each ball having two colours, one black or white and the
other red, orange or green as shown in the following table :OrangeRed Green Total
Black White Total(It means there are 3 balls which ar,-^ black and red, 12 balls are black and
orange and so on.)If out of these balls, one ball is selected at random, find the probability of each
type of ball being drawn up.(Solution)
3 12 15 307 3 10 20
10 15 25 50
UlfilcbrlT-diRdchi {Probability Table)Orange-Red ■ Green Total
0.60 'BlackWhiteTotal
0.06 .24 .300.14 .06 .20 0.400.20 .30 .50 1.00
1.5 (Probability Theorems)
yir^sbcii 3RW % 1^. fl-MHl’,ylf^ctidi-^q.Hi^ "t, ^glcTT "tl ^ ^TRPRT: ‘ yifqqjcii-'Sn^' ^ Tf-^ tl '2mt WxR^ TFT tl ' ' .
(1) TT^rr-TI^ (Addition Theorem)—'TRTRtt^ SffWsif (mutually exclusive) cT^ qcii ^ MiOlehai Pj TT^ mcii % ^ yif^ehdiPg -ETZ^ % -ER ^ mtatn Pi+Pg"^!’' ^ ^
W I 3TT^ ¥1 TUfern ^ t, 3 ■3RT 3TT^ ^
^ ^ ^ t F8TT 5 "STTr yiPj^+idi ^ ti ^ rrtt
(odd) TTTgTT (1,3 Tl?TT 5) 3TT^ ^ yild^dj ift’Fl ^
yifqq)ai3Tl^ qfe'ii qifsy.! 3T«rf^iR7'4’'fsRTTan^wiRchas -l + d + .i = i.or4
fWi■3Tfr ^ 'm ^fT
(ar) "'TTTRfR? arr^ (mutually exclusive) f’I
C^) MIIWRcT) atTq'jff UdiiQ. y^iR '^i
wfi^fcffhf 5.•S^WctT
(Illustration)What is the probability of drawing a black card or a king from a pack of ordinary
playing cards?(Solution)
Number of black cards number of kings number of black kings26 4 2+
^ _4___ ^52 "^ 52 52 “ 52Hence (p) =
‘TRFltw (mutually exclusive) % TTT^T-'^fTs? v3'i=m 31^(belong to same set) "ti (Von Mises) % 'PTH ^
Chop'll 1% ^■*T35ZT % 40-41 ^ ^ TTT^ ^ yiRl^tidl 0.011 ^8TT 41-42^' cr4 fqqie ytPl^di 0.009 'tl qd-iiit MlwRch 3TW5ff f. ^ Ref)71l
% % 40-41 1R ^ cr«n 41-42f ^ -4'^ yiRl^bdl .011 + .009 = !o2■f, qq'lTqi 31^ ^ ‘tV
(2) (Multiplication Theorem)-"'^ ^ ^(mutually independent) ■qzqr % yiPq^hcii ®1^3nfq^Kn pg ^ ^ % 133^ ■RT^T ■qft yipqqiai p^ x p^lWtl ” ”<s«\ie<wil«f, T3«F> f^qq>i^^1^ qft yiRididi d- t, w 4 wn ^ yrRj4.'di -i. ti ^
cf^n -qM ^ 1T^ -RIST
^ yiPqqiai^ X-i =-^ #fll
(Illustration)An aircraft is equipped with three engines that operate independently. The
probability of an engine failure is .01. What is the probability of a successful flight if only one engine is needed for the successful operation of the aircraft ?WSITT (Solution)
Since the flight is unsuccessful only when all the three engines fail, then the probability of unsuccessful flight is :
■qft
^q?pq
^ ^«fT 4 3TT^<s«sivii ohI
.01 X .01 X .01 = .000001.
The probability of successful flight = 1 - .000001 = .999999. yjfqqidi % IpJR 3J^ % ^ 'q? -cHiqjfqeh "f 1% qd-iiy, ^ 3Te?rR (same set)
^ 1R 37ra7q^KrF ^ (Moroney) R ^TRlt ‘Factsfrom Figures’ R Oqq> "t fR 3R»R't- '
“■3R cqPtd % rR ■^’ tHlfqq. ■qft 3RRt Rr# RFT RTRIT.■§■1 qilRiy. ^ qi5di f Rft y1q>qiIyqT ■^, ^rnsif
^ ip. 33^-3RRT T<^ sfe '=ff^RRl Rft ^ mf^Rnn
\
3^ RT^nfr ftsrf ^i’Rs4«hl t^d ^ ttrcTmT
fR ^ 3^llq=hdr'^ THR 3f^ 3Rfrq ftrqr ■R^kH RRt 3T??rq-3RITTWT RJt ti
6
(3 ) MiRiichcitait (Theorem of Conditional Probabilities) —(sub-events) ^ 3^ 3TTf??fTcn (dependence) ^ ^ TT
'SPTI’T "Sf^ ^[pT^ "Sn^ ^ (corollary) f! "fF 31^ 3nf^■s^^HT3^f % ^ 3Jir^=hdl, ^ ^ 3nfii‘«b‘dl 318m ^ m% "^IZ^ 3rrf^mKn ^bfll ” p(A and B) = p(A) x p (B/A), "^TBr
H yi|i|eHdl ‘A’
JiffyWirn
p(B/A) ‘B’ ■8iz=n it ■srf^ it ^ ti
ZM im* ^ ^ ZRT PietilC^ ^jf!^ ■'TT qi«A<|l5 ^ 3nfqet)'ai "f, S6<n( ZW ^ % Zt '5^; 3m> ZT^ ■'TTMifq«t>cn "^Wtl 3^: '?TT8? it ^Ksfif^ Pi'bvi'^ yifq=hai ^ ^ ^ ^ ?TcbpfHifqqjai37if 3riz«v^ ■% ^iiT^i«t> 1§T^ ^ yifq=hai (Probability due to partial exhaustionof a sample) it f I
1.6 ^ ^ (Bayes’s Theorem)
■ 31^ (random experiment) fnci'^ ''R yifqqioizff i%8TT ^ tl 3lir^l4)dl3^i i ZTT^ ^ ^ ■srPTT-RHT t, ZZ c^Pdd^ ^
^ 3nftfZjZl 3nz 't-'^ qmiq^^i "4' "f "Zf? 3m?R'm hihI'JiI, «f)T qMqii3< TFZT 3^)^ '^Tl i Mifq«t>ai-t^F^l ZTT^ 'f! opU Z82Tt 3iqeiW'i Zn[%, FR3nm-'^8i i '^tfftdTdisi! i -mrlsH ^ f alh: zrt st^ ti yir^^diaff i ^efn zt?^ yqViirH«ti >j|iiq)iO ZTf ^Md< zmrbT «RZr tl ^^Rsp) STlxmil^ sffq^ (Reverend Thornes Bayes) "% zm 31^ (Bayes Theorem) ZiFT znZT f qqlfq) 18^ ?Rf(«0 ^
ZiT HWiq fZiZT 8|TI3TFT: oHiHiO jq^lq ZZZT ZT ’'HHWf ZT zrfcTfmZ znzZiKt T13ZI 'fl Wt ZT it ■srfZdZZ
fq;fqi« % 31T81R rp; ITepZT "t 3T«mi ZZZT % T^^-oqqgK % aTT^fR '’TCI 3mbT hI<«ii*i oErfzCTTZ 3T5^ ^ arWR "ZT 3IlfzZiZT (nPfdd "ZI Mifqq^di (PriorProbability) f I
’Zmi^ Zit yiiq«bai ‘fzyfmr 'ZRZT, «qqT^<5 yifqq>cii % Z^Tfmr "f I % Pi<!ffl "% 37T8TR "’TC "Sm yifqqiaiit zfl '«ITrfl't ^ Z^ ^I'nql^* yiiqq>aii< (Posterior Probabilities) t’larfzftzz 'jiii'mO ■% atwR'm oqqstRqi oqiHiR^ TWTma^ % 'mTT^inz ■^' 3r^ ^cM-n it zmrbfl
tl .,
’jzzTRftz t^sft zt 3nfwzT zrzt zt wtz wt fzftzZ^lFT’ZTqZm,
Z^fftW*!, Tm*” tz-3rab7 (random experiment) % Ej, Eg........ 3TTt^ 3Ttz> ■’TRmfTZ?^izzp? t fet' 31?^ zzzT zt yiRidKn P (E^), P (Eg)....-'m f i y'lRj^idiat'
zt ^qT«<i5 (Prior) yifq«baii| ZlFI '^sfTfn t, qqlT^ i cji^fqqi 'Tft’ZPff "Zit ’^sfTZt "t ’’T?^it -ETzmailf ^ zz^ % amrrff zz 3ifM^%icz ti ar^^Rriz % aqtzr Tmm mRuiih tf TRFt t, zft
't ftii zRt fi 3rrtzr zfmrm % ftiR (fsti Rt'Ti^zT^ mn t) TT?ityiPi+dik P(R/Ej), P(R/Eg)......-zw^ tMi mRuihT -t ^at' yird'cbdiat ^ 'zzr-z^ ^TRltfMz fzmi TlZRH tl ^T?fif^ Hilq«hoi‘t ST^WqRi* yilq«hai'< (Posterior Probabilities) q)evjin1 t, qqlTq> i zR^iihT zt 'JinqiiO it znt % Z?ZTCf elPl. ttzt tl aTgzzfR^ 3nfzzTznt posterior probabihties) Wt qiwq ^ P(R/E j), P(R/E2) % zt 3n1zziZlt (Conditional
i%/zzf 7Zwn<
J77/Z7^c?7 Probabilities) ^ ^ ^ ^pn f l 1^"'^12=11 (E^) ^(posterior probability), !T#TTrH^ ar^^RITT ^ (R) % ^TQ; ^ ■JRTR ^ ^ '^n^-
P(E^)P(R/Ej)P(E/R) =itz P(Ei) P(R / El) + PCEa) P(R / E2)..
Illustration. A can hit a target 3 times in 5 shots, B 2 times in 5 shots, C 3 times in 4 shots. They fire a volley. What is the probability that 2 shots hit ?
(Solution). ‘Fire a volly’ means that A, B, and C all try to hit the target simultaneously. Two shots hit the target in one of the following ways
(а) A and B hit and C fails to hit.(б) A and C hit and B fails to hit.(c) B and C hit and A fails to hit.
The chance of hitting by A = -I and of not hitting by him = 1 - f = f
The chance of hitting by B = f and of not hitting by him = 1 - f = f
The chance of hitting by C = and of not hitting by him = 1 - =
The Probability of (a) = f ^ f ^ j
The Probability of (6) = -I x -I x ^
The Probability of (c) = f f x ^
Since (a), (6) and (c) are mutually exclusive events, the probability that twoshots hit
27 12 45 9-------------= — or -100 100 100 100 20 206 ^ X 100 = 45%
■ (priori) ^yiPi=t>di 5ncf ^ '5iid1 ^ (unbiased) I 3T^ ‘^THT^ di;fil efit '■H'c.'ql (fair) "SRSK yiPi'^dii<,W
■5PThT ^ ^ tl ifT ^ ^ ^ cTW
yif^«t>di'< 'Jil’idi ^0 fl yi|i!<^di>| ?TPT % 1^ ■SPTTRPRr-'PP^' cFT y'cfieU
^ t, yip4«t)dl3ff ^ Tjupn
■fe' 3TP^-W<s^i ‘4’ TPTPt ■yfRIP? yipH«tidi hhI^x
^ f^dd) >111^^1), hRuiih yifqcbdi % Pichd 371^'t to 8 2 "hR yiPMchdl 5
m ^qT-5 m "q^ ^ ^ t, 500 qr 1000 m iwn il' f^rr qr q^qft jrm -i'% 3Tm i\ fw ^1
qF "qqirrT
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8
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tl ^ (365 W) 13 ^ ■3:^ tort ^ ^ 3TT^ ^-sntam ;
nif^fftni
. 13;^ sTRft-'srmt
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^ f^ ^ ''^IMcbKl % 3TTIIR. ■qr ^ ^ fttjfftcT ^^jfltft tl ST^WC
% 'BTS^ ^ yi|c|qini
f ancR-’^^m■qft "^jirat t ■stH^ aiiyR "qr ‘^nt 11 "qp! qp ai^HH cpifit,tad'll
^ tlPT 3TTf^ ■% Pi^Iq 3IFI: oqiVi'ici ai^niPici STltq^^ f’l STEPPcrffT ( iiin^aidt= 0) f^nf'-clddi (yifqchai = 1) % qwf yiP^ehcii STJ^lTd ciqiqi ■qTRTT tl yiRelnil ^
P«q ^ ^ q»M t-
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0 .1 .2 .4 .5 .6 .7 .8 .9 1.0
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1. qqtq (Random Experiment) STTq ^ qq^rt t?
2. uitePTS^' qq ■qtq’ qq qqr aitjf t?
affh^^hl^ /q/qqf 9o^qfK
iiil<4'=hai 3. ■OT^^-37To[f% jilf^e*idi cbir>ii<l
1.7 wf^r (Summary)
• uif^Rmr ftorl (Theory of Probability) % im■f^I^ Mil ’TT^ (Goddess Fortune) 3T^ (super-stitions) STTPTTc||^ ^5l5ql 'jy, M 'JilciA ^ ^ «h<l^% dwicild ^rwicW) % ■'TR? Ml
• Hlf^etvdl .ail'd % Iq«t)l« ^ «t>K«J Mldd % Sc^eft' ^7?c|^ M
, (Random phenomenon) ^ dnlVifd "ti ^ 't it^lin 3?^^ixj^ ^ ti ^ ^ w ^ t', 1^ 3PiW M ■Eie^ ^^ ^IIH #TT. ^ ^1
• ^ ■Jnhr (random experiment) 'M '»in<tiiO iHcri*^ «if^«t)ni3Tf Mf ‘hMHpT^ ti yiHj+disil M ^ ^ ^ -^Ml % ^-tthi t, sn
Ml ■fsT^ yir<^4>di ''jM 3T^^ Ml diaiq^^i M TF^ "t ^STFT'««!«< ■’^T H-tMlMl qlciqi<ni TFcfT ^ 3flT Ml Sid'dlA M Unf^J^TdT-'PrM^ ‘tl
3T«TO-W^ (Exercise Questions)1. xifM<»jai- 3tM lq<^W Mlf^y.!
2. Mif^^di % ■*lMl ■^i
3. Mh 3iM^ (Bayes’s Theorem) Ml fq^'d'ii MllMi^l
(Reference Books)
1. t#s!iM1^ Wfit afiT iiMhT-s^Tw fm ^2.3. ' ^Mlfl SfiTp (ijsq^^ T^tjfsjpf 577 ^7¥/4. yif^<*)di fqd<«i'-*///5'f %. Rinh, fW^K
5. «Mlfl-W* W/«7<V; •TRIw 5/%^?77/.
10 3^gc7T WfisH<hln fsM'J
’ite
(Probability distribution)
■HT^RT (Structure)
2.1 ^3^?^ (Objectives)2.2 Mwiiqii (Introduction)2.3 ^c5.iPa<^ fqa<.wi ^ (Utility of Theoretical Distribution)
2.4 ^ (Normal Probability Curve)2.5 -ssmVi
(Use of the Concept of Normal Probability Curve)2.6 MIKliebT % ^
(Normalizing a Distribution of Score)2.7 «PO^;i (Summary)
• (Exercise Questions)• in? (Reference Books)
2.1 (Objectives)
IR FFif % 3T«m ^• ^«&iPo<T» fqa^wi ^ ^
• f9)^ Rw<.wi 9?) 9>t^ *^1«nt«l
2.2 Mt<u«ni (Introduction)
9n^-^9FT 91. 9n^-9lfrI9)T ^T99jf 9)^ 9^ 1^1999 %9I 'Srrai ^ 991 yc^9) ^ ^ 39^9191 tl 3n^-^?R9 9it 1991 ^ 9991 9J1 91 H9>(ft t-
(1) qi«ifq«o 91 3tqcilf^a 39^ 19919 (Empirical or Observed Frequency Distribution)(2) 319919(9lf^ 139^1^919 (Theoretical or Expected Frequency Distributipn)
«/73w«?V /q/«»«#r 119«9o1
qiwfqsh ^shct^ "^IT 31qeilf^a ^nq*!
3WR ■’TT ^ 'Snjft 't^l <i'Ai5^'J|i4, CM<.<31^ 100 tll'<iil6«h HvAijO % 37WH ’’K
31T^f^ ^iRriq)!, qi«ifq«ti (qa<wi qigcii^^l TRJR ^t^-
^nKiilsq* H'»i<jO (? '4) :
9ffe1r ^ Jtm :0-20 20-40 40-60 60-80 80-100
10 15 30 25 20 100 .
qiwfqqj 3T^cTt^T^ 3TTVR ''7T 3TT^[%-f^cR’^ % fq^Od, yi[q«hdi .(Theory ofProbability) % 3TWR ''T^ S^l^friq* ^TRT q><‘^ ^ STT^f^-t^clT^ ^ ^ «n tiqTtll "tl
IRf^ 3Tl^f% fsRR'^ ^ ^«5indq> !nfWflT fqa<«l ‘^fT# 'f I “t^% yr^qj ^ xifqq>di3Tlf ^ yifqqiai fq?KWi «t>eeiidi fl” T^ep (random
variable).% xc4«t) yifqqidi tITcT «ncit "t, 3Tcf: offt xifqehdisflf ^ 1
#rT ^?cuii8f, -^rf^ ^ ^ 128 ^wt-
^ ^ ^ ^ aro 8 ira»R
6 7T' T H
1 • -2 83 4 5
fSdt'4 rtiqq)i : H
Uiqqii :
H = Head(f^). T = TaU(^) ^^iPdq) 3T«Rr ylPqqidf WR ^hn-
Riqq>l{Number of Heads)
H T H H T
H TH H
H T H T ' TH H TT T T
xcqiHfid an^f^ (Probability) (Expected Frequency)
illfq«hdi
±0 168
3.1 48e
12 486
13 168
% (Total) 1 128
2.3 ^i^ipnch IcrrRTTT ^ imilPmi (Utility of Theoretical Distribution)
■tef^ an^ aiT^f^ Rffer^ RFWjjif amiR ^ t tT«tt ’ag® .an^fqat'Ji 31^ ’ifiR^ldqT RFRefT "tl 3TT^f^ lqa<«t ^ 3sqlPiai "PTH ^^narif■4' ^ tl-
(1) . PHfifTid ■qp^graff W ^aif % aiRrfg an^ -^cR^ ggy ^t ^ -sflftiq ^qratPiPfT^aai (qj^c^qui f^RTi y+dl fl
(2) UrmPfia an^f% fqa<wi
(3) McqiP^id tiqVf RFRRTT 'jqljHi'i f^ '^n Rqr(4) "ara RTRifqq) -aij«l«JHl "5^ arfqqi "siq ^ itcqiHfia 3TI^I% foRRq qq RnniH'd ^l^di 'tl
rcr [T
t tl
12
(5) 3TF^f^ ^ 31T^f% f^a<.wi ^ '^rT% "JT? W >iMiqi oIT "t% 3?^ ^ (fluctuations of sampling) % ^R'JI t ^ 3p!|■^TRTTff ■^l . ’ .
(6) 3T^?RIH ^ ^ 3nw t ^ t 1^ 37?^ ^ ^ #ftl
(7) . 3T^ WR 3R7 4iH<^|3Tf ^ ^RTWR 1^ f^cR^' ^ ^T’^R loF^TT '^nti ifwcuiisf. ^srm
37NR ■'77 371^k1 q-ii^ ^TT^ etiHsl ttM Pil^'cia «=t>ai ^-Pi<4-5i«i,^ :5f%TT 3TTf^ ^ ‘Q' ^ f\ t^TzRR f^tcRTH % aTWR '^T: "RFr^ ^ tl
^efilPnch f^d<ui ^ ychiT
3TWTm7f 37^gft 3T]^I%-^1^rR^’ ■SRTRRf R<d<u|, fg-l^ lqfi<.«i,l^cR’JT (multinomial distribution), fRrR^j|T4f^ fqcK^ (Hypergeometric Distribution),
3TTf^ ■JRfW i^rqr «fTcTT 't'l fq;f^q'J| "4 *^1^1 Hadf^d-CJl 3Tfu^ WT«i^ ■!■-■.
(0 (Binomial Distribution)
(ii) WJRR rqd<“i (Poisson Distribution)
(Hi) TRTFTPT I^TR’JT (Normal Distribution)
%r«q>dT
3TT^f^ (Normal Distribution) %TRTTHRq
% 37r^f^aqn<
^ "ti 3ifdn{-qd ^rMRll (Expectations)■’7T 3TTWft^ ^ 't’l qi«iq 3f|c*d ^3ildi^'qd d4><i^id 37^RH ^ «*-'Hifqdi ^ S(fqq)dl (Probability) % 'STTF otTdd f^rm■^RkH tl ^ ^ ■STRl t ^ ■qr '3^ (Head)f^ ^ 7TC'^|ef?f 1/2 t, rjj
y■0: ^7?^^ ■qi^ (Six-Faced Die) % %% ^ -qr qra % 37T^ ^ ■ROTmT )/6 t. m 2
f«Hici ^ ^ 99 yld^id 'll ^27^ yifq^di ^*7^ (Probability Statements)
■t ‘l^^qdT '=7lcid ^73^737) 37TW "qr ■nlq'^ ^ '^73373^ % 1^ cimI^ STJRFft cqc^ aTTcT t’l WTH Ti: ^ qan 3pq qifqqRTT q;«R 1^ it -qzqi % it^ it^ ij qi i qrt^ rnfv^d(Definite) efni f' ■^;[q ^qtrl 3i^hH ^Tqq qni i qjt i) f^, ii
^r^iddi qi ■grfqqKTT fticsi-d qrr fqfyqd sr^qqq qRi qf! ir^ ^q^ql Trai^ i "ft TTqq qfq % •■^37Tftqt TTStftqt i (Goddess of Fortune) qft ■^Hcidi i f^RRT itqR dretudld■qpq?fit i 3Tqqt ^ ttr^ttstI’ % TTqrvH % ■qn*??! #n ■strt^ fqrqri qq Trf^nqit' i^rrqt q3qT3i % qfel Ffi qtr tqfvqq 37W7qq f^Rf% q^cRq^q ITTfqqRn qq ^3711 Iqi^ 37r yifq=t>di [^.fill'd qn qqqW ■^sTTftqf qqi ■^Tstftqt qq? Tflfnd qit "fi qf Ri^i-d qRqq i qq
• ■Rit ^ i sRqqi H^Tq'^i q«n qqqWt i qcnv? artqlq^ itit f qqi ^3q% qlqui i tTfe^ iti % i Si^HM 0qi^ qO’ 37TqqqqiqT ?t?ft ^i
tiTTqiqT
q^qiq ^ q^ q^ it ii qm i %qR qq? it q^qi q^t qrq qi qrmt tqq mvi qq^r (Simple Event) t qrq qq: wq qi qzqrdff % qfeq iri qft ^ qft qnit i qq iit q3qT37t qi "H^SFr q^qr (Compound Event) q^BT qnqj ii qqr ■ftrq^ qi -dtsiciA
. qi qqi qii qq i»qii qi uid»4 i ^qqt% ■'^ qrq ^ fiqqi qit qi q?^ irit qttqft TT^qq q^qi qq 3<(6<wi ii q^qiq T^qq it it ^7q>d1 i qqr q^FTT q^rq
qr 3Trf9fq it ii qq q;qr "^nq qfeq iti qi# q;q» qrt qqrf^ qit q^it i qq qi''MCdi*) (Independent Events) q>Bi il IdnOd qq 7^ q^qi qq 'H<idi37t "^R qqjq
TSqjR /q/yqf 13
■'T^ ^ ^ 3l7ftTfT (Dependent Events) f l ^ '’ETT^T
cT'^lci’ii ^a?i Tj;^ ■% tn ■’73 ■f’H^ ’’TT 'R^TT^
mP^^hdl /yrfiW
^ '^p\[\ ^ m\ ^ ^ ^ ^ ^ '77^ ^ wr
^i3^’ "t =mlT^ ■’TF^ 7§fcn TTIT ■'TtIT ■R^kTT 'll 3Tlf^ '^73^3^'' Hif^ehcii
^ ‘77127 'trfd^ 77^1^ "t '^TTRT Sm^'JtF y^iii,
^ ■'R ■f^ ^■’73'^ 3TPn ■RRR; SiHqvjiI
37Tr?7F
yPayf^ (Conditioned) Ft^t tl ^(Mutually Exclusive Events) 'll 17T^ "% dw[^
■273=n^ f W\ 37FTT ^ ■5(^ ^ t t^T77% ’SFR’^ T^SF 77T27 371 TTFT^ tl fRF
■'TT^ ■^Ft ■’77 1,2,3,4;5'^6 37^ '=FT 3TRT M<es<. 3isq'4f "t '’77^ '3‘STcT^ "R
'%Zqdnl
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q(3d 3727^ qfdfl ■% '4 yc^iVII ^ ^ldq> F^cft "tl "^3^
RWTTsq 37% 'St33T3iif ■^' i%ft '27337 %f^nl ■2733T
’■?% "^Tfl "^n <H«Fdl "t "TTHT^ aif^«t>dl ‘offfeF ■2733T % 31^'^cl nRfVqfdqT ^ ti’om ‘77127 ofcT 77^^77^ 273373^' ^ ^ 375'7Tcr •
%\ 373: Mifqqiai ‘277 yilqqidi 375*773 (Probability Ratio) '3ft 3T77fi373 ‘7J3 ‘5773 ■f%77 '37 R373T
t-qiTisa ‘3337377 '^ft Wom
«iR^chni 3RjHld (P.R.) =^ R«77ST ■333T3ff ^ ^T
1%Tt 'SFt ‘397% ■R 3i%73 "TT^ ‘t—‘3T (Head) f%T7 372737
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f^qqil 3ft ■'^ 77727 ‘3^7% "37 ‘3R 77*^7733771; 77%t t-
(i). -qFTF F«77 5^ %f Ft 1773% 1^ M,
(it) M6CII Riqqii 1^ 3277 5J377 177333 '*73 fh).
(Hi) HeCJl I77333 ‘*73 3277 5^ f^qchi
(io) %t ^ f7T3% 33 1^1
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HHT, HTT,HHH,THH,
HTH,TTH, TTTTHT,
33% HHT = HTH = THH 3277 TTH = THT = HTT, ‘?77f% 3773 7T07T33I^ ti HHH ^ TTT ■^.'^--^ 377 3«77 HHT 3 TTH % #7-#! %7 3T7t RTt 770773% tl 37cT: #11 W 37 "dHt ‘33 "3^ yiPqqiai 1/8 t "331% f% ‘*33’ "33 "qi '^■‘33 'f^ 3ft yiPqq>ni 3/8 ‘1^1
F77 3337 ^ ‘S: -^t .37771 (Six Faced Die) 3ft 33^ 7773 %37t 37 ^ 36 7F3T3% Ft
■7737% tl ft 7777*^ 64 "t’ 3753 1%37 "nRl tl
14 d Wn/ ^iT<s^'fil<i f^fiFJT
wift 2.1. ^ tirof ^ ^ ^ 36 ^mrra^ tot Mjf^«#»dii?X77^' X77^X77^xn^
3T^3Tm3?^ 3f^37^ 37^
I IIII II I III I II II II
1 6 6 1 72 3 3 4 4 1 5 51 1 2 1 1
2 6 2 7 6 2 83 4 3 2 5 4 51 2 2 2
3 3 8 6 3 92 5 3 3 6 4 7 51 - 3 4 3
46 106 3 7 4 8 5 4 91 5 2 4 44 4
68 5 9 5 10 5 111 2 7 3 5 4 55 6 5
10 6 6 6 127 2 8 3 6 9 4 6 5 111 6 6
Pfi^qa ^ %xiT 'sn «chai "I— .
TTRXJTt 2.2
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ft 2.2 % ^ vw t % ^ -qf#' ^ ■?rT«i xn: ar^’ % 2,3,
4. 5, 6, 7, 8, 9, 10,11^ 12 ’SnfxRFJrn sFxm: 1/36. 2/36. 3/36, 4/36. 5/36, 6/36, 5/36, 4/36, 3/36, 2/36 ^ 1/36 fl '
37cT: XT^TT '?n7-3nfx7q? (Equally Likely) W■ET^qi m ^ ^ W n, ^ w:^ % ■^rfel ^ ^ yiPl4idlm/(m + n) cT?7l *3^1% "SlfeT "qf! yifqq)dl n/{rn + n) "^RTTr % '^TfeT MliM=hfli^ p cl*7T qfka ^ ¥lfqchoi q ■?l%cfraT fci<a^ "t l 3TcT: Ph^l%
yf^ vufqqini, . p —--------m + n
nqfcTd xf ^ !4iR4ehqi, q =m + n ,
i /? ?«IT ^ ■qW ^ TT^ % #ni sm: p t!«TT q ■4' ^ xx^ %■^n yqicfl 'll XTJ?l ■% 52 ''rat I (Randomly) xr^ Tdl
fqqrml XR ^ ^ (Any Ace) ^ yiPl+dl 4/52 1/13 ■=! |1^ ^q<tq< |i MiPq^di ^T^qTcT.'iTFT
yiPi^i sraf I ^ ^ ■#■' I x3t^
yi(q«t)a> % x^ "qT! I Midi ■% Tjpf ^ii-'Hiqdl 31^ PiPfMddi li fdi^ yifMMidi "%
15<STi-c/di
TfpT (Extreme Value) f VirA|<4»^il ^ ?TT ^ 3Ts^ t% ■Effel ^ ■ZTT ^ ^ ■^TT^ T^’ rHPv^d 375*T?T. % Vltedf % ^ tl TITl^raKrT
q«fiHcic< ^rfcRTH % ■^n '<l4)dl "tifqi^-qi yifMq>oi ^ifwmlq, ^ c!7f%^ yr^q y«^a ^)TcTT t"! ’'T^
tl t' ^'^wTh wim ^% STTWr tr ^ yifq'+’H ^ 3T3*l^ cRTRfT ■SflrTI tl ^f^^fSTT (Mortality Tables)
%-3TT<jR ■’R "^Irt inf'll arr^ oqnw<^T % >Ji1fqci 'wt ^ yifq^iai w ti •.(Birth Data) ^ ar^^TR ■^JlTcn 11^ ofT^ % ^1f^d ^ int^T^Tn ^
(Statistical Probability) ^Twt fl ^ RRR RftftsrM n W-’°i\^ R' ^ ^rffe^WdHI r ^ qfid t ^ ^fq'^q t fR qiT^a ^ THf^RFcTT '9)^ ?i % 3RRTaik ttt ^ W<T r/n ^ rIrt qzra?T ^ tl am: t %-
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= Limit —p = nn-Kf<
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yilq<^iai 45 cM "% yiPqqiai .55 tl qi'jf^«b (Random Variables)"% 'f^ n %amm aik tft "r aTT^HiR^qi ■RTfEmmr ^ ‘^rm mf^q) yif^ehdi % arrar mmi ti
• yif^qi'ni ■^^Hia ^im % 1^ ^iftsa ^mm ■% qHia ^ W<sqi 'cTETT q^iiarf ^w\ ttm arm^ ti ^ ^mma^i ^fl%?T ■Emmarf ^ mm ^ %
■^TT^ (Permutation) ^*1T (Combination) ■% VcqqT ^ "SmbT fmm mi R^cTT tlam: ^IFT ■'R shnqq «q4 ^ yKf^«b 'JiiiquO "af^ tt
TWl
chH’ciq (Permutation) dlc44 "aa smt (Orders) 'm (Arrangements) R ttl ^ #T■’T^ ^^at "m "am^ ^ ^ ^^at (Arrange) Htiqi ^
A. B ^ C ^ ■®: f^-f^ ^PWf ABC, ACB, BAC. BCA, CAB a«TT CBA -f %eit -sn R^mr ti 'am: ?a tfta ^D "^ ■?! "SRI "SRlt tli cm 12 fq-^ra AB, AC, AD, BA. BC, BD,CA, CB, CD, DA, DB'ciEfl DC'^i am: 4 "f "t % 'gmi 12 m^i ^ (order) ^ t ^ ^at Bp=T-fRm "f R§t mtl t % AB ^ElT BA^ f ■^''t tt t ^ ■^RT f«FTttt % ^JRR ^TWT amm-amm ti n aroma ^arf ■f ,'^ r ^arf ^ mET ^ m shHT^qT eft tror ^ ^ "Tim ^ mr y«tid11-
Rqmr
tfti Trog -^if^ mR A, B, C ^shH-qq
5hH-qq
■mrtshH'qq
/n!
fbH^<4 (Permutations) = '^P^ = (n, - r) ! '.
mm n\ ^ qiH^P'ici TR (Factorial n) mmmT 'mmT t §<Hebi ma 1 ^% sj<iq< 'f)ctt tl am:. '•
n.! = 1 X 2 X 3 X
■q^g ! % .3TqF gro -f llni t ar^ifti;n! = «- X (rt - 1) X (n, - 2) X
... X n
X 1
W/'/csVeftlW16 '3«-qa<
41 = 4x3x2x1 = 24 51 = 5x4x3x2x1=120
3Tcr:
sfnrjPncT (Factorial Zero) ■grR’jpiTfT (Factorial One) ^ "iTHcRT^ ^ ^ ^ ^nr^‘ ^ 6720^ tl ‘ara: ^ 8 •^' 5-5 ^3# ^ ^«h< sbH^
8!«P5 = = 6720(8-5)1
^3Tf '4' ^3Tf ^ ^ ^ f^RTraf ^ If^^T(Combination) ‘tl ^ ’9fK A, B, C ^ D ^
,' ^ ^ ■<R GTR ^ ^ ^ ^ AB, AC, AD, BC, BD ^ CD ^1f 1^ ^rer 'H’qql sF^r^ “Ft tioHi ^ srwt t' =MTRti 'acQ*^)
ft AB^m AB cr8TT BA^ ^l n 3TWT ^3Tf ■^'-^ r^3Tf Vl TTm ^ ^ -^f^' ^ Jim -pRT ^ ^ ^ "^sn fii~
n\«^<4 (Combinations) - '^C^ = (JJT7)T7T
3FT: 8 ^^3Tt’ ^ 5—5 '^^STi' ^ c^«h< q-ii^ ‘^fT^ ^ ^ li<s<j| 56 "fHI
81®C5 = (8-5)151
% 3iq^ri^et)»i "FT^ ^ 1^ yqql tiVs*4i sh^T^ql tio^jaiH-qq
^ tl^ ^ ^ ^ 3Tf^ ■'RFR STwHf (Mutually Exclusive Events) '^’ ^
(Either of the Events) % '^ifel inf^T^icn, 3T^-31WT ^TfeT wf^qidi3Tt'^ tl 3T^: ATT^IT B STT^ >M«i'ii>< t ^ A 378^^ B % '^rftrT ^ yiUl«F^
PR (A 3T5rar B) = PR (A) + PR (B)^ TTi 3T^ % ant ^ TJTtem i/6 ti am: 2 ^ 4 %
^ snt ^ •artn^^ i/6_+ i/6 = 2/6 = 1/3 iTEftmF't2.4'^6'f't %nl % ant ^Mifqq»ai 1/2 tttll
■nr ^ 3if%i^ ■^TEnrat'% ■^n«T ^rfen ^ mf^RnTT "an^ ■^m-37^ ^rfei ^Tnfn^jtiart % % "^irt ti 'SR: An^ B ^dt ■^rsnn^ t cr a B ■nr^r
qfcd ttt ^ Mifq^idi
PR (A^ B) = PR (A) . PR (B)■tt nft' ^ ^ nr ■7?# rr 4 ^ rr 5 ant Rt RTfnRidi 1/6 x 1/6 = 1/36
tHll ^ ^ RR m (Even) 3Rt nRT ^ RR f^R (Odd) 3^ ant Rt RTfRRRT 1/2 X1/2 “ 1/4 #ti
■jn^RuTT % <sw<l«td qpJid fR^IRT RiT RR^R RR% ^niPi'd R^at % Rfen ■% ^RR^ ■t a^pTR WIRT RT yqidi tl ^Idqi RR^ RF^ ■^tf^RR ’’J^-qurHH (Presumption)
, aTRRT- RTRRT (Assumption) % anVR RT fRf^ RRRTat % RfecT Fit ^ WR^ nfqrtR ft^
/qA/'V* 17<sv^ai
V//4<*rti' Ihdi (Mathematical iVjodei)^- yfdHKH ^ f cT«n (Model)% sttvr -qr ^^ 31^ ■’qzqisrf cFi 3f^iTH d'lirl ^^uP'i'hi % ^ yfciHif^d "q ’'J^Tcl: HftdiR'Hd (Hypothetical) ^ fi FT -qr^.W sT^qrat’ ^ "q^T (Trials) ^ (N) ^iT% "Eizqts^i% ■^rfer ^ qit ^^nfqcf •sn^f^qf ^ irra qfl ^ ^qqTcfr ■f i [^•^1 so qRqr fel 3TT^ qil 37Fjf^ .25 ^ ^ 3TT^ 25 # Wfti PR(H) = 1/21^ 50 R TJn RT 25 qiR! ^ pr(T) = 1/2 so ^ ipqr ^ qr 25 w
i•^1 f> % 50 qp qft -sm^ fcra (H') cT«n qz (T) 3tr -qfr aq^ 25-25 ff wtTi ftF q c^ fwjq' q«ji rfiq ir-rqq^' qq tjcf xrr^i -qp qr qra ^ qr^ qfqmqf qfF ^nqfgci
qqrR,I
arrqfw iPfi qq qi 17^ fi ^rnii 2.5 q ^ iTTqqq' qp qm :.4 R' #t fqqqfF qri 200 qp qr<f FTq rtpi qf^ncltq y^o fqrq q^ ■f—
■RTTqft 2.3 Wfuft 2.4
rfh? («c*«bl 200 FtJIH '^JTr^cKhl 200 oirr FSiH qfwidl<4 fq^vf
-i^cKhi ^ f^srfq^T5F3[^ f^srfq qTf^raRTT 3TT^^yiHjetirtl 3Traf^ /.
qhrF ■fRrfIftw i HH HHH 1/8 25 .50/ 1/4
^ f%rrr rim Tr^ q^.
wr HT HHTHTH2/4 3/8 75100qi
THH. TH
qr qq? qsRTT THTTHT1/4 50
^ q? TTH 3/8 751.00 200
qNf qq TTT 1/8 25
1.00 200
^ rRF q qf^ qq flRRSi'' qfr qq^ qiq '3?^ qR CgfqqT % %q RHl % j024 FR) 'S^m ■ . qn^ qq qifqqpn feiq % artyR qr qrqr fr qnq qwqf^ fycRq? qii qRnft 2.5 qqr ■| ft q^qrl^. fqcpqi qfi aq^fq q^^ % ?q q fqq 1 q ir^ 1%qi qqi f i qfqqtq Iq^ (Mathematical Model) %.3nqR f '^RR pqi
• WOiil 2.5. qiT q?! 1024 qiT RT TRqrf^ 3TT^f^ fdd< u(
f^ct«bl q?! fwln yil^chni 3TT^f%
qqt fqrrr
qfi fqq^ "^rTr qqr qq? .I^Tqqq q?
3TTq l^rqqt f^nr qqr ^ fqq% qq
■^qq i%nT fnji rftq fRq% qq
qj: f^TTr qqx ftr fqq^ qz
qfq fqq% fqrT qqr qfq I^<»^ qq
qjTT fqq^ f^ti qzR qq
1/1024 1
H^T 10/1024
45/1024
120/1024
10
H»T^ 45 . .
hFT^ 120
H«T-* 210/1024 210
252/1024 252H-itR 210/1024 210
W/)^4>?4 /^/qf18 q»qcR
r«Ti tTOT WcT TT«r
^ HSlt 3Tra f^ISF^
l«e»ehl rISIT ■'73'
xi^
H3T7 120/1024 120
H2T8 45/1024 45
Ht9 10/1024 10'plO 1/1024 1
1.00 1024
300"
252250”
210200".
iE
35 150”
100"
4550"
iL 11
hV hV hV hV hV hY hV ht^- H^T
1. «HKu[l -sr^^ f^cRTn ^
"SRiR % diitjpTi fqd^.’JI ^saiPnch 3TI^f^ f^TTTTJT (Theoritical Frequency Distribution) xtt virqilvin SiJt^Ph (Expected Frequency Distribution) xn3Tr^f^ (Ideal Frequency Distribution) "tl 3Rf: 3TT^f% l^cR’^ fqa^.'jl■f ■'^ Hi*^ai3Tf (Assumptions) % 31T^ ‘1^10)^ cI^tr f^qi "tl1PdcRui diwfddi' 3iqciich^T (Actual Observations) XR STWlftci ^ ^ f I
<ipqd <3t9l<ncbS’ll
^<5,iPd=b Pq^Kui dP^rdl'il Hifqqjoi xr aiRnftcT ‘^IcfT ■'Rxg qiwd^ ^ WcT I 31^^ infjTff ^ ^ -itSKnl W<sqi3®IctT dq
■% sf^ xR ^ 3iq(riird5d tSRRX'T (Real or Observed Distribution) yiPq^hni xr'STRnftd ^iSlPr1«R 1^fR^ ■% ‘'STIdT 'll ^eSiPd^i 3TT^1% (qd^^t q!^q ■4
% m f 1 3T% X|r<p4^rdql' ^ FT^FT Wf ST^lfwl ^ il ^^iP-dd) PddCJlT % wfbT PHHddt- •
(i) ^TltT ^ xjI Pti^-qn % Si-d^fd % Iqn^yi Rff y^fd ^
(ii) ^ Piufq RiT 3TfyR "tl
(Hi) RT^ 3T^pB WTf^ RfTRRj f I(id) qiwfqq) rqd<.«il R>t yiPd % 3TcRfRRr oZRRIIGT "qr 3RRRpf Rt fRdTR
[qci't^tl % ^MiM'-i (Substitute) 't’l
(d) 3Tcr#%d fRcR^ RtI '^^iP'dq> fqd^'jfl' ^ RR% yPaqqi <ST5qiq<dl (Sampling -Fluctuations) RRRlt 'f I
HfR^°fft^ 193^Wo<
(vi) qi«^ch cT^IT 'SR ^ 31^*Tn ^ ^eiqqj 'f’l
(uii) '^«slPa«t) f<Nd<'J| ^Pr<^ >511 q'd ^ % TWTVPT ^ fl^TeTw
37^ f^cfqi'l ^87 3?T^ fqf^*^ ^^fq?i ^ ITlrf Mldsbl
«Ft ^^*-‘^qa; 'Jifici "TTr■f^l fH.¥«♦)!< ^ f^fd ^ yifq'^fli fsH«{lq fqt<1!< (BinomialExpansion)^ ^iH'ialq«h1«♦’<«! ^ ¥qVi xjTi tl ^ ^ lT«7f? ■^' ■?7f •^i
(1 if 2-^2
/, n2 ifJl 1)+ 2 —X- +2 2i1
U;2J~ L 1 L" 4 4 ’^' 4
^ ■'T^ % "^ft ■'7^ 9fiH5;[: (Two H), (One H) cI^tt oR^ f^^ (No H) % 37T^ ^ yifq=hfli ^?TI^ 't’l ^ TTT? cfN f^qq>i ef^ ^ '■HHlcb^uf ?Wt-
nvif _2A2J ■'U,
/ » \3 \3if n+s - ■ - +3 -^i + iT -,2'^2j “Uj
12J 2J
1 3 3 1= — + — + ^ + — 8 8 8 8
THI' rH«r=hl •'77TS7 f§[T^ "TT^fNiT^ (Binomial Equation)Pi*^qa 'wHt—
\4 riVlf fl +■4 - ■- + -2) ^ V2y
nfcif ^ i +6 - ~2) V2J v2j v2j v2,_J_i_A±J_“ 16 16 16 16 16
^hW^tjiT yr^^t) ^ (Denominator) W^37t (outcomes)^E7^ ^ W -sm (Numerator) ^ W-’>^ tm ^ T?! tl (Head)37T^ ^ aipT^di ^ p ^ ^«n ¥3 (Tail) 371^ oFft yiR^^ni q rR n 'TCt3[fi7R -Mq^ f^’d (Head) ''73 CT^il) ^ yifq«bfli (p + q)*' fq«iK
sim ■5TT 73^ t- ' .
1^ % 1^77^, (P Q)^ = P + Q
^ tefgjf % 1^, (p'+ q)2 =p2 + 2pq + g2
■% (p + q)® = p^ 4- 3p^q + 3pq^ + q^■^nT f«qq)l % (p + q)'^ = p^ + 4p^q + 6p^q^ + 4pq^ + q**
/, \4 \4fl 1 .2'^2;
1+ 4 - ^2A
, , n{n -1)= p"^ + ?ip"-iqi +---------~pn
d^iddd qnlqi^'JiT ^ 'TTTXTRT "TTXTt^Txrj (General Equation) fdHdd "3^ %X§I
n f^qqil % f^, (p + q)"
■^IT ^Tqidi "t—
' - '‘CoP"qO + '‘CjP”-iqi + »C3p'‘-2q2 + ...'^C^pOq''?77 HRRT d^lch^ui ofit ^ fW t 'f^ n. = 5 % "fST^ Iq'Kid (Binomial
Expansion) ^FToRT ■?)x7Tl
^ (p + q)^ - ®CopV + ^C^p^q^ +^C.2pV +^,C4p^q^ + ^C^pV = lp5 + 5p^q + lOp^q^ + IQp^q^ + 5pq^ + Iq^
20 HlVs^<*ilii fcffil<tT
•V
(p + qY % r<^WK ^ f^ri<ui (Binomial Distribution)tl ■5rf^RK=r ■nf'HM sRf# (James Bernoulli) ^ «fT ■=Tm
^ f I ^ 3T2f t,T?^, TT^ ^
iiip4<frii fcRi'i'JI
cf^ (^d^< ^^ WT tl ^ 37T^ ‘(Head)% f^ ^s?T g ^ f^ 3?T^ (Non-Head) %y^=K1 f^qi. 'tl ■§■ ■fe' fgR^ tqd^^'l (n, + l)’^ f I fg^ f^rTFT ^ Vt^ Ig
HTPITcR^ *j'j|i'^ (Numerical Coefficient), p % Hidlcb (Power of p) cT^?! g % 's^idl^i (Power.■^ ¥te^TrH=h f^<sl^
^ -Rra ^ ^ t W fi■^' p^ w 5Rm: 13^-:^ wl wf t w ^sn^fl ti "qg ■j^’ p^ ^ ■?pT
i Q'^ 'm ni\ t W q’'if ti 1■3?^ -^ ■^‘p wp^ -^rraf ^■zfW n % tl fg^ f^d<*J| % Trt^ TRrfRrfb? (Symmetrical) ^ "t ^2" ■% ^ ITt'Hiq'iIsff ,'5)1, "5^1 ddidi ^1 fg*?g l^rR^ '^’ p g^TT q ^ RH
^ ^ ?Ttf ■5^' t pgan ^ tl fg^ pft«nRPT Rgrr ■'R fgR^ [qwil ^fd'Mi TRlfRcT f "'R^ ^ p cT«TT q 3RRTFT ^ l^' fg^g f^WK % ■q^' 5n I^RR^ aRTHfRcT fl ^ 1^5Tfa -il’ n % RP! ^ ^ "RTR 3RTRfRcnTT R7R -gtcfl''j|(^ f I Idd<*^i % ■'T^ % ^'<9MlrH=h ,^'Jii'ch'l (Numerical Coefficients) '5il
(Pascal Triangle) R^TRcn ^ RTPRfi W f^iRT ^ RRRfT t’l
'I'l "SIRR RRT Rgof q) ^ TP' ^ RdT
RRPT
Pitched
fl 2.6. (p + g)'* ■%. % idR^ '?R3IRR^ T^TTS^ ^ Hit<t)<n
rIrRTtT TRT^<n)
1 . 1 1 22 1 ,2 1 43 1 3 3 1 84 41 6 4 1 165 1 5 10 10 5 1
6 15 20 15 6 132
6 1 . 647 1 7 21 35 35 21-
1. 8 28 56 70 ■ 56 - 28 8 . 11 ,9 36 84 126 126 84 36 9 1
1 ,10 45 120 210 252 210 120 45 10 1
17 • 1288 2569 51210 1024
RR^TPf gKr ■% iJe^sf) ■'Tf^ % gWl fd>ii<l % T^liW) TJep’ % sRld<t 3Rq -RRt TJ|f57 -3^ ^5nR ^ ^ TPT^’ ^ ^ t) M
6 = 3 + 3, 15 = 5 ?•. 10 rlRT RTcT^ Rf 21 = 6 + 15 371^1 ^ RRS:■f n. F 5 ■% RR RH shH»l: 1, 5, 10, 10, 5 ^ 1 "t '^, '^dd '^ITci TPT5^ "RRIP
RRRRT
flfgRg fdd<u( tRg?! (Binomial Distribution Model) RR "^URgriw t^^iidT.'^’ 3RRRf 444)^11
Tig RgrR^Jof ■^RH 'i’l t^l. RSRTSff gfl .RRRRH-^RlRRPn ■^' STRJR ""R Ig-f^RTSR(Dichotomou's Classification) lRRfT ^ R5RTT t R?T Ig^g 5R 'SlRfR fl ^ lRg?fR7I ^iq^Ach 'SRRFf ITSF ■RTcT RR? R'^l ^ Rfg I^FRTI WR RtI f%Rl 1%5R RT 10
• RcRRRR R?^’ RTPTT Rfleinn ^ ^fR^' ^ 1^ "Ml RRH TRlft RRRRT t 1% Wa "SR IrRR R’
a'^a< /5/WRf 21
’ft ^ wit ^ TtR qi<|f^«ti (Randomly)'^I ¥11'^d:’7T ^ ^«t)iq "3^ ^ .3fk 3lfy^ iW 3 ■'TC WcT "^Nt »«hl<. ^ 3^^ afk ^ ^ ^1 ^ f^ ^P?T if W % ^ ^ 3^ f I
37«T^ 3^T^ ^chiq ■?r^ 3^ ^ afk "t, 3^T% qi'Wifqch ,yr^Ti<T «F>t ql^Pcwqi yr<{dO’
■^Wtl 37cT: fqd<y| (p + ‘^f’jpoyqi sP-q^c^di q-iidi qi'jf^qi 3^ 3^
^ 3^’ % ^ yiRl4>dl ^ W ^1 f^:^T% n = 10 ^ '^, ^
dHl ^
% #n, 5 wt! ^ ^ vit^r^cfi ^raff^ (252/1024) #Tt, 3T3: -^rf^t5T^ 5 WT WT ^iTcTT t fR "^iFT ‘^fT -fH^hdl 't 1^ 3TI% 3^ 'SI^f% qi^R^e+i 3
'^gt^ 3TRf ^ arfii^r ti ■^' 3=^ -^t^kit t % w ^ ^ ^ ^'Jit'idi tl fqd^wi 3Jt
fqqjQTH wit % ■'ift^ ■'R p = 1/43^ q = 3/4'?t3TI 3^ fqd<.wifqqq "4 3TWI-
3IT <:tehdl tl3IT^' % ’ft TI^
Pi^^qd ft^y—
\2 ^1 Vae.
313: qi^poyeh WTt' 3it TT^ 3?T^ ^ft 1/16 'gt’ft, WT ¥eT ‘^TT^3rt ■yt-’HIbjdl 6/16 it^ 381T W[ W3¥^ 3IT# 3ft '^’11331 9/16 '^l ^ 33T^ ^fr033ft oqiqeiRch 3Wf^ 3ft h^Iki f I 'jHy’teMI ■^’ ^33-'Cf%^l 33 ?t^, 331731^ ^ 13^33^3lt 33 siq'iyf 3 3t3lf%3 ■5t3T S'glrH'b 3331311' ^«iReT3 o33Frfl3r 11313T3ff.% 31*iZI33’ft fqd^^'i 33 asqld 113731
(I) I^Md-f^ci<«i (Binomial Distribution)
f^33-fqd<«i % 133 fl31T dRid^i diTdl (James Bernoulli, 1654-1705) 33 3T3 ^33 t 3«3 1^ q'iT«nl |qd<«i (BernauUi Distribution, Bernaulli Trial or Bernaulli Process) ^ 3)^ 371^ 'tl fH f331'3 33 3133133 33f3ft .f373i 83 3ltl 11133 337T?13 ■333ft% 8 3^ W313; 1713 133 831 f|33 (Binomial) 33 3l2t 3m' (Two names) t. 313: 33^133173 3^ % 3R3f3 3331 tl 1133-1331^ IgR-^d 3ll^f% 1331'3 ^«*iq) 13JF 3ft 1T35113T 3 3Tin73f31 % ^ "4 13^11331311 ■31"31T3lfl3 '^131 f?7T3-l331^ l3?t3313Pi^^Rnfiad %- ^
(1) yqldt (trials) 3ft H133 RiRqd 'gtdt 'll
(2) 11^ % 313331 (mutually exclusive) ’iR’^ll^l ^ 1^1
(3) 7J3t3 3ft 11W131 37T 11%31^ 'p' ■?t31 "t 331 3111373I3T 37T ‘q' I 'q’, 1 —p, %
(4) 7T3t3 (trials) 1333 (independent) ^ f, 3T3ffi, 1^'03^ 3J3t3 % mRuiim 371 3f’lT3 3TPtRti4 '31^ 31^ ■^ hR'JIIHI' 31 H-Sdl t"! .
111333 ■373 f37t3313lf % 3511^ 11133 '513(3 31^ Wisft 11W13 OT tl31' Iqd^^I Rh^l ’ll 'Wl ■SST35R '•f 'efT^T '31 113731 f "3^ thrift ■t3-'5r3t3 % ‘n 133^ 513t3 Rif^d
a, "si^ % ^ in^’13 qRuiw a(failure) 31133731 ‘g' "gt # ‘1 -p’ % 3331■f % 313(3 133^ ■gtdl 3T%3, 33111373131 3ft 'SlTf33731 (313: 311333131 yiRiq>dl ’ll) '513(31
^3^ qscridl 3lf^l ‘11373131- 331 *3111373131' 37) Ill’ll hR^iihT % '373 "^Hl 31131 3 1^■33% VilR^cb -3181 % 1573 ^'l
(1 -3 4 4
o 1^9^+ 2 — +lie; 16V
^=Sl?337-1^ra733t'--T137
^ tl3331
1137^131 (success). l31T3ft iflTqqTdl ‘p’ 383 313373131^t!3gT311331331331^31 3373T
22 3«qd< . \
r
iy<W»l '3®l^ % ^ 11;^ fqci^in *i^dl■f^l (fair) ^ ^h^'jIT ■’Tft^'TR Pk*!! (Head up) ^ '«i^ PK'ii
(Tail up)—■f I .% "pR^ illfflehni : p = % cT^ 3^% pR^ viA=hd) :
(p + qY T^' ’fte■f%RT ^ t<«hdl ^1tl aRT:p + (j=l|^ ^ ^ ^ ^
•^^iraiR ^ ^ (A^«7r B) ^ ^ Rs*nf^ Mf<^ wr it^ :oq«w
BAHH
T H(H = Head, T = Tail)TH-
TT^ ^ RIST -pR^ (H) ^ I t ^ ^ pR^ (T)
^ -p 'P ^ RpT^m w t : .^ i h' H^RI PT^Pi' -iOM^ P (H) pRp = ■^,
faolq Pt^% (iesiciA P (H) pRp R^^l*=fdl = -j
R«TmT
Pt^* ^ T^ RI«^ P PPP pT Pra (H) PRP ^ ^ 2' = 7 Pl.'^PRP
(T) ^ .R»ii5pn Py -J- Pi
^ pR^ ^ Md'ii.'^ R^P^TcIT RHI "3^^ R^MTq’ii RiT yP’ci ‘p’ TST 'stiy. RRTT Rp SRTRiefcIT RHT 3pt RRRfr RCRIR^ RiT rPkT ‘q’ V3\ '51113; ''PPmPl RppRp Rp RCTi
PPT IpRTT ■5n ■R^hTT P-
TT.HT THHH. QQPQ QPPP
R^3p Wjf^ «*-»Hiq’ii Rporp2
p^ +2pq + q^, (p + (j)^RiT fRRIR Pi 3Ri: pt wa-qfSRp (p + q)2 Pi
pJ JtildA "qr ■3R% to PRP RP Rrawn (p) = ^
2pq
pT iR "SRp? R^ pRp "Rp RiRTRR (q) =
3RT: (p + g)2 = p2 + 2pg + q2or (i + i)^or j + ^ + \
^ WR ^ fRRp>“A, B, C RT 8 R»PfPcT mR^iim Pt RRi"PP-CBA
= p3HH H. H TH
= 3p2gT HHHT HTHTHT T -
= 3pq2T TH= q3TTT
. 3RR<R «/7^^?4 23
f^tRW . ^ ^ fnr# (H) ^ = -^Irmrl' ^ i^ (H) ^ = 7 ^ i ^ 2 ^ I
#r ■4‘^5^ ^ (T) ^ ^TOTf^ =-i.x 4 X-I = "I
' tt^ ■ftr^' ■sFft 2 ^ ct^t 1 ^ ^ = (^^ i x 7 = i)
#T f?T^' ^ rff7-#T ^ 2 ^ cT^TI 1 ^ ^ ^ -| tl
p = i- g = i ■ . .■ (P + g)^ = (7 + -^)^=P^ + 3p2q + 3pq2 + g3=l + | + | + l
m -aTryifeT f-(a +6)^ = a^ + 6^ + 2ai) '
(a + 6)^ = cr^ + 3o^6 + 3ab^ + b^ ' . . .^ ^ (p + g)" % ^ T^' ^ ti ^ mRuih! ^
3TT^fW ^ ^PHT f ^ iV (p + g)'" ^'WRrn ^ ^ ^ iV(number of trials) n ^SRT^r 'H<shi‘ yRiPifVt^ ct^Tdi t’l ^
■stW- 100 i\ ciaJT ^ cfl TF’Tlf^ 3?T^ ^ -mK iH~N(p + q)“ or 100 (p + g)^
100(p2 + 2pg + g2)ifp = 41 . 1 -,<? = —2 2
+-1=100 fl^ fU - + 100 -U; ■ UJf l^+ 100 ~= ^““4^1 47
= 25 + 50 + 25 = 100
Then4J
37^ =25 foi two successes for .one success and one failure
for no successes
■ri^ N ~ 100, 'E?3^3Tf TTSSIT 3 ^ ^ 1^^ t^R^TR |R' ^-
50
25
ifJV(p + g)'^ = lOofi + —^2 • 2
100(p3 + 3p2q + 3pq2 + q3)
f 3\ y 3^ + 1.00 - + lOOj 4ri= 100 - + 100 - = 12.5 + 37:5 + 37.5 + 12.5 = 100v8y {S (8
1^7^-1qdTJ| 4, 5, 6, 7, 8 n ■^1^^311’' % (p + g)" % fq'WK ?Icl t^RT ^ ^t«F?rtt- ..
(p + qy = p^ + 4p^g + 6p^g2 + 4pq^ + q^(p + g)^ = p'^ + Sp'^g + lOp^g^ + lOp^q^ + 5pq^ + g^ip + Q)^ =.P^ + 6p^g + 15p^q^ + 20p^g^ + ISp^q'* + 6pg^ + q^ip + (iV = p^ + 7pPg + 21p^q^ + 35p'^g^ + SSp^q"^ + 21p^q'^ + Ipq'^ + g'
24 P^rfl TtffWF^fM f^fiPjf
^ 31^ f-.n(n - 1) „
1x2 ^n(n-l)(n-2)
1x2x3 ^ ^n(n ~l)(n~ 2) (n - 3)
Ix2x3x4 ^
-2_2(p + q)'^ = p" + n.p"“ + q +
q^ ... + q'^+
^ /I = 5, ?IW (p + q)'‘ = (p + q)®
5(5-1) (5-2)3x2 ■ ^
5(5-^!) (5-2) (5-3) (5-4) , _ 5^5 5x4x3x2 ^ ^
5-.2_2(p + qj'’ = p® + 5p® “ * q^ +
5 (5 - 1) (5 - 2) (5 - 3)4x3x2 ^
= p® + 5p^q + 10p®q^ + lOp^q® + 5pq^ + q®(Combination) % I^T^' % 3TTitp: ^ ^ ^FT WT ^
6-3q +.25-4., 4q ++
t-(p + q)5 = '‘CgpSq® +.''C^p‘‘ q^ + ”C3p3q2 + '^C2p2 q^ + '^Cjpiq^ + '^CpP® q®
= ^CgP'^q® + p"* q^ + ^Cgp^q^ + ®C2 p^ q^ + ^CjP^q^ + ^Cq p'’ q®= Ip® + 5p* q + 10p®q2 + lOp^q^ + 5pq^ + Iq®
(q + p)" = "C^qV + "Cjq"- ipi + + '‘Cgq"-^^ + '"C^qV"= q" + "C^q'^' + "C2q« - V + ” Cgq" - V + - + P"
^ p = .i, q = -irT51Tn = 8, teTR ^ ^-
8 (7) f ly1V2J ^2J. (1)(2)UJ U
8(^X6) fl (1)(2)(3)UJ 12
+ 8(7)(6)(5)(4) Qf (l)(2)(3)(4)(5)Uj U,8(7)(6)(5)(4)(3)(2) n V 1 (I)(2)(3)(4)(5)(6)(7)i2jl2;
1 1'^^ ^ lA*«2 2^ ^ 2
\7 ^sn+ —j.
\5 / jxS 8(7)(6)(5) riVflV (l)(2)(3)(4)l2j
8(7)(6)(5)(4)(3) rn^ (l)(2)(3)(4),(5)(6)l2j A2
2;
if
\7 /-,\81• + +2;
^ ■qr-
; x81 1 1 8 • • 28 56 70 56 28 8 12^2; 256 256 256 256 256 256 256 256 256
8 qfjq qjt qrfqqnn ^ yc^K
H T P P
18 0 .004256
87 .031’1256
■ 286 '2 , . .109256
3^rt< ffllWJWhj fqftRT 25
m/A<uo/ /q(T<w 5635 .21925670 .273 ' ‘4 4
256563 5 .219
256282 6 .109256. 81. 7 .031256
1 •0 1 .004256
256Total l.DOO256
(II) IcTcROT(Poisson Distribution)
% 5lf^5 (Simon Denis Poission) (1781-1840)1837 3T1^1% fsRRTJT ^ afaHK-i Iqa<.wj "tl
%\ ^ fH 3TI^f%-Iqa<,wl ^ ^ ''7^ %f«ql'JiiT '4',cIT^ ^ ^ Tt fqcu'Jj *lt "3^ HPRTI
oich4 1''sfPTfR '5Pft^ ^'fj^nai cl«lT ©MqeK ‘511711«qa»q 3^lT
'511711 ^', 73^^ ^ 'ellTit ■*TT^ 3Tf ^ 371«illft71 % ■'77 fqa<,u||
■f, ^^ciqi'<
■5R1R ‘SRpI -if 41Lbddl ^ yiRl^idl WJH tl ^
■% ^ '511 ^ebfli't '3iqfq> n 37^^ 37^7 '511 '7?T (n, —> oo)'7i«7i p ^ ^ 37k 5n TWT ^ (p -»• 0), fT? TT^ 'f^ ‘3^ -5^3^ krf^ Jim (m) i\
m k8R ‘7^1 37^ 7i^‘ ■4', ■t^rwER ^ ■377 ^ ■4‘ t-'5i^ ■qr 37^ ^',51?T «'+>cidi ^ yirq«bcll «1^ 'tt 7W1 «qVil ^ '77^71 «t»iH)l 37fu^ '^1 '?77 y«t)i< % 377^75)711't^TTT'q '5)7 oqqsK '5k71'% '5lkl, '^T^lNt 5ri "STRl, 'jrik <iq>wi-^f«:qT, ui^qiT '5)7 371W7 37lf5 ■^5 ^ -k 511711 tl -MT-!17H fkk? -k ■#171 ^133137f -f krf^ 37R71^ (77^75, ^5)7) #1 ■5T%t^ 3i»fl<iciT 't '5131 '317 tl. 5877 -517^ 5t 1^5) IT# *5157 '311^1775)717■35-37^717151 (sub-interval) 5)75t 'ttt qifey. ■f577^ 35-37R715T ‘t* 'k 37ft757 775)51717 5■53 77%l ■35-37R7I511 5t 7n§q7 tt ■535137! 5t 7T^ tkftl ‘5? ■tlct t ■53515 ■tPtiHi-qcii «qnoi
(qn^^t ■^JT *
(Form of Poission Distribution)V
fg55 t^TlT'^’T 5)1 #7 ‘’^75775 (variate) oPuso (discrete) tit t,3787f7( t ■’J#) (integer) tit tl 0. 1,2, 3.......*775)51717371’ 5t 377f55)711^ ^ t) iqwK ^ W5t 517 7755711 t-
W^TTTn
, m1+ —+-----+----- + —1 2! 3! 4!
—.r mp = e"'" r!
«
26 '3'qfl< «/7<w«»)?«/
f-^ yet>dlNo. of successes ; 0 1 2 3 4 rProbabilities (p): me ni*e~'"e-m
WT--^^
fmrrJT % 0,1,2. 3‘"-..... ■'W4»dcn3Tf ^ t- ■
(i) «hV1 (m) r^«hicii '^JTRTT "f I
(ii) e-"' ^ T=TH W ^ tl
e ^ 3im) ^ W\ 2.7183 itm%i
11 1e-m-m (2.7183)”' Antilog (Log 2.7183 x m)e
1= Reciprocal of [Antilog (.4343 x m)]
Antilog (.4343 x ni) g-m = Reciprocal [Antilog (.4343 x ni)]
{Hi) % IPflTT ^ troiiT 0, 1,2, 3,4'\'
P(r) = e~ r!
(Characteristics of Poission Distribution)
rcjci<.ui -if idHr^ndd ^—
(1) T^firSrT "fecRtri-f^ fqa<«i ^ «[1 7§fo^ f^clW (discretedistribution) f, 2R«rt?[ ■3«=hi "3^ f^d<u|^ ««t)ai 'f I ^4i<sHi 0, 1,2, 3...^ ‘^'pTcf 1%^ "SIT^ ''7T qani ,'3t^d
?bTT ^ 0,1,2r..^ 3PR1 ^<s41 f* ■^*t-0. r, 2...«l.
(2) ‘p’ ?I«IT ‘g’ ^ l5fOT ^ ^ t ^ ^^ (p) ^ ^ ^ t W3^ ■E13^ % ^ ^ ^ yipT-^dl (q) 3Tte (1 %mw\) tl ‘ri ^ FfcH tl
(3) '^[13T iffMicft—idd^ui % hNIVi (Parameter) '*TTWT (rn = np) tl ^m ^nd tt Tit 3RT Tm^ ^ ^ ^ f I
(4) 3T«r^ % rqd<wi t’ 3T^' (Constants) % '*1^
^ . X or m = n-p, a = V;;^ = 4^,
f
^t-yctiK
. . ^1=0. ^2 ~ ^3 “1^4 ”^)-(5) 'BKr tqdT3M ttm tl m
^ wr t CT81T fWTdT ^ tt# -sirat tl ,
(6)^ ^ 37^ % ^ ^ ^ -ET^ ^ ’Sr^Tlf^ ^JTcJT tl (ii) ^ ^ WT, ^ WT ^ 3RRI^
f % ST^qid -f tl (Hi) ^ 37f%igF ET3?n % ^-41-rRld t ^ ^
!'^d'll ^ '4Sdli
^ ^ FTFT ^ ■grftf^jfTT iJ|uq ttrrt ti
C1*<(l
(model) T?rn f ‘p 3icERT 7TS^1 n ^ hUhi'ji ■?), ■^, f^-qqW
■qr ^dl'+iTi ■^fI^ 3Tt^ <H<sqi, f^fficT '’TT^ ■^’ ^ ■h'ohi,'^’<^, iV?ft 'T?
3TT^ o[T^ yie^iT ^ «<£q| 3TTf^I
^lf ‘t>'fll /%f77W
2.4 '^TnTRT Mifqehn( 0^1 (Normal Probability Curve)
■3)^ qf^iq fqci<ui oPusa Iqfll'Ji (Discrete Distribution) t «^qcrt
■3^’ Hr<.r^Iaqli' T^’ f^r^TT ^«t>di "t '5T3=fT % hP^wiih ^^<ndi^ ■!■ R 37^ WcHli), ■!' Pfld^) (Continuous)
W^ ^ f I ^ R7 % yikllSfi’ ^ 3TSM oijpcW^i' % ^ 37T^ ^Tm tl ^ ^T3^3Tf % R ST^etr % Irtt ^TIRPT -Sflf^T^ t^RTR fR?f
(Normal Probability Distribution Model) "f^PTI ■^STTcTT tl f|:Pt ^ p ^ q■RR "REIR ■?lcn "t fT«TT HiciiW) (Exponent) ^ RB 3RR (infinity) ^ 37k 3TJRTT "^kn t era ■RRPT yifqchdi lqd<*^( 'RIRT ?kfT ^1 W tr^ 3(<gl«sa, TTcR k TTRfkcf fqc««( t"! 37T^}k^ «/'Rs4=r7l k 7TWR? yiPqchdi fqa7;»j| eFTF ch-^lq 7^«7R 'll W7F1 yiPqqxil-lqd^'Ji ^ ^.siPd«t) tkcRR tl m % 3PRT 3Tk 37Tlfkc[ ^ ■% qiKUi (qd<w! liSlpq^l^ 3R '^IcR "^kn 't^l 5^Pn<< W^TPI
Hifqetidi ^9B (Normal Probability Curve) 3787^ 'i^. kt. kt. (N.P.C.) % 3F7 ^ ■HJ^lPiR f^rqr BRT tl
vufqchrii "^fF x
(History of Normal Probability Curve)■4 73;^ ^.siP*^^' (Theoritical),
(Normal Curve) '41735. kt.WIPT ytpMqial (Normal Probability Curve) qi«iq
3TR^ (Ideal) R87T ■‘iPui'ilq (Mathematical) 'll 'RRPT
kt. (N.P.C.) % 3m k. *41 ti trtp^ infwcn ^ w 33 ^ #33 k x?tmr ■JFkrkft TRvn^ff (Abraham De Moivre) 37t RRT
'nkncikT xmt^TRT ^fl Mli 4n oqiqsiR^ 333)3 sBrokrt 3iR35T kkk xnfiR(Pierre Simon), ^j) rr^XT dlM147 (Marquis De laplace) R) 3t?RRT 31. 331 "343 odloilq^i
1733 k 1XT 397■f Pji'^l’) XPl
3rfcf RRT (Carl Freidrich Gauss) 3RkT^ j?idi«Pl % 31Tt3 k P+iqil 3RT k 3T3I<aMlcrirq<l 'gRT ^iT Bik 3T^ 3mR ^(iq) (Recording Errors) 33 "feRR xrmR? yiiq«hcii 3sF %•STJFX 3kn f 387T xrmPI .yilq=bdi 397 ep) fgq^ fkcRR (Binomial Disitributidn) k qldiqi n 3RRf (Infinity) 33) 33133 'SR3 1^31 RT X73i3T 'f I 3X7% '^Xl 3f) '?3% <^ll3 XTHTPl91)33)31 337 37) RlfxxxR 337 (Gaussian Curve) 3733T XTTRPI 337 (Normal Curveof Errors) kt 373T 31^ 31371 X7mP7 9Tf3373T 337 % oqiqeiP^-^ 333hT 3k kfRRRR % k.slvsk
.(Adolphe Quetlet) k 3Rk7% ?lcn<sql % I7t3 ^ k' 37r3f^ f3X3R f^l 3X7^ f33R 3T f% 333)3 37^7333X3 331 HI-iqT^ 333137) 'k XR3k33 X73X313k' % 37WRR k'
f3?31X7 31 f% 3lk 3lk 3X fkP^R HRPy^l 331 kf337 -JRXimPT Mi[qq)dl 3sF7 371 (Model) % 3R kf f3731 Rl XT3731 (Mental and Moral Traits) XTIRPT yiPqqidi 33) % k f33lX3 Rlk Rikkl 3X137T 3g 13?3IXT 37rviT3R k' XTr3 '^3711 37TRP3X k 313 3131X737 3 ^ HlHd X7E33 ^371 33 33371|qd<o( ^impi 91(33731 337 % 375?R ^ 9131 3311 TRk7%’ % 37(319 39f k XR 9»?fk7X7 31333 (Sir Fransis CJalton) k ^3f3333 f3f9R3137)' k XTirI^r 379k 93)3) k w 137 373^ 9R1r37 331 9rfX(f737 I3^33f37( (Mental and Physical traits) % f33XR XTRPl yiPqqxfl 337 % XmR glk fl
Hl^
tl 3^3) I
28 3^3cR «//<sq<*?q (3(331
(Natural Phenomena) TfnHFT.Tlrf^T^ ^ ^ yi-'HMiT ^ ^31^711^ I^l' . ' ■ ••
‘iPwioIm ffe ti.iHi-H yir<<=hai ^ %irf ^ t n ^ ‘'TH 37T7T
(Infinity) p q '•TH TT^TH 3TSlfj^ .5 ^1 «mIT^ 'ylt^<*cii fA^ldl STTUTftfT "f'TTPTRT yiP^eftni "Wl h1§«?7 %, 'gHT yfrlMlR^I #r 4,ku| iTt^en'
^?F) (De Moivre Curve) ^ ^TTcIT "tl ^iRsqcbl "A WTP^ IJTf^T^Kff ^ STf^RT
Hersc^yf "tl 13;^ 'SToFR % As&iPflch '^sR’ (TTTRRT ‘STlf^raTn W^T) %A' fqciRd ^ ^I'RsH'dlfqsT "A <5oi«hi 'fTT '^sF' ^ ^i-sd 'HH 3TA^ TTlfeT^fhT fsfftpft
'A ‘WIFT yir<^?hdi 'mr (Inferential Statistics) ^3JM t^l ^IHI'^ yiRWd! % tfr^R ^ [d=bici i^RTl cff Pi«h'^frH'h ^
^s5 'A A\\M^ TW' '^iiA^iTI
A’ yifA«hdi iiTf^ t <wlT+ ^yiRH«t>di ^ "A IqdRd ^ rl^TTfA SWvIIRRcT ^ 'HRTFI infAehdl % 371^
^ W^ ^ ^ ^ tl N ■5i^ t ^c|diRt)d 371^ ^ ^ 371^ ^7RF7
yiRi'^idi ^57 % 375^ ^HcIT 't’l SiefcAfVd ■ysrfrT % qiKwi N ^ ■'77 «<iq5iR«t>
% 37%3T^ A ■^iT y^di "tl nR^iiHd:777*71^ yifA«hdi '^sfi'"% 3i^^h t^dR^ 777^77R7P7 f^AWT'STf ^ TTFI^ A 177 y=bi< yflWisA fe<>^i ^n TT^kTT "tl PT*^ A’ 1%
■77n7F7 yifqchdl ^ t[5fr ^771 A.&iRd'^j, ^R'id1«< 77«77 37RA cTsF f, f^RT^ oqqgK A "'jA TTTIAt 3777^=^ if A "77^ f^iRT"^ c^iqeiRcb ■J^tAAt Sick'd 37f^7^ Al •
TTRTT^ MrR4chdi ^ ‘ .
(Characteristics of Normal Probability Curve)yrfA^TTTT ^37 (NPC) Ai TTAf^ ^ A wet>7 ^ sf^d ^ lAn
lAi 177 ^ "ni^ 37^:^R7=7"fAR7I '^l 37cT: 777*77^ yif^cRdl ^y^«Ki «i'RsM«tn A wTr^ '
377^^ f-
1. 3ii«y>ld (Shape)—TTTHT^ 37TfA^ ^ (NPC) qiwq A ASTOT (Bell Shaped), ^ TTiTffTH (Symmetrical), T(W<iH (Unimodal), ITHTTR?'^^rlT (Mosokurtic)^^'ll177 ^ A ANf fAA ^ 3TT^fAAi ^ if^ t w Ars ^ stA 37^7^ aTT^fw Afcff f 1 37R7W7 3Ti^f^ Ar^ A t ^ ^ ^ AfA ^ Ffm ti _
pIT*
1 »-3 o -2 a -1 o 00 +1 o +2 o +3 a
2
Hi1&4‘t>l*4 fkfkkf 29
2. (Mathematical Equation)-^TT^TPT ^ (NPC) ^‘ir^mT^l (Mathematical Equation) f—
N -x^/2a^ a
x = q%WT -irr^ X''^ ^ 3T«^f^i,:x: = X-M . ■ .a = yi'<ii’«t>) ^ Hii«h
N = 3Tr^f%
1TR 3.1416l^+i TfH 2.71828 tl
3. ^Hnal (Equality of Measures of Central Tendencey)-WTBTTntetH ^ (NPC) ^ 'sr^ % TTB titZTXTH, .iTi2rf^ ^ WT ^ cT«?I ^ -f^ ^ fl’ara;
M = Md = M^ -
4. fqcjHdi «^uijct) (Coefficient of Skewness)—"HFIFT yif^'t^ii ^ (NPC) 3T5T(Symmetrical) f =hR^T ^ [q^Hcii .ipTfeFr (Coefficient of
Skewness) ^ t[H ^ t 3T«lfci_ . •
S, = 0
5. ci5hai ^pJTf5) (Coefficient of Kurtosis)—(NPC) ^■^Hdi 3^ ^=hl«rll 't'l ^ 3T)^ <jTc||^ 'qi? ■!■ cT^TT .1^ =(5t>'al(Coefficient of Kurtosis)' ^ .263 f 3T«l1^
Ku = .263■ 6. ^H*dWVn (Agymptotic)-'HFF4 yiP^^d'! ^ (NPC) ^ 1^3Tt' ■^' 3FR! (Infinity)
^ -ark mm ti sttur % 1%^ cfl 3# f^^ "i^l TI^ op? ■% yiHI'4 3^lf4=t)dl ■^5’ 3TTqR 3T?RI ^=(1
■^nxTT FI
y ~
[q-etoil
71 =
e =
TTlft^TCIT
7. cij^ni ■f^TTT xrft^fN (Point of Inflection)—nRqd'i ^■f ^ 3prft qst"cii (Curvature). 'tl ‘HFTRT (NPC) '^T^^FTH
g 3?qk( ± la ‘'TC 3Tq^ ■^TTT qR.c(f5^ «t><di 'tl "^F +la - Ict % 3TFIR 3TtT 3iqcicri (Concave) F^ +ia % "SFR —la % 3f8?fcl^FN)1^' (Tails) 31TUR ^ arlr (Convex) F)ft 't’ldncn
3iqa«n
%# i 'H----- —I-+ +
+1 o-1 a
3
30 ■d-^rti
8. tiHin 3r?TT (Equal DifTerence of Quartiles from Median)—yiri<+dl (NPC) ^ -^mPT ^ tl
% ^ITRt ^ 1-sjfe' (Probable Error-PE) ^ tl ,'3T8^f^I^
Qg - Md = Md - Qj = Q^PE9. 1oRI^ Trar Hn«h (Relationship between Q.D. and
(NPC) Iq-cici'i ^ tVq<rl^ % '*TR cFT^FT 2/3
Q = .6745 oW o = 1.482 Q10. ^ciT-oei ehlffi (Highest Ordinate)—■^TTHPI ‘Sfli^T^irn ^sfi" (NPC) ^^«ZT ■*7«rtTFI ^ ^ t W*TF ^
3T«?f?( N ^ .3989 tl ^ ^ tl
Sif^m l^tn'^itilHI'^
3989
fe4
68.26%-34.13%
I1M +1 a
.47.72%42.72%
M +2 a -2o M -2<t +2 a
99.74%49.87%
-j
M. M +3 a -3 o
5. t* ^?afw»n (Area Under Normal Probability Curve)
~<i^ai W7W/'*?<9 31
11. (Areaunder N.P.C.)-'^TWFI'5nPT^^ (NPC) ^*11^ WTFT SlHWal «t>6Cimi "t cl«11 '^^ cR^ iRjZ
WTPI IJriV'+fli (NPC) ^ n*)'^1 'Tt ■% ^3)4>crt "3^ «+il(<i4l % X!K»i«fil% 3f^ opt ji«^Rja ■f TfMI "^^F) PiHf^di\^ ti irKRTH ^ Rmi ^ rt ftsm n tt«rth fw?! ^rtfe ^ w^ % 34.13% yikil«4> f I RWTRH ^ ■JTH^F ^’SfrR ^ ^ m 1^ "^tfz %^ ^ lITRTf^' % 47.72% yimi+ ^ t ■*TK!TITB ^ ^
^ ^ % 49.87% yimi* ^ tl iTWmH ^ rm 50%-50%■^tl
h^RnT
Pcl-cleH TR ^ R«mHHii't)
yj'<ti«t5
Tq^ t(i) RSSiJ^ ^ ± la ^ ^ 68.26% W<n^
(ii) RHIRR ^ ± 2a % "^el 95.44% UlMIdt $1
(iii) RKWH ^ ± 3a % ^ 99.74% WIVil**» ^ f I
± 3a % ^ 99.74% R«ft W^ SH ^ f I TRFnsfl' ^RR f^RT «TRn ■§ 1% RRPl yilM«hai "^5? ^ ± 3a % 100%) 'yi’<H4j
3TT^tl.
RTTep RHIRI Tnf^raRT-^pR
RThPi
(Standard Normal Probability Curve) ■RRPI ■yif^RRI ^17 (NPC) RRNR’JT ^ 1^ RTRRT ailMehai ^sF %
RiJF % aiRTK (N), RHRR(M) Tl«TTRrR7:f^raRq (o) ^ t,■4 H^cro SRI ■'R ftsRRi (Constants) 'll "R ^fNt % ^IR "qr RRRT 'yif^RRI ■^37 tFt 1^31 ept W R37(it tl % %T^ ^FT STRTR, RWRH TTH37 UR: ^
t, SR: ^3% WTRJ yiPlehdl ^ ^ fUR-f^ pidimT 37^ ?7q ftdfldTTH37T ^ WTR UTRt ■^■, RRRT UTfU373I 37T UUtU ^RT ^l■f "^sF) '3>t 't>lr«:<<i (Ordinates) % UT URTR^ 3»t 3!^ qTR
, ■qft U 37feq WFT SRRR RT 37Ff
RhFT
^1 RmPT yir4<+di R37 uft j^uqTR qpgsMt f^vlqdi^if -^w f^ ^ i % RwruR ur ftsR 37tf3 HNeb uft 1^*13 «t)lfi:4i ,% U7T ^5141 Cl ^1^ PiPfqa yfflJjm "tl '^’ 37?T ^ ■f UTRRTt "37^ Mll«t) 'UTRRTf (Z-Scores) % '?7q ©qc^i cR fqfWH cptf^ % cRT J^UqTcT ^ ^37 fdl^'dd Uf^Pd
URfR^ % ■^‘
R37ffl
aq^RT sq^qR^tl %^ (Standard Normal Probability Curve) "tl 3R:"URPT UTfU3RT ^ t f^nq% 1^ UWRH ^ UR ^ % UTRt UUT URUP 37T UR %
UUUT ^1 cR URU7 UTURT Milq«shdl US? U>1 UUl3>TU
Ulf^RTUT URtf7 UTUFT Urf^RTcnUPR7 yc^a UTUTR qsF
UlfUUTR Usl7 T^Hl'l«ti UTUPT
Y = -^ ■<l¥n
g-**/2
"UU N = 1 11R UPR7 UTUF? U#R7cn UsF Uit Uluru yTpleb"dl U37 (Unit NormalProbability Curve) U7?l "UTcn fl "^W % f^ 137Tf
% «ncRT 'skn "tl i^ttIUJpl+di Usf7 U7T ^^fUUTR fgTT^UTURT
UT^RTcfl UsF Uft UulUTUq "t—UTUIR
1 ^-z»/2
32 <i-^'=i(1< «/7is?V<V3?<V
Rdi'JIWTP7 yipHqioi ■^, TT^THn, Hn=h fq-qci'i "m "% sn^ t ■q;^ ^ wnt ^ t fW'+ii w^ %■^T^nMB ^ I <H!TVt=h1 ■’7^ ■^TTHF^ miPh^ Wp^FTTTJT^; Ffst 1%^ ctilfcHi' ■% % ^5l'+»d 3TI^f^ ^ aT^qic!(Proportion) % oq^ =b<al ^t Hi'i<^ WTPI 'jnf^T^rcn or5) ^ ^emdi ^
Sjfqchdi’ ^ Rft'el ^ ohlftf'tf) ^ ^^^T3^/37T^f?pj) ^ ^OrT Pch^f Tf^RTT
|^T% 1^ 1^ iT%:znTR TT«TT (^-elcH ofT^ f^cR’^ Iqa<>«i M^qlao "t 1^R7^
R«mH ^ RH^ Ti;^ ^.- ^ ^ f % rh^ rtrt^ uiF^^di ^ wv^R1H«I> yRH=b) ■% y^«w R>t 'H«t>”dt ■§■! ^Rf: RR^ RTRRi RTfRopcTT ^ RTRlt cFT ■inql'l
RR^ % fRt^ R^ f^rf^ RTRTRF^ RRRT yiKii'ehl (Standard Scores), yiRTfeP
(Z-Scores) 37R^ ■f^TTRT RH (ct Values) ^ 't, tl cfr’T^RRT yi^iVl RTIrRRT RHI ■#>' Rl^^ WTR RTtfeif ^ ^ RH ?^RRR7 RR# R RIR R=R ^ f 1 RR^ R^RRH RRl'-R yir^4)dl RRT Rft SnRR ^ % RiR 1^ Rt 1w ^ t RR[ RtRRH #> %R RTRIRT .RT fRRRI RH R^ "t ^RPeiU. Ri¥T RT RRiRT 1% RPTRT RTRPT infRRRff RsF 3TT^ lTP<rf^.(f^R7R7RTR) RRT ^ f RRT 3TR^ RTRT^ (ftRTRT RH) '^JfR arfRRT ^ tl RfR WRTR? RT
fRRRT RTR RRTcRRT ^ t R\ R^ R«TRTR RT RTRTT^ RT PRRRT RIR % #ft RRTT
.(Right Side) ^ fFTcT "t 'RRf^ '^RTrRRT’ TTIRTIRT RI fRRRI RH RTRt RRR (Left Side)
T?RT 't’l RRff^ f^^T ytKiiqi R^ RTRITR’ RR^ % 1^ “SR^ R«RRTR RTt R^rfRR RTTRi fRR^ ^RrT^f(Z = (X - M)/ct). ^Rf^ TTIRTTR- % R*RRIR ^ R:^ RT RRIcR^ ^ RTRTIR’ RT fRRRT RTR RPR ^fcTT 'f 'J»qIV RTRURT R^RRTR ‘^'[HT RT RJRTcRRj RTRTfRT RT fRRRT RTR RPTT 'BfRT ■f I fRTRt RTRRRT RT fRRRT RR Rif R^RRTR R RPTRT fqq<nH % ?TR RT yi'^lW
RRIrTTRI (X = M + Za) Rff RFTRcTT RR^ ^ RR5TTT "tl
UlfqchriT R3? RTTnff(Table of NPC) ’ '
Rfrf^-l R^ RTRT^ yifR+dl RsF R>t ^^RTcT TRR^ RTt^ % RRR RT^ (Column) Z RTRTfRT RT o RH % WRR RTR ^?IR^ % RRT T^ cfRT R^ t cTRT Rf^ (Row)
fR% %R. R¥TRcTR RR TRH fRRRT RRI tl RRR RPR .0 5.0 RRT % 3fRT cTRT RRR Rf^
.00 ^ .09 REF % RR 3T^ fp; t'l RTTRlt % ^ RTR ■4‘ RiRRiR Riffe R RFR-f^sRR Z RTRTTRjf (RTa RTRf) RT RlMzRf % ^ ^JrfRRRf % RTRTlRff RR RT^RTR ^3TT tl RBT RF RRT «RH ^ t f^ RTtM Rf^ tfRT tl 37Ri t%Rf RRRT Rt RTRR tH RT Wft ^ RTR Rt R^ ftoTtift ^ aiRdlRR R^TRT ^Rlft^l 1R RTT# R^t RFTRRT R RKTRH RT fm Rtfe RRT f^ fRR Rt Z RTRTRv (RT o RTR) RT fT*TR RTffs ■% R^R 3TTt RT^ RIRTTRif RR 31^4Td RT^TcTI ?nR fRRT RT RRTRT tl Rf^ Ptitnl yiyiich RR a RTR 1.54 t RRT R^RRTR R 1.54 C RT fr^ RtfcTRl % RTR RTRPR^ RR RTTRR/Rf^ RTR RTTRI Ft eft RRR RP=3^ 1.5 RRT RRR f .04 tri^l RR 1.5 RT^ Rf^R .04 RT^ RR^ % R7ZTR RT TT^ RF ^1 TRRRf TR^ t f^ RF'TRsETT .4382 tl ?TTRR ST^f t f^ RtRRTR Rtf^ R 1.54 a RPT RT^ RTlf^ % yikffRTf RR 3T^RTR .4382% RTRTTRT tl '^T^' t RWIRTR R 1.54 o RPT % TOt^ RPRIR? % R^R 3fRT RTRT RiTt RT^ 'STR! Rt TT^ 43.82 yPd'^l’d tl RRff^ TTTRPR RRT tTRT TTRfRR RRT t ?TTf^ R*RRTR ^ RTRI 301 R Rfot 3TfT RTTRT ^ RT Wtfe
3T^’ RRI RKTRP RT f^ R^ ^ ^ TJTRTTR))' Rif TRsO RTTRT Fftt tl 310: Rf^ a RR RTRTTRRT FtRT tf TORT^ RliRROT RRT ?^RRTR R%RRR % RTRf 3[fT ttRI. RRT Rf^ a RR RRTcRR ttRI tf RKWR % RTRt 3fk FWt, ^ ^ ^RRT RTTRT tftl , • .
^/i^qS?R 333^WT
^IciReKi Z ^THt (a 'RT^) % ITT^fiJ TJWP? Viif^*'fli ^97 ^ <+)')n»i]l^ -ai-qil HRr^i'<i-3 ^ t^^TTjR % cT^Tf yHisl'fn HiKii'ebl(Z) % -RH 1^ tl
’rte43.82% 43.82%
-1.54 a M -1.54 a M +1.54 a
^ 6
2.5 MifMchflt cI3> % ^
(Use of the Concept of Normal Probability Curve)
'^«i Rti^i ^ ^ H'llfq^ti'i, 4iRi'*i 3Trf^^% STRi^rr^ tttrp^
% RfdRa I'l TWT *^i«d'i q7 ^ t|‘«ipH«dai ■^97H5Tq "tj sTfcqiHchO'Ji 'MiHiW Hif^«dai "^gr ^ •csh’hVi ^iq?iR«h % ‘^RT^ % "grr
tl ^ ^ RT W^ % f^tTT^ ^
RRIR ¥lRig7cn % yrqq ^ 3Tr®Fcf
iyiR4«t)rtl fqa^*^! % "4 ^IqiK Rd^T ^<ft RRRI >iiRi«t>dh'^57 WRcfT RTR U^TR ^ "gn RRT^ Rnqi ^ ^ :
1. 1^Rt ■^' W( ’T^ Wdi't*)’ ^ aTftRF RT ^ -at^ ^ ■^' gTt Rsgr ^ ^rti
2. IRJ? TTIRngrt %'^Nf 31^ UTRT ■gR^ "ST^ ^ R53T inw/grRTI
3- f^Rft Rjf IgRft Ri;(lq 'Si^' % ’STRlfgr ■m1mii< ^ ^FTBTI
4. R^sm % RiRrT ^ ^ Wf ^TRTI
5. ^ (Subgroups) Rfi yr^qiqV^di "giT 3RTR (Range of ability) «rcraT TFI
RTRPq
TR7R gft ■% ^ ■RTRiP? TnRRRn '3(715 'gf)■RFTRcfl ^ Rfiqi 'W ■§! 3iRiRq^ % ylaqq'i % RTRRR ^iRchdi %
31^^ ^ ^ ^ RRIR yifqqidi^ tl ^ 37nt ^ arwTRif ■4‘ ^ nf ti
qg7
1. f^KRt ^ ^ 3T^ TOT ^ eyteiJi' ^ W3J\ 5TTR ■9TRTI
(Determination of the Number of Individuals getting Scores above or below the given Score)
^RTt -RJf % f^ W ■RWRH ^ -RH^ ^ tTOTT^ 3{fe -m R7R.3(^7 TOI ^ ^ tl ^ 'TOTf^'
■RTRPT yilq«^ai RPRlt ^ WFRIT fR "SgTR
-qiRlRT ^% 1cici<.w| Ri) RIRPT fqcKui ^')eh'K 1^RIT ■RT eft Rit 'HHt-marf rt) ^ rr "Rgit ti fdij; rr^ "q?^
■gTt a RPT t sRqfSa gR t cf^TT |R RR RT R 'R^RRH Ritfs % rH % yi‘^l«hT RiT >frigid
HRf RR tl dcM^qid o RTR RT 'STR % TOTIR^ R^t Rf?RTcT ■w'om W RR tl
34 W/^qfi?«v fwfinjj
^ (Continuous Variable) ^ ^ f,
iWTT^ ^nf4«iia1 ^ ^ ■-Hei-Hai ^ WT Wt^ 3llkH'«b1‘ ^ (^
^ "iH^«W ^) ^ 3^ lT2Tcf iili^d WiSHl ^5F)T^ "t H f^ WcTT^
■q^ ^ % ^-qr ^ sik ^ ^ 29 ^ 2^^ ■qra
^chi'll ^ WTPT Tnt^T^Kn ^9F 28.5 % ct qr
^ ^ fm?T ^ % cFOT ii ^ ^ ^ 29 3Tq7 ^ t28.5 ^ cte 29.5 ^ ^ fl 3TfT: 29 ^ '-^ ^ ^ f ?Tt arsf f -f^
, 28.5 ■^l ^ WR ^ ■51^ 29 3TftT^ 3T^ ^ -SJsff ^ ^ ^ f fft 37«f^ i^ 29.5 3T^: "^R TTRIT^ 3R? 'SITRT oIT^ t?Rt ^<s4l ?TTcf
q=R=ft ^ t ^ ^ yikjt^ 'SFft fTR ^fbn ■qt f^r^K ^ f cT«TT, ^ qTRR? 3#R? 3R? TTTRT
'^1^ ^ wf 4)<’fl "f rit ■qrqrf^ ■qft, ■^t*n ■qt f^RR f"'!
Tcri'^<qT-200 '07^' % 13:^ qft^rif mi. 1^ qq 50 ^«7t rpr?
12 ^1 tF^ qft'^ qr yT‘^i«til % Tqa<'Ji WTRI fc^{q hh4j"< ?n?T
(i) 60 3Tf%Rr 3f^ TTM #t?
■ (it) f¥rr^ t5Tqf ^ 35 Site 3f^ TTR F^?
(iti) tsiqf 55 qRT 37^ qiR #t?
(iv) 30 ^ 3^ qRT 1^ #f?s . FH--Fq^ t qr. N = 200, M = 50 a = 12
(t) 60 ^ 3if^R5 37^ qrqr qR^ ^hct ^n^ft ^qiqpr qrfqqRn
60'^ tm #Tr srsTtti; 6b.5 % a qp qr 1w qrife % qfqt 3tr qn ^qqrH'^ qft qRasn qri ©qq^T qi^qr f^Ff% qrqrfqr 60 srfqq? ■f’l 6O.5 qrt a % qn ■^' qfRf^ qrrq qr
X - M
nif^-^ai fmrw
t^SRT q^ % qR^ '3TJ^
yiKii«t>
xm cy RR =CT
50%
,18.94%
+a50% >1
31.06%M—►«— 18.94% —►!
■to 7
37q.: Tirqpq qifqqirfT qqr ■4. +.88 o qi' qi^ % qf^ 3ik "f^qq 'STR q(T«d wq 3q^ tiqT33f "qqffiq "fq^iT "qqi 'fi ^rrqpq qrf^
■qKqrfH qi 0 a qf^ 3:^ "^ = 50%
qwTRB qT0CT'^.88aqq7'^ = 31.06%
qqr qTTRft ^ ^ i %,<^«t)dl
cf^d< 35
3^^: .84 a % ^ ^ W 50 - 3\.06 = 18.94%31cT: TO 13n -^TTO t % 18.94% 60 ^ 37f%l^ af^T TO 1%^ ^1
r
100 '4' ^ 18.94 W 60 ^ 3f£F TO ^ fl18.94
• 3m : 1 ■■ 60 ■59 3Tfq^ 3f^ TO ^7^110018.9437?f: 200 ^ X 200 = 37.88 60 ^ 3jf%mr sj^ TO ^l100
WHt) 15m WoHi ■'jyif^ ^«hdl "t ■?5R#m ^ 200 38 h 60 "5^
arT^ 3f^ TO -M ■#^1
(ii) ."swTfV 35 3im '■yrm ur^’ ifft ■^im 'roft 'ti ■?iFnm^ ■4‘ 35 iftm 3^^ 35.5 % a 1TPT ^ 1^ ^ ^ E^iTTO FT."^' ^.■^ imffm f^FT% TOT^ 35 3lf^ If 35.5 a tTR ■^' MU^^lcId "91 .
35.5-50
Hifq«tjal
35.5^ CT^ = - 1.21 CT12sm: 'SnftTO '^ '^ ~ 1.21 a T’C fr«m ^5^ % ■50^ 3ik ^ mffecT ^^^^1 -Fre f -f^ -4' -1.21 CT itkitth % mf 3tJi ^ ^aif ^
■f^mr mn 'i’l wTim Mifqcuni ^kuB ■pr^ 't 1^-
-1.21 o38.69%M
88.69%M- -M
f%nr 8
, ^ ■m0a'59^3ik'^ = 50%• - 1.21 a 0 a ^ = 38.69%
- 1.21 a 5^ ^ 3Tti ^ w 50 + 38.69 = 88.69%■ 3m: 88.69% ■^‘ ^ 35 5^ arT^mr amr to #ti
100 ^ 68.69 35 ^ 3Tfe ^ TO ^ fl
88.69sm; 1 ^ ^ ^ 35 3T5mT 3T^ TO ^1100
88.69200 Wsti X 200 = 177.38 W 35 ^ TO ^1
: ,. 3m: TO.mi immr 11^ 200^ ■9177 wmf % amr 35 ^ 3Tnm7 ^i(iii) 55 ■^ 3^ ■gra <h<^i sim ^<d1 "t s^Riy. 'm*FPi mlTicFcfi
■^■^’ 55^1%m 3TE^ 54.5% c'mm % -suf arfr ^ ^mro ft
^ «im%ii % 55 5^ 31% mm '^7% "f I 54.5 %t ct "4 sjE^cid "gr
.100
36 a^(t< fwfir^
54.554.5^ aiTPT =
.38 a ----- M 14.80%
W N- 64.80% -----
50%K
M- •N
f%ra‘ 9
'STtT: .38’’77 % "^rf %Wd i\ ywi-4 yiPj4id! ^ "5^ "FT^ t %
^ O'a ^ arfr TO = 50%
■RKJiTfH ■m 0 (T .38 ^ TO = 14.80%
3TcT: .37 CT 3qk TO 50 + 14.80 = 64.80%sm; 64.80% to! 55 ^ 3f^ W ^I
lOOTOf"^’ ^ 64.80TO? % 55^ ^ tl '.
qiT«<i©M
64.803T^: 1 TO ^ 151^ % 55 ^ "^l .100
64.80200 TO ■^f ^
3l?r: ^ ^ t % 200 TOf T^'129 TOf ^ 55 ^ #tl
(tu) 30 3TEF W<T ^ ^ ^ ■^nsqr TO i, ^^PtrUj. TOFT TTTfqTOI
30^fTO'#TTT 3T«TfTl,29.5% aTO sik ^ ^ TOf ^ TTTO■TO^ 30^ 3T^, ■fl Sl'^iiqi 29.5 ^ ct "TO q<^?iA '’R
29.5 - 50
X 200 = 129.60 TOf % TOTO 55 ^ TO .100
29.5^ aTO = = -1.71ct
-1.71 a50%W-=-
W ' ' .»i^ 45.64%‘^4.36%
10
'WT&i^lV 37o^voT
\
- 1.71 (j"^ qiTtsti ■^, I^F^(o<.wl iV^TT "w ti tiiHi-H yif^=bcii ■^rai' w-o^n
■m 0 CT % ^ 3Tk ■^‘= 50%
TT%2TTrpT 77T 0 G - 1.71 a ^ tJTiT = 45.64%
3Tfl: - 1.71 G % ^ 3^ ^ W. = 50 - 45.64 = 4.36%
sm: 4.36% % HTW 30 % ^ #^1
,100 '4‘ 4.36 30 ^ ^ f 1
nlf^ftidl /WrRW 3T?T: WTR? ilirqchdl
4.36% yikIH 30 ^ ^3T?T: 1 ^ ^
100
4.36X 200 = 8.72 30 ^ ^ ^11^#n3; 200 ■3T^' ^ 100
^ ^ t 1^ 200 «5T^'9 30 ^ ^1
eft Uiyiieh) % 3?^ TirTT. cF)T% cT^ oyr^<il <H<S64l ?rRT ch<»f(
(Determination of the Number of Individuals getting Scores within given limits of Scores)
oR tT^^PTR ^ HI1=h fq-ciel'l ^yRi^ra ^ wtpt
2.
TC yiKiicfil % 3T^
WPIcIT 't’l
yiKiict) ^ t^TH Mi'^ii’ch ^ ■^frtTT. eFl ct '*TT^ sRcddci '^TT^ ^ «Cf^ g ’’7^
5TM %T 't'l W yi'Kll'^ % 3T^ 't><,'^ '=fic^
W5Tt yfd^lcl WteMt cR^ o4<^ ^^1 yiKii’qil 'RtR ^ ^ '^ihTT
^ ^ yi‘<n’«h'l % ■^rrar (qi«iq 0.5 ^ arfu^) yi^iicb ^yikiichT % 3^ w ^ ^ W5rf %' 3TT^i -ZTR m ^ ^cj Picriyil-^ 3rri^ 3T^ (qsi-iT w^rfeFf ^ '€\ f?rwT ■
^ MRc<ffid ^ ■§ (^<3q> f?T ■^TfcT ^eno "fl cA<gq) W\ cT^ t
^ 40"^ 40^ 50% #ET W 50"^ 3tfu^ 3^ -RT^ ^ %t ^ ^WT W7FT yiRj4>dl ^ ■4‘ 34^4^^ 39.5■=ft^. 40^ 50 %-%5I ^«TT 50.5’^ ^
% -sir^ ^ 39.5 ■5^ 40 % ^«n 50 ^ 50.5% #er ^ ^
■3^T^TW-'q^ ^ 400 ^ %t f%TT W 1%^ -R^rRH 100 ^«TTfd-cf^d 15 RTRT "Wl % g,R&ci®qi% %t himm ^ ^ PqaRd Rid^”< ^TIcT 1%
(i) Rfid^ 115 130 % '^?
(ii) 1%cT^ ‘SricRIcT ■^T% % ^fecT«^TT^ 80^ 112%
t f% N = 400 M = 100 G = 15
O') «wTRt) 115 130 % W% %1 %@11 iTlcT ■^R% "f ■RTRRT
TnPr^ ^ ■^'115 %t f%=T ^ 3T%cI, 114.5 fl8n 130 %t ^3^ ^ 3T%d. 130.5 RT %«TcT %tf^' % ^ FT ■y<^f?i(i ^%TT f%T% yi'<i!% 115 ^ 130 % "^i 3{cT: 114.5 ^
130..5 %f G ■4' RT
38 3’6^rf<
ftcT<W
2.03 o.97gM—M
33,40%H---- 47.88%----- ►
N- 14.48%-W
■to 11
114.5-100 = .97 a114.5 cBT =15
130.5-100130.5 gJT = = 2.03<J15
-STcT: ■^TFIFT yi[q4>^ '4' .96 cr'^ 2.03 "tot'tiVii, ^laT'STf oi<eKi 'ton "w "ti ^
0 a ^ 2.03 <s^ '^ = 47.88%
0 a.'^ !97 ^ ^ = 33.40%
3T?1: ;97 CT 2.03 a ^ "eim =-47.88 - 33.40 = 14.48%
3ra: 14.48% ^ 115 ^ 130 ^ '^l
eWlT^' 100 ^ ^ 14.48 "tot % yfxiieh 115^ 130"% f l
14.48 % -Srito) 115^ 130% it^\37cT: 1100
400 tof ■4'
3Tcf: yifqchdl ^ f f% 400 58 ^1% %t115^ 130% #ftl
, (ii) ^%f% 80"^ 112 % ■%^ 3T^ yld^ld W, "t,^TTITPT yiPil'^-cn ^ ■^' 80%t.to 3T%tl 79.5 ^*TT 112%t -3^3TS%1112.5% aTn% ■'TT ^ "3^ ^T% %f ■to% yiyii^ 80 112% tl 37cT:
79.5^ 112 CT%f CTT7PTf % ^1 • '
X 400 = 57.92 tof % 115 ^ 130 % %^
79.5 - 10079.5^ aiTPT = = -1.37ct
15
112.5-100 '= + .83o-112.5 W a^TR =15
WfW^fhl f^fk^ 39
-1.37 a
29.67% . 14^71.14%—M
.83 a
12
■STcT: ytlM^ini ^ra> ^ — 1.37 o .83 a'^TT fWc? qiT«fl"f^n^ fflitsl ipn "ti tiPiRi wk^iI
0 CT- 1.37 ^ ^ ='^41.47 a % /0^+.83.CTCI^ W= 29.67%
^ “sn t - 1.37 a"^ + .83 oW^^ = 71.14%
arcT: 71.14 % ir^ 80 ^ 112 % #tl
3, wlttl?!?! '^51^ ^SjftRpif % THWS^ "3^ ?ITrT oRTHI .
(Determination of the Limits of Scores which include a given Percentage or Number of Individuals)
Hin^) fq-qcii ^rra yWi-q yiPqqjfli % TTc^ ^61^'5n’<lf^ "5113 ^ <ieh^ ^ ufd^fid Wsfll % ^l
■??? y«t>K ^ yifq'hai 151^ ^ qt!V^ iifd^ici ehlfdql %% sim ^ ^ •3irai t Ctan ^ % -^aj <st^ ^ 311^31
^9FT% ’JcT HiKii'qjl I ^ cliTba ^ ^1hii< "SRP^ti f^ oWT ^ Tj^ ■4' ^ ^ ^nrtrii%r?iT '^ntn t-.
W'<n<S, X= M + (c'*IFf X S.D. )■^3^ ?T^ CT '^' q«rei^ % 3^ HH=h "5^ tPT%
■iT«imR ^ ^ tl
500f^n«hi -HWi'q ^TR "3^ "RWIR 403^fclTjdH 12 w ^ani m f^-
(i) ^m^t^ ^ TK 20% ^ ^ 5IT^ ■3^ 3#T^ af^ -^7
(ii) yiHl-q 5IR "3^ % 50% f^)^-f^iTr HTViiVl ■% 2fT^ 'STT?! "^I^?(Hi) ^TRRT -m -qr ^ % 350 ^ ^ ^?
t'% N = 500 M = 40a = 12
(i) 4qlT<^ 20% '31^ ^ff»d 151^ "t, §t<Rny. WIPT yif^+TTl^ arR ^ 20% ^ 'ST^ ^ ^IHI'^ yiRl4)dl ^
^ 1^ ^arf ^ 1^ -m h 1^ 20% ^ 80% ^
■^‘ aik f^ ^ . k'
40
31^ •5FT t ^ 13^ yikll'^i 20% ^ % yikll«t)T ^ fT*=T #ftl 37?T:5IRif^ WTFT yiP^eRdl ^ WOT WT ,mi‘<iiVi
■4 ■'T^ <HIHM 'yif^«b"rn =tsF '4’ 0 a''TT ^ 0 T[ 1WfT"ftw mi'<hV)T ^ yRiJfid ^ ■’tI "f, ^'wleiy. qiT^d 0 ct %"Sri^ ^ yfflifid ^TTcT "^^1 't^lT^ 0 CT% 3Tt^ 50% '?T^ "t '^fferT «hlf<i % <ii41 3^
.20% "t, 0 a^ coifed ^5^ % 30% 1?r5f ■?i^l 37^ WTPT '^sF' WFnTT^ ?=T 30% (^ wfTwr fOT tiRtw) % c WT ^ tl ■^' 29.95% %■f^ R7MHI WT .84 a'tl owllV) ■^ffecT «F^ 0 a ^ 3rtT fWrT ^ ctWT WIcW> ‘^hTTI3m: .0 a ^ '+.84 ct % WTW 30% W cT«TT +'.84 c ^ ^ Wl^] 20% ■^'^137^ +.84 a ^ ■yrmf^ w^Ffi ■?^—
30%
♦0 a + ,84 o
iw 13
■*3^ yixii'ch, X = M + (a WT x S.D.) .
. 3Rr; .84% 1^ yikj|%, X =.40 + (.84 X 12)= 50.08
' 3m: WT wmr f% % 20% % 3nm% 50.08'^ 37f%mi "^i20% ^ ^ ^ 50 m 3w -gro ^\
(ii) % 50% W ^ t, 3m: 3TT^ 3T%^];-25% tt^^ittpt m 0 ct
, 7T«7I 371^ 37%cT 25% m 0 a ^ 3BT TfTTOT mf^OTT ^ fcTTiff^^3?f y^Rid wi 'll f f% wirii %f wffB '^7% 3mR
• %ff3% % yi'<ii% qll^d yi'Cich 0 a qtTtSd %tfe% % 25% "t,3Trf: W7FT W Woft 25% % a iTH %3T ^n%n. % .67 a tl t f% 0 a % %%3% ^ % %iT W + .67 a #TT ^271 %% 37k %t %tf3 % f^ W - .67 a #ITI
- .67 a + .67 a
14
41S^waT
— .67 a + .67 cr ^ '’T^- .67 o ^ ^ Ulk1l«h = 40 + (- .67 x 12) = 31.96
+ .67 a% Tin<TT^ = 40 + (.67 X 12) = 48.04
- .67cj^ + .67q% ^ ■RW? %.50%^ 3lcT: -,.67a^ TTT^31.96 % 50^1^'^ cI«TT +.67a'^ yikll4> 3T«rfcl. 48.04^ % 50%'^'^ -^ft^TT ^ ^ '4' W ^ t % % 50% ^ 32 ^«IT 48 % ^ 3TSF "STT^^1
'(iii) % 350 qiTbn fl '5^?^ 3t4. ■!■ 1% "% (350/500) x 100=,70% ^ ^ W f TT^n ^70% ^ ^ 1T#RT ^>^1 yiP^chdl qgr 0 CT^ ^ ^ ^ 50%
yipil+dl ^ ^ srtT ft«R 3TcT: ^ ^ aitr ^WTFm
"t, 'flldd 50% 0 cf ■% 3ik tl*71 20%"^ b a% arR ^1 ^^ ^ ^arf ^ W tl i fN> ■=fH % 70% W5if ^ ^ 30% ■^' ^ 37^^ ^ ^ ^ -sn^ % 70% "SHfi % w^' ^ Teg TfRn ^ o a ■qr ft^Tcl "q qiTan ■#!■ ■^g 20% 't afPT: WTPT Mifq=hfli 20% u’^TR^■| t % 19.85%% a■RR .52 oil ^%f% %l^ %ffe 0o% ^ aik 1^ t aR;i
%fecr %tft ^ CT^ + .52 a^l
0 a + .52 a
IS
3TW + .52 a %t yiynq> '’PtI
+ .52 a % ^jR Himicb = 40 + .52 x 12 = 46.24
3R: yiP^cbdi ^ % angR -qr wrt ^rrt t' 1% % 350^ 46^ ^ ^argr -sitr gj^i
4. Tif^ % 11?^ w(Determination of the Relative DifHculty of Test Items)
yiPq+di ^ % ^EiRPT %t WPRT ii 1%Fft 'qfl^ % 1%%=T -q^' m +PdHi^TRft ■gRR ^ %t RT ■fl fH% '51^.% '517^ %t "^R '^iT^5R R ^ %) -q^T ^ 3TRg %tf2 RR %l RTcft tl ^ %)fe
% o iTR %r ^ 3i7q Rg 4)PciHi^ T<R ^ RTfri fi ^ ^ argfti^ a itr IrcRT aifggrt, TITq ^cRT ^ Rtm tl
agi6<ui-f^ % %g ir?^' %t ^ ^r %t PHn^^id, %i 317^ % ■=bP3dl^ ■RR %t ^RR ^1
RR RRTR«7T RFt
Rft "^R "^>1% W ■jrfflRfT
. 50%12 71%3 40%
42 <i^a<
1.^ 50% ^ ^ tl t % ^ 50%^ ^ W^ sfrr #fl ^ 2'^ 3'^'^m ^ ^ 71%.'^ 40% ^ ^
^ ■?Tf 3^ ^1 trmp^T ^rpTrft ^ ^ ^ ^ ^ ^^ im 37^ TOt t, -a TTH BRT ^ Trf ^ ^sIh^ -^cH: ^ ^ ■sn^l 37^:
HS'I 1 ehfcj'll^ ITIT = 0 a^2-^ ehPdHlj W = -.55a ■m 3 ^»fcRTf W = + .25 a
■FT^ "t cfNi TTH 2 F7FT TT^IT 3 F^ ■<+niH' 'll 3IT^ 1 W'T 2 ^ 37^^ .55 a 37te-'5Ff3^ t 7T27T17?^ 377?H 1^ 37^577 .25(j37f«7^ tl -^fSff^.80 a arffeT^ fl
njf^ftJiii f^ai'J.iFhM 0^7
3 ■sr?H 2 ^ 37^^sTFi
■to 16
5. ■fe# 7R^ ^filrfT % 3Tmrr FT ^ FPft FJFTT(Division of a Group into Sub-Groups bn the basis of Ability)FIFTF "SniFFicr FsF % "SIcFF FiT 3m4Pi «*^6l FiFF^ 3'*70*i^T
'=h<^ % ^ IFtFT '371 «=hal't <j'7F*^6 qVqcii lFT<fR 3TFf^ "% yiyiiWlFFF^-FFFT I %Tf «PT^ "% "to; FTFP7 Hif<<-^ai FF7 ^ "efri "ScT^ ^ «s«i«i< Fpff "Sfe ^f 1^ 3HF1^5 FFT^ 7TFT IF FFTf F*^PFc7 cfilfiMl 'FT^^ a FTF ?M ■sfTFn^' ^ f i ^ yikfi4) ■f^*FT Fit F to fIft Fcn^ f I fIf
IMf^ "SFF^ ^ ■??T^ Fit F7^ 'FTTFt ■?!# Ft f^fF^ FT^‘ ■#)■ "FIFPr FTfFFTFT ‘FFi 'FTFHt FTT yRiffia ‘^ITF ch<4i "fFf^ "FFFi^^ 'ST^ Fit yfa^ia ^<sm '^fTF FiT ‘fl FF)^ "FIFP?
yif^Ficii FFi ^-Sa"^ + 3a'% FtF 37Ff7I^ 6 a Fit 99.74% "^TF 37T FH^ f’’, "^Ff^ ^TTFFTftFi FFFqT3n if -FlFPr yiPFFidl FF7 Fft 6 a ^ FI^ tot tl ^ tof if FH tto FTTF! t %-3a-^+3a%#FFFt 100% ^ 37T to f I
FiT ■JTFk
^3FT?TFT-'tof % ■QF^ "FiJ^ 'FTI FTF7 FTt^ FIFFTF 60 F FPTFi 10 FTI 37«ITFF7tot FTt ^ A, B, C, D, EF FW FFTR to Fr?FT t f^-toFT F7t WFT F7T# to to% ■yiFfto 'FiT IfTFR ■snFTTFJt Iqa<u| “FTt 'FTFFF FlfFFiFT 37^<^M Fto ^ ?fTF «t)R^T7*7n FFi
1^(i) ¥rto ^ Fit Ito^-fto^ "STtoF "iJIF ’TTTFT Fif?
(ii) ’STcto "fto-lto^ 'SIRfto “FT^ WF 377^?
t to M = 60 FFT a = 10FFtto FJntofFi ■Ffttoto ■^' "FTFl "Sn 'F^ "I to 'Fito "FIJ^ - 3o ^ + 3o to "to srto.
6(7 ^ tocTtor tor 'I'l STF: ~ 3 <T ^ + 3 CT FFi Fit 6 a F7t totoF FFFiJ^ '4 FFFT'-'FFFl 'tol
«/7<2flF>?e/ toV^f 43<r^wai
'31|4|mi1 ^ Old'll,t" '3TcT: «r4^ 1 3^^;■5nfq^ ^ -4' ^ 1^ 58 % sr^^nr -^i f%ra 1■
A^f + 2 "3^ '^T^ "STTsT $l4i
B4r + ict'^ + 2 CT rrai %
■3troi|^ C'2f0o'^ + larT^%W ^1
D^f-la-^OCTH^^^ ^tl
E4f-2CT'^"lCT3q^T^ F 4f - 2 a ^ ^ 15TW #tl -
jr//wwr /qa<of
I
D c
+1 CT +2 c +3 a0 a—3 a -2 a -la
H---------►H—----->.34.13% 34.13%
>1>K!♦ 47.72%47.72%♦K 50%50%
►H ♦I►H------ ^—►H2.28% 13.59% 34.13% 34.13% 13.59% 2.28%
►HK
17
(i) yr^ef) 4^ ^ ^T^ft yfa^fici ?TTcf =fi<'ii ^ <7 %
^ WTFT yir44>dl ^ ■^TRTlt 4 4^. 1^ 58 ■^’ Vi^ ^ ^ tl % 1^A^ 2.28%, 4^ 13.59%, 4^ C^ 34.13%,^ D.-^ 34.13%, 4^ E^ 13.59^ 4^
F-^ 2.28%^ ^ ^1(ii) 4^’ "^iT^ ^1^ .■^1^,% y'l’^ti'ct)'! ^ 'PTR ^friTf 4t sM ^TJTft
I'wld^ G ^TPrt yi‘<tf«b 4’
+ 20"^ y('<n’'=ti = 60 + (2 X 10) = 80
+ 1 a iiTkiiV = 60 + (1 X 10) = 70
0 a = 60 + (0 X 10) = 60
- 1 a yl^i4i = 60 - (1 X 10) = 50
- 2 cr ^ ^ = 60 — (2 X 10) = 40
3FT: ^ ^ t f^ .80 4 arte uicfit<s Tn% ^ w A't^ ■m44i70 "4 80 % ^ 3T^ Tn% B
44 3V^cj< WfWl¥N
60 ^ 70 % ^ 3T^ ^ C TTTfJtl50 ^ 60 % ^ ^ Tn% ■gi^ ^ D •CfT^fitl40 ^ 50 % ^ 3T^ T7T% ^ W E xntNti40^ ^ ^ ^ Tn^frtl
44l6<'J|-400 ^ ^ Wjf^ ^ -Wl ■^' % ^ ^ ^^TTTt - ^ 3lf^ WTP^ "^fe, 31^ rI*TT 8JWT "^fe
"^fe yi^lfchl ^ [q«iK ^HHM Tt^l "^^FTPr Urf^T^TSl W ■4' f^uT^-l^ W 3TT^?
■^eT—'^TTOPT ^ifHObai ^sF oqiq6if<.«h ^ 6 c ^ "'tN *fT^ ■'TT yc^ef) Wf1.2 CT ^ ^ autrfti 3TcT: 'm^ yiP^i+di ^ wfi ^ t^sTf^ Pp=m #Tt-
+ 1.8 c ^ ^ ^ ^T^ 31T^|
+ .6 CT ■^ + 1.8 ^ ^ ^ 3Tf%T^ ^ cn^■4‘an^l- .6 o ^ + .6 o TRi "^fe 3Tf^|
- 1.8 a ■^ - .6 CT ^ % W 3T^^ ^ T^ 3TI^|
- 1.8 o"^ ^ ^ ^ SifhJT ^ ^ ^TTJF 3TT^|
Wt^ ^ IS"^' W ftWnt ^ a<c<dUH if tOa^ .6a^ ^ 22.57%^ W Oa^ 1.8 a46.41% #tl 37?T: W^n t
^ t f¥*RWR
Iqa<wi yn.cbCTH’^l iTH^
f^^ ^ ^ ^ T^' 50 - 46.41 ' = 3.59% ^ ^13Tf^ ^ ^ ^ 46.41 - 22.57 = 23.84% ^1W?RI ^ ?n^. ^ 22.57 + 22.57 = 45.-14% ^ #tl3T^ ife ^ TTTJ^ ■4‘ 46.41 - 22.57 = 23.84% ^ ^1^ ^ ^ ■4' 50 - 46.41 = 3.59% -SR
37^ ^ ^37f^ ^
^ ifew /
••IV'-
ri';i-''-1.8 C -6 a +6 o +1.8C
H22.57% 22.57%
K ♦I>K46.41% 46.41%K >K >150% 50%
♦K :H♦K >k3.59% 23.84% 45.14% 23.84% . 3.59%
18
d'i^ai Wf^Rffw /w/wf 45
400 3Tcf:
,• 3.59 X 400 = 14.36 3T«?fcI, 14 W5f #tl '100
; 23.84 X 400 = 95:36 3T«tf^ 95 #tr3Tf^T^ ^ ^100
45.14 X 400 = 180.56 3T«lff^ 181 ^1■^TWf^ "^fe100
. 23.84 X 400 = 95.36 953T^ ■^fe ^100
X 400 = 14.36 3^sqf?i; 14 ^#TT ^ ^100
2.6 mM/chI % ^ cht-fi
(Normalizing a Distribution of Scores)
t ^IHH yiP^+dl ^ (N.P.C.) % ^tichcll ^ c^T tllHRl yiP^stJCll "tl
f' ■% ^V)Ria fqciO^ WfFT yif^«hdi f*F7
^ ■*?rnT ■=r^‘ ^ w ti 'W % :?Tr«? wiftt 'tiK'JI
vir^4>di ^ ^ 'SR^ ^«iT arrarfrci wnsif ^ ^ ^ ^ isf^'ll fcF^ [qdl^i ^ ^TTHMd! PiJj-qq '^iT^ % '^^ aiT^T^ra? WdT ■!■ ftdi.'Ji %
WTFI b4irM=hdi iqmy| f=Fn «n^l l^cR^ ^ ^TRFRIT (Normalcy)^ 31^ % ^iTteMdO^ C^ %. T^?T. '^m snf^)^' amlt^f^d<'J| ^IHH yif^ebdl ^ =bT«^, ‘^STRIT ^ FIT ■SFR ^ ’
^1«trK f^<Hi ■^RFdT 't‘1 T?I% '&iRiR<w ai^O'tlfPietidl -ST^hI 3T^VPT. tiHfMi
■?mT%TR 3T«?^ 3T5HVR ^ ^ f^F#3nfqRi^ y^Vi lifers: "t >iWcr)\ [4d^^i "^t^n "ti.
WiPT
^Fcfit ^ 3T?TWF? ^ l^fjRd ^ fwfd ^ yifq^T^ % y IHRT wR^jhT^ 1^ 'sn ti ■^‘ d^f ^ 'rh t i¥ -^rq^’
Rial'll qt^a; wni-q 'SiqdlRtja fqd<wi % yiHi’^i ^f«*dl "I cI^^T ^ ^FRF 3iiyRia
1^5^/SR ^q>dl% fij^crlHI 3TRTf^ ^<qi 3iiq^qch ’RSn ^m^VH ■ 'fl % 1^
^q(ad SlRnRid 'HFFI
3TOFFf 3TT FTT "f
f^TdT'R ^ 3TnT^ (Fitting of Normal Distribution) fqrt^'Jl qiftsd yiTViqilq 'sril^RFTl TPftn
■fl q>K^i1 ^ ■RFF >5iq^liq>a tqn<,y| «RFF yifqqiol tqai'^f;
yiRr+di ^ %, 3T^ t ^«nf^y'qil T^tqHid, ^'I'^fqqtri'i fT??! o^Rki (N) lqd<.W % <hhjc^ e^dl ^1 3i<Hiqr*q .
^ fqdRd 3ITRTf^ ^1% WTPft^nTJT % ■
Iqd<ui ioRTT’R "t cit WTFT
T^lfci ^ ^ 3t^ ^ fe 19 ■pn ti f^nrd yikH'dil ^ Pqd^yi cFFR ‘HFRT tHf^F^RTT "^sF ^ "tl
■f^nETt ■yPT.-aFfFTFT 3TT^ f^dT^ (Observed Non-Normal Frequency Distribution). d>T yiHMl4><u| (Normalizing) "dT «|hi'<F t^dT’d 3ii«'dd (Fitting of Normal Distribution)^ ■fqftpif fdi^ ‘TTdidT ■!—
H/fis^fl^i RfkW46 •S'o^ai
(i) (Method of Ordinates)
(ii) (Method of.Areas)^ fi ■51^ ^ W7==! iThFff 3n^ t ^ ^
^Rm % 37^ ^^ 3n^{%^ ■^' Tn y«h.oi 'll
^Jfe3T?Er-^(Method of Ordinates)
■^5^37^ % "gRT fVi«i1 1^ 3R7RR7 RIRPT fqa<w| ^ 37RT5H %1^ frcf ■#7rf[‘ ■5F7 sTjRT^ ■smn t-
(0 fgrR^.% fm R«RTH (M) ^«7T ,(S.D.) ^ WTT ^ 'STTrft ll
(ii) WtH ^ % Rtarf^ (X) WM ^ f I ’
(Hi) (X) "4 4 Rt^RR (M) RST ■STRT 4 1^RT4 Hl^lW) (x) TITRT■5n4 f I
(lu) f^TiPrifl Wkii"^ (x) ^ii=h (S.D.) 4 RPT 4^ mikii'* (Z) W ^ 44 'tl
(u) 444^ 4 4 Rf RRFT yifM'^di RsFT 44^ ^ (Table of Ordinates of Normal Probability Curve) 4 Z hikHVI’ % (Hights of Ordinates)a7«7ftf y W ^ t'l
r4 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 60-64 65-6955-59
Oiqcilf^n0 13 42 36 100 168 70 40 24 7 0
3il<^
2 9 30 66 107 117 93 50 20 5 137T4^
elHi-ql^ci lqa<.«i .
180.168,
160 !\/140 \
\ i:/ •/ 117‘120
^ 100
lE 80jp>-
. 1560
40
20
07yi'di«h ei4
19. wHi’<i)cttivi ^ ITVT? (Effect of Normalizing) Mifm 473«wa<
(ui) Chileqi ^ (y) ^ ^ 3RRf^ (i) N ^ ipiT °F)T% Hiich ’TFF
^ 3qT^1W (/l^) W ^ ^ f I
^ diicjpTl fqfl<«l % ■^TPTRT fqdV^I 3TW^ 3^8^
nir<f=hai Rt'di'Jl
3nrff%?f d«vi6<.'^i
etfi^^ui—I^TR fqo<ui % 1^ WfT^ fqd<.w| 3Tra^ 3?^ ^ftf^l
2.7
3TT^f^ f^ri<ui
3u^rii f
155-59
4 .50-54
45-49 ■ 12
2540-44
4135-39
•• 2230-34
1025-29
320-24
15-19 2
120
'5^—H5cl ■*?^HI'1 TT^ Hh^ fq'ctQI'l ^I'Ji'll ^ I ’’T^ ^^Hli *-ll'1
37.25 cTSn -RTO PciycriH ^ ‘‘IPT 7.15 m ’^TK^ X, x, Z, y cT«7T f^'m ^^ 2.8 ■4' W ^!
■WTtrt 2.8
cKt1^3T^ ^ fcRTTOT ^ f^a<wi 4f oi'^wHI
'^\ft 37^ ■•‘STHTTraJ «iHJ*<Hehn 3TI^1% _iX Nxy
In----------------
UHdl*3TT^fw 1IEZI
zf X yX S.D.
57 2.76 .0088 .74 = 119.7555-5950-54
1
.0478 4.01 = 452 ■ 14.75 2.064
13.-28 = 131.36 .158247 9.7545-49 12
26.93 = 27.32094.75 .6640-44
35-39
25 42
-.03’ .3988 33.47 = 3341 37 -.25
25.64 = 26-.73 •. .305632 -5.2530-34 22
12.04 =• 12-1.43 .143525-29 10 27 -10.253.47 = 3-15.25 -2.13 .04132220-24 • 3 •
.61 = 1-2.83- .007317 -20.2515-19 . 2
N » 120S.D. * 7.15M » 37.25N = 120i= 5
48
wrpft^ 3ircjf^ ^ "t 1% 3i^, cinWTF^ ^1 37T^ Pc^di'JI % M = 37.25 S.D. = 7.18 f
■^sfe ^ fc<d<'J| % wn^ f[ tl WT ^ W ^ t % %% 1^ Z ^ +3a t" ^ 3Tfj: 3lftiReKt 's)'ild^< ■cid+l TTM ■3TT^f%
■*TR ^ f I ^ •% •gif % ■f^ Z ■*TH - 3.00 ^ ^ 3TfglT^
31T^ ^ ITH # ^ tl ' .
^sPTTH
(Method of Areas)f^fg ■f^ sTOFiP^ ftd<‘Ji % fim; ^ihi'<i icid^yi
Pl*^lTVid ^^fl^TFff eRT ST^WT ^fRIT ■!■—
(i) ^ fqa<.'J| % 1^ (M) HH«b fq-cKrn (S.D.) ^ mumi ^ "^sn^
■^TR^Tt 2.8 n WfFT yjf4«R'dl .%
niRt^tfW f^crm
q'1l':h< d’l'+n
tl(ii) M’TH ^1hh< (Upper Limits) W( 'll
(Hi) ^ ^ ^3Tf -qsqiqH'gnigR R^^Rrid W^ xW ^ t'l
(iu) W<[ xgpff ^ TTH^ -m ^ ^ tTPT (Z) ^ 1^ 'W f I
(v) WTBT Hir<<«hai ^ ^ (Table of Areas under Normal ProbabilityCurve) Z "RPTl % -yil^'+icii <hihi-h 'Snt^T^RIT yRisfici
(A) ^ 't'l(ui) ^ ■^: 37^Th^ Tfem (Ac) W ^ # tl
{vUyMw^ wfi % 3T#T?£T (Ac) ^ N ^ cigT 100^ MFT3TT^Ri (/’im) '^ fl
[qa<.wi Igdl^l T^RcjRid ^ f^rfg (Method ofAreas) ^liiiRbci <s<;i6<y| ■^T%^1
>i«05Vy|-i%T=T ^K«ri 311^1% fqd<u| ^ ^IHMl-^d 3TT^i% fqd<y|
'HKun 2.9
16-17 18-20 21-23 24-26 27-29 30-32 33-35 36-38 39-41N
31T^f^ 2 16 254 43 30 17 8 5 150f
gWTHH g«TTTTHgr 4.90 TJRl ^ tl 3T^ WJfl WIT ^ Woft 2.10
1%^ ^ tl ...
Ri^dH gff wn ^i wtt ■qi 28.62g«nHM=h
3Wc7r w/<&'vqb']<v 49
2.10^ fgrRTJT ^ fqntuj ^f ^SfrRT:iy»cn
cnf %■•TOT cFtfz
cRT
3n^
37T^ •
/ =Ax.JL^ 100
a: = Ul - M S.D.TfftiTT ^snT>H ■^JT/ 5«mci
Ul Si^Hin
A Ac
3.26= 339-41 5 41.5 12.88 2.63 .5000 .0217
36-38 8 38.5 9.88 ' 2=02 4783 .0591 8.87 = 9
33-35 17 35.5 6.88 1.40 .4192 .0134 20.10= 20
30t32 30 32.5 3.88 .79 .2852 .2138 32,07= 32
27-29 43 29.5 .88 .0714.18 .2378 35.67 = 36
24-26 25 26.5 -2.12 43 .1664 .1844 27.66 = 28
21-23 16 23.5 -5.12 -1.04 .3508 .1007 15.11 = 15
18-20 4 20.5 -8.12 -1.66 .4515 .0369 5.54= 5
15-17 2 17.5 -11.12 -2.27 .4884 .0116 1.74= 2:
12-14 0 14.5 -14.12 -2.88 .5000
N= 150 M = 28.62i = 3 S.D. = 4.90 N= 150
2.10 % 3iqcilch'i ^ T'T^ 1% ^ |c|d<y|
WTFT Fqa^wi % ■?T^ f I ^ PcIdl'JI ^ 28.60 ^TPl^ 4.90 f1% 't’l '
^JfT Ul<<nch) Tjfrsrf^ chUI(Converting Raw Scores into Normalized Scores)
tinW)! % 3T?TFTFTIMv w\ ^ ^ ti -irafq 3T^ Wf^Pd^T ^
HP<R^PiqT ^ 3Tra^ "t^l qpJia 3TnT5H 1^rfV ^qcillqia f^TWf
fST % 3ira^ ^iqj^qchdi flFTf 31T#3R <iHqVll '■ft
% TT^zpqH HMch l^'ddd ^ ¥lHl'41'J)Cl fqfl<w| ^ ’’ft ^
W ^ ^^TRft f'l ■'IT^ ifl'fllWt % ■^TTRRft^
"W 'tl aWT^lRT fqdRd yikllchT
yRqfrid 3nq^q«hdi "t nRqfdd
aiyiiqil % ‘d’^TF
% 5M ■5PITOqd<wi
^iHi'ql'j>d yI><11=^1 "ft ^iHt-ql<jia yiMni'qiT ^ cTSh hmch ijd
Hnqi fqqci'i.q<iq<. ■^’I ytyii’chT ■^rngrr ‘SIFT: ^TFR %3Trc(?^«hdi ^ ^ URTT^ ^-y(mi’«t>l "ft IFFT MNHdidf
'dTt ■ftdT ;tl 3RTTITPT ^ f^dRd "^ptWeF
'ft yq)i< Hp<.rVqfa 'Sch*^ ^qidl 'f’l yr^iqil % ■^T^^Erpj ^ '^TFTRft^ yiKiiqil % ft q^ddi 'JiR.ei cT^ ft ftPT 'ft 5^1 qft <4Hqinicii 3TT^?^ra7cn ^
yiKii'ehl ft fftfv ■f^Fn TfT fti ^rft
yiyiiftil WTPft^ yiKii'qil ft nRqfdd Siyildi'd ylMf-iT ^ ft—
/q/V-vP50 >sWfj<
(i) -rp^ -mfM % (M) (S.D.) ^ ^ ti(u') 1^3% <iH<H TTFcTl^ ^ ebi'i 'SR^ itIRllqi ^ (/) c?8?I 3^T^ (c/)
^ ^ tl .(wi) dcM5?TMlfl yr4«n ■^rRTl^r ^laiVU'H ^ (PR) 'STcT f^RT ■^jTRTT 'll yi'^’eh
(■??[ sFH) ’4' 5 Mer'^< 100/N f I 31^ ’W^T ^ sindii yimi'ch % fV^I ■RtcT^RT f^SRI tl
(iu) TTcllvft^ ^ (PR) ■4’ ^ 50 ^ ycll=h< TJTRTT^ ^ ■^' f^s# W ^ ^ f I
■f 1^ PR-50 ■^cTT^nr yir<4<^ai 14^0^1 R*rf^ 3(iRl4>TT%2pcrH -giT;^ -f fxT^ 1% R^^^iRRT 1tsR5
■ti
■^l
(v) 'HRFT infeifn ^ ^ ffH^ ^ ^ (PR-50)% *tft ^ Z rh ^ ^%4 ■§! 4 Z RR ¥iMil'll ■^TT4^ 'BRRTtfR Hii«h yimi=h (NormalizedZ-scores)
(vi) "SR Z R'l'ti' 14(S.D.) ^ "^pTr =ti<4i ‘4’ ’^41^ I"' t^fRT^ ^i^<j>d(Xj.3ITR ^ ^ f I
3Tt4 3R^ %4 ■3^T?TR Jfiyil«til ^ 'HlHi"4l‘jid ■5riRTf=pf R ^R'^fdd '^>7^. ^ l^fRf %•iM<l'^ T3RR ■^>1^
•5^1'^W—"PlR yi^{'=tiT ^ '^’ yRqfdd Rfrf^—
t
69 65 6566 72 66 69 65 67 68 68 66 64 68 64
70 45 67 68 64 67 62 68 70 64 64 6971 50 65
64 68 67 6661 72 62 69 58 66 67 65 67 60 65
64 69 60 71 67 65 67 68 64 68 69 68 53 6864
6767 7064 64 71 61 69 64 5372 67 61 71 68
70 55 68 69 68 6659 70 54 69 66 69 45 68 66
5665 69 69 65 61 71 66 70 7260 62 65 70 67
6768 50 68 67 69 68 62 68 59 68 53 6960 62
•61 69 61 71 66 58 70 47 69 62 70 67 68 71
52 6568 61 68 46 66 52 70 59 66 66 62 6269
yiKnVl 'iidii HM't) f4^cnd ^Tlcf IVi^i 'aRmi
^RTI: 64.9267 tT«lT 5.65587 f^TT ft aiRRR ^ Tfunr RRf RRRt 2.11 ■^' "SRgd■f^RT RTT f I ^IRqt % ^c|ct1#,4 f 3TRRR ^ ^ 72 -3^7 ^ ^ ^80 45 3T^^^^^52 3^^ ^3n44|
d^rt< fkfU^ 51
WToft 2.11
Tf^ WcTf^ 'HIHI'<<l^d Mf^cifan Ch^dl
TrafcB^or «iHi’<<i<j»n
Hncd TITOTcB3TT^fw ^ ‘WwPR-50
itz X'-M + ZaMlKUe»j
PRX f cfZ
80.31 = 80 75.06 = 75 72.73 = 73 70.81 = 71 68.49 = 68 66.23 = 66 64.76 = 65 63.57 = 64 - 62.44 = 62 61.14 =61 60.12 = 60 59.05 = 59 58.37 = 5857.80 = 58 57.35 = 57 57.12 = 57 56.84 = 57 56.56 = 57 55.59 = 5654.80 = 55 53.67 = 54 52.88 = 53 51.75 = 52
49.67 46.3341.67 35.0023.67
9.00- 1.00 -9.67
- 17.00- 25.00 -30.33 -35.00 -37.67- 39.67 -41.00 -41.67- 42.33 -43.00 -45.00 -46.33 -47.67 -48.33 -49.00
2.72150 99.6796.3391.67 85.0073.67 59.00 49.0040.33 33.00 25.0019.67 15.0012.3310.33
72 51.7971 . 7 145
1-38 . 1.3870 101.04,69 17 1280.6322 111680.2315 8967
-0.03-0.24-0.44-0.67-0.85- 1.04 -1.16- 1.26- 1.34- 1.38- 1.43- 1.48- 1.65- 1.79 '- 1.99- 2.13 -2.33
66 13 7411 6165
5064 128 3862$
. 3061 760 234
1959. 358 2 16
9.001 14568.331355 17.671254 17.003 11535.0052 2 83.6750 2 62.33447 1
3 1.6746 11.002 245
S.D. = 5.9907M = 65.6667
N? '5rr<rr^37^ yiviicti
% ^ 65.6667 ^ iTH 5.9907 t # F? yiKiiVl'
% 6t<!c« f’l TJR cT^TT
WK ^ ^Tcft (Rounding Errors)% 7TS7T HMcb
fqTetri’ll %% ■fl
1. ^ 37N ^ f?
■ 52
2. shell 3r^f^ Iqeiiwl ^ MR'siq "^H^l
3. fllMM ^JlfqshdT ^sF ■?ff^r<f -iere^y
4. ^iKiish ^ ar^ TjT’^ «’<shi ^nci ^7
2.1 ^TTTO (Summary)
• 3TT^f% f^eRPT 3TT^f^ yi]<2««hl RFWJpf ®TRTR TIFB "t cl^ ”9^^ 3?T^i%f^OT 3T^ WFI^ f I
• ^ 3RFR ^ tl'm ^ ■^' T^^F it ■R^=TT ^ ^ 'sneft tWpf xi^^TT (Simple Event) f ^ 13^ 'HW ^ ■Re^nafl' % ^ifer. ^ ^
^ ■^iTeft f ■RHHiarf ep) ydii (Compound Event) ^FT '^FTelT %\
• ISF^-f^eR^ ■RT«T ^iPuid^i «<*ifcil (James Bernoulli, 1654-1705) ^^3TT f F«F q-iTeil PqeK.ui (BernauUi Distribution, Bernaulli Trial or Bernaulli
«/7e^«h‘3</ 53
Process) '^FT W ^«&ir^«ti fqa<wi TJfFTRF suldl ^ ^
3J^mF ijr^ % 8 ^ 1713 ^3TT «Tn• WTP^ UT^T^Jcn (Normal Probability Curve) (Theoritical),
STT^ (Ideal) (Mathematical) 77RF7 (Normal Curve) ^T35- (N.P.C.) ^ "STmT f I
• ^ ^ ^ FT Ft ^ FIFTf#' % ^ 3T^ W^ ^'^1^ Fit iltWa tioHi Fit TTTFFF MlPl<*di FFT F)t TTFTFclT ^ fFiFT Fn «=hai ^1
• fsFTTt TT^ FF F^*iii Ff Hlddi fF^TTF sHF FT TTIFT^ FlfFFiFT FFT % FrFF F?t T1F7FFTFTTriF) WF fit TTF^ f ■f^Ff%‘ fIf fqi^"l F^ yfci^m FT W<sqi % '^IF STFT '’JT^
^ f I .• TTTFPT FlfFFiFT FFT (N.P.C.) ^ FcFF FIT %F^ FlrfTFf^ ^ fFTFI- FfT TTF»FT
FIF TTF^ F5T fqa<«! TFFT^ FTlFFiTn FF> %. "^tFI "tl
3T^3^RT-W^ (Exercise Questions)
1. TTRl^ yiPlFidl FFT % FFtF WFI^qf FF ^ ^ FH TTFiFT t, '^FFri F^fF ^1
2. ‘^*iPdF> 3tl^f% Iqa<ui SF^^PTFT TTffeTFit Hetq'^ul STTFIT FFR FiT^ 'tV fF^FF F^f^l
3. ■’FfFTTF Fqa<wi Fit iF^tFFI^ft FF yWl<a Fitful
4. TTTFPT FrfFFiFT F^ FFT ■!? ITTFiT O^faeiftiF)- HR-qM Ft^RI
5. FTFITFit % Rtitfl IfFT^F Fit fFTF FFFT TTTFRft^ Rt)4I FUFT ^?
TPVi (Reference Books)
1. TlifelFitF f^FFT ajftr TfFtF-FTFTT im T^ T7?FF F^#?7F/
2. yiRssiqilq f^FFT—^^FTFTF 37? FiTF/
3. ®Ft^t 3#7r 37? WNI
4. yifq^Jdi fFFT^—fii*i)i, fhim Ff«cf^77F7
5. FT^W, ik)w F/ScF^»?7F/
- 54 ?»FFT /f/VFT -
?^-3
(Correlation and Regression)/ '
■HT^RT (Structure)
3.1 "3^?^ (Objectives)3.2 (Introduction)
It
3.3 JHitvi* (Partial Correlation Coefficient)3.4 (Multiple Correlation Coefficient)3.5 ^-TicftWR (Multiple Regression Equation)3.6 ^ if 3lfvcF ^-'iratWR
(Multiple Regression for More than Two Independent Variables)3.7 ^ <Ht4«t>nl (Significance, of Beta Coefficient)3.8 RRf?I (Summary)
• (Exercise Questions)• (Reference Books)
3.1 (Objectives)
^ 3T«rH % ^-• 3TfiSfl=h 37k "^l• ^-TnftWTT ■^fl
. • ■^tel ^«ii«tiT ^ 4’l
. 3,2 THfOcRT (Introduction)
T^«f> ^iToHchlij RN (descriptive statistics) t ^oIT^ ■*71^ ^ 't’l ^^15 "SRiR % ^ ''TlTPTft^ ‘^sncTT
^ W ^ t % fel tRJK t¥*m cI«Z7 MKWRch ^
55
^ 'STT^arf cjl’-qi^ql ^’, o2T%Rft' ^ ■^’ ^ ^‘, ■fsit t^‘ ^rf^fTSTT 3T^ ^Kq^ T^'. Pittid "^71 4l*fl ■^TTcn "tl '^-cKpi’fen
(interdependence) ^FTcT tl ‘g^PT ^ ^ ^ #r it%zt ^'m^ tl
0^ mR^imi
(Definiton of Correlation)^ ^ cr»Tti -ii -^8^ ■^,,# ii 3t?mt f^srf "qlr^ 'ii
‘4; HPWcfH ^ ^ ^ "STFcn f ^ -^szif W?W%f tl(E .Davenport) % Hdi^<HK, '^TT ^•HKt^R«t> 3^ f 3Tg^ '^\m cT^ ■?fT«i-'?TT^ HRqdd ^ H«jfd
■fl"'3f<T: 3^nK^'1^^T?^ (interdependence) ,3T«T^ ■gri 37«2T^■^iTcTT tl '^^ ■^RffeT^fttf cikh-ilkh t 3TRi^ "% "R^ ^^Pd "R '^TT^ MiHdl tl
"t' Rifl'd d'+i'il'^ H6Tci tl 1^ P^«&i^ % g^ cTc^yPdHi'A'i ’spra % '^Tt^T (Bravais) t f^-m «TT,
W^T*i«RT d^idlkh ■^tT 3^^^ '^TT TFIRtH TfTeCT (Sir Francis Galton) ^1 1896 t’■jrfes ■^R^nTTR^ Ph^i (Karl Pearson) ^ ^^n'ch (Coefficient of Correlation)■gKT qft Mp^idl’^ "Mv yfdm^^d I^T^I 1^ ■^' (‘Tlir-i'i cTSTT Phm^i)1d-^^'h «5i«idi inPp^TT^ (Biology) ;d«lT ■^5FH-%SIT (Genetics) ^ 3T^ "^RWlTSTf ^
f^i^l 37«f?n^ "t' ^ W d=hdlq^ tl 3T«f?TT^ t.'il<Hq'»i< (Neiswanger) Pno^ t, snf^ cqqeK ^ t?n t,iTFWjyt ■q^ 37^ ^ f, ^ y^iqdl trlT t; 372f?TT^ ^ ^■^W t Pjid^ , t)?Trft t '3^ "3^ 341^1' g?n^ t?TT t P»n4> 'gRT fWdT eUt ^iPtdqT tt ^qidl tl" Mdl^dHi (regression) 1%^T’q-3TgqTcT (ratio of variation) % fq-qi< git "gN 'Rl tt 3TTVTftg tl git gg ^ 3TT?g^<T gRgt 1"^l^RgcT grt
37R1W*H 3lggT qigJd'Jid gg iqjfq^-flq tWlI
% ygJTT
(Types of Correlation)
Hp.q^l'll’ git fg?TT, 3i^4id 3TTfg '%,3TTgR gT PlMfciPad ggTT^ giTtt ■ggigr t—'
(1) gWTgi 3jggT q^wiicnqi (Positive or Negative Correlation)(2) 3TiTVigi 3TggT ggg'^ (Simple, Partial or Multiple Correlation)(3) 3TggT 3T-.'^^^ (Linear or Non-linear Correlation)(1) sigpggi 3T5!raT.g?gIRgg> (Positive or Negative Correalation)—^
gfg gg7 ^ 3T«TgT tggftg f^Srf f gftgcfg tt^ tf ^ ■3g%- gtg tJgl tl gftgr-g^ gr fgg ^ gt 3TggT ggr gr-g^ ^ g^ gr ggn g^-g^ ^ gt ^ ^mw^ gglrggr (Positive) ttgT tl g^ gg gft gg g^ g^ t gt ^5ggft gft (supply) 4 gg ^ t 3^ g^ gg g^ g3^ gr gggft gt w gg^ tl gp^Tcggr "pg g?TT t’ ttcTf t gfg "'1^ gr-ggg g^ fI ggr
gr-gyg % gg^ gr ggt gr-ggg ’4’ gi^ ttdt ttl fg- gg^R ■gFFRg^ g^t fg?ftg (inverse) ■gFRTg^ gt gTFt tl JR ggrR\g7T TmR^ g^? ,gg gfg gfgr gTgr ti 1%^ g^ gg g^ ggg gr ^Jgg^ gfg (demand) -grg tt gTcft t str ggg gTg tt gnt gT gPT gg gRTt tl
■gTg^gTgTgmrti gfgf^ gigggiK giT
HiTisi^4>l^ Mmi56 3Wc77
(2) wr,(independent and dependent variables) ^ ^ohi 5TO
1if?d % 37T%IR 3iiR[^ 3^5?^ «<§'pR "ST^ "FI "tl■?R^ (simple correlation) f l 9riTt, 3TPTR
9riit (subject series) “t, % (independent variables)“t cT*?T(relative series) "^TF^' "t, % (dependent variables) f I
stH^T^ (Partial Correlation) "4 31^W 1^T?n ^STM f I (Multiple Correlation) '^’ #T '^TT 37f^ %
■ HK^R^b ■^TFttraF%l 3TWPPT fetiMi ^ncIT 'll 'SFFFtrif^, "FTSf, cfSTT 7§T^ % hRwmIhT % 's|§,'jVil 'be'nmi "fI
(3) 3T?raT (Linear.or Non-Linear Correlation)—3T21^3T-l<aTfl 37^•f’l ■^-'*5^ % iTCsr ^ t '3^' Ft^l 3i=fii
73;^ ^ "^7^ I^RTf y«bni 3i'cbi|Rjra1«4 (arithmetic progression)^TcT^icft’ ^ 3727^ ■^jriTrF^ «ebal 'll 'FT I^T^TT
Ft fsF^ TJqF "^Nt ■% "^l 37-i.<a1<H ■^“'Correlation) ^ f, ^ ^ ^ ^ vm
37^FRT Ft^tl ■'R yi’Rbci ’’R 'SfTFT "tl fFR FlRi=bi fRH«|lrl "Ftt fRtF ■^RFt "t- ‘ ■
■%2r
3T-W^R RF^RSRT
X YX Y
50 1020 5055 1240 100
60 60 22 ■15090 3480 20098 45100 250
120 . 56120 300
3T-t#^ 773^ cTT^ FiT 1^^ 3TS7^ 1^ F?T fR R^TR
YY Y Y
O X o X o X o XNegative linear
relationship .
■R^T RF RT^ FRFT '^f^d FtFl 1% oqcteK 3TftI3Tt?T 3T-5.<glq R^R^ RM RTfFT "t, R^ 3T-'t7^ RFR’^^ d=bHl=b 37cRf^ RtfeR F^^ % RRRR, FRlft'RF RFRFT 7F^ f % fR’RRRF ^ % R«T Mr R7R=%T tl
Positive linear • relationship
Positive curvilinear relationship
Negative curvilinear relationship
HiTi^<^y 57
■3T^ 3?5HVFT ^ 3^flT^ (More than two variables) % "Sfc}^■HSfifdd tl W ■JraiT % ^
^1 ^ f^T y^'R % ^ fq^Q^'i|ui et><«^ irfb^ ?M^ f %rl ftsrfh ^ ^ ■^rr?? ^ arTt^r ^ ^eqfdd ^ f
■% <SM=b (Multi-Variate Data Analysis)'t'l foff^ M'hK ^ ^in’ch fq;'C^H''l ^ yfciHI'^'i
^iT<sH«t)irq<{l' ■% SRT ■§■! ■% (CorrelationalAnalysis) % 1^ y^=Ki Wft .WRJ^RITFT^ (MultivariateCorrlational Methods) 378^fcT •SlffTRi’ (Partial Correlation) fT^TT(Multiple Correlation) ^ 1 f^Rfr 3lfh^ ■% ffR 3?«?m arl^
%.1^ yi'<i’i«t5 dHd®J ■§' arsM ■% '^-■^ ^wii’«t)?TTcT ;§ ^ '^tt 'i'l
RFf^ (Multi-Variate Data)
3,3 3Tff$rai Tj^aii'ch (Partial Correlation Coefficient)
«<ami ■!■ ■'TF^ cpT-RTT RFRPT (Misleading) ^ R^RTT 'll^ % R^. RRRR Rtpnr ^rc^cn -^j^r ^ t rf^ cft^t
RT RT fR^ (Depend) RiR^ % RRRR '3R rNI ‘RTI % RItIrR RT RFR^R^ "^pRi^ RTRT ^ RTIRl fl (Height), RR (Weight,) ?TRtlTR5' 81RRT (Physical Strength),(Vocabulary), RprfRRT RhRRT (Mental Ability), RTRRR WT (General Knowledge), RR STTRTR (Size of Shoe), Ri^TIR? Rt fRRTT '^TR (Money Spent on Dresses), 1rrr f^Rfl interest in Opposite Sex) 3^ "RT RR ^RR rIr Rfl SR^ RRi 3^^ (Age)% RTR-RTR R^ "I’ rIr 3TT^ Rft ^ RrIrT RRR (Variation) RI^ fRiRTl RlRR?f % #R; ?R^
IRF^ ^ ■R^ % R%R R^RTR^ ^^ii'ch R^t RRRf Rfl RT^rI RR '^FRRR: RRTcRR> RRR^R^(High Positive Correlation) 'JRfR^' RTR ■?hn RFg RfR RfRRR "RRH 3R^ RT^ RrWI' "R "^RR , RfRR?f % %T3; r1 R^ %■ RWJ yewwi'R IrtRI RT^ RR FwiRR: WTRR ?pR (AlmostZero) RT RR^^R (Negligible) -rpriRT ^ RHR ^1 RRR'ftRfR ^ R^' ’^Rl^ RRT m%■ RIr RFR "^RR R'licH't) 1T?F^R^ RFRR 371^ % R^ RT 3i-c||| RRT RR rN^ % R^ % RTITR RTRT TFT ^1 RUfR RRT RTT f^TTl Ir^ TFR^ RiT 3itfR?R ■§■ RT^ RfRR?f Tri^FfW RRl^l Rfl 3Tf^ RRfRT IMRRRT F^ % «hKU| RRT RR % "^R RRR RRTRTRi TTFTT’RTR RIRT V RIRT t r1 ^ -Wr 3R5 R^ (Constant) RR^ RT ^ (Eliminate) ^ RTRT tl STR: RfR FTT RRTR % ^ R^ % rIr RTTTif^ TTFTT^R^ (Real Correlation) Ri^ RRRT ■!■ RR 3T1^ RT (Age Variable) % RRTR R^ TTRTRT 3TRRT f^qPsid ^TRTTTTR R>^ 37Rfft5R RT (Undesirable Variables) R^il^d RR TTRT^ ■§■ FTTf^ ^ ^ RTTcifRRTTTFTFR^ RTT 3TtziRR RTT^ % ^ 3TRffecT RTI RRIR RT)RT y^«td 3iqiTt9d RT dicy4 fRiTTl RT ^ R^ 'f RT qT^d: 3iqiT©d ^ RT^^ %RTT FRR RTrR^
RF RT 3TKTRR RT T^ R^ "% RIR ^ TTFTFR^ R^ifqd RTT TFT "t RRT TTFTFR^ Ri^ THH$ld % %t( "^TT^ RRTR RTt TTRTRl RTTRT STTR^RRT 'll RT^: RR RRT ^ R^ %^JRIRT 3FR SiqiT^d Rtt (Undesirable Variables) % RRfR "5^ F^ RR TRi '^TT^ TT^ V
RTTRT TFRR R^ ^ T(%RT1
STIR^RRr RTRT i\ RFTTTRIRT RiTRT
WPs^RfhJ /RfVRf58 R'^RcTT
. 3T^ ■^^TTefr ^tTsfi eFt ■^fTT^•^fei|of ^ ^ctJdi "tl ■^" % qiwfqsb ^Tlrf F^ tTn^lPTSF (Primary Variables) "^iF^ "t <H^iTo'rt 'mF^ f f^T^rf^ (Controlled Variables) ^iF^ t’l 71^11^
^VKtfqqi IpHTSF 3T^ffe?T % ¥*TT^ "5^ -^qllVa ^yql'iirHqi (Experimentally) F^TT IgFpT, ^iT<sq«h1q (Statistically)
Piqpqq feRT W tiqidi fi 1%^ ^ xqbiirHqi ^ PiqP-qd % 1^ "3^(Constant) VSl t 3TqfF 'Srf^ % •5PTJ^ ^ t f%> M ^ ^
'Ft B’TI S^?i1'^1 % «hH F^l iq>«1 ^ tuT^eblq ^ Ptqr-^a
3Tff^ ^ 3PTW 1^ ^Fmr ft ¥«TTT Wt ^ ^TBH yikiK ^ iraWqii.'ii chf^i F^ TB%Cf oJfTofFTft^F "^fe 31^ ftfv ^ ^ciii l^fv ^ 3Tfi^^ ’’TO^ f^qi^STTcTT 'll tiiRsq'+lq PiqPqd % %Ti ^ «a«t><. ■=^' % iTwr ITF^ftSRI FHcf
Tehqi "^JlTcn Tf^ (Partial Correlation Cofficient) tl 37F:% #g 3T1%^ J^-W^ tjutT^ ^ 3T8|^ 37f%J^ % 3I«IR g=i^ WW^ 'SBT^ %■3=T fNI ^t^^llP^d ^wiiqi % ■'Tft'TiftF PhMi ^
dHVIfl
TRRT tl ^ ■^TfF 3TT^’
<iH<i'd qici^l ^ Aqi^ F*1T ^ 'H6^«t*^ ’pTT^ WF f^RT ^5n^F^ 3?T^ SR^Irdt FTt^ 3ir<T F^ ^ ■% HTF 3lffFlF> FTFFIR^ ^jw|IF> I
^I'Rfiqi Fit oqi^i ^ ■RTFIFIf (Homogeneous in Age) 3T^ yPmJfiT■%FvFrf F^ ^ % F^ TIFF ‘FF^FFW Fft "FT ¥«trcfl ‘I'l
'^iT^iqj FFRF^ ^pFIFf 4»qcrl T^FT 3T«?FT 31^ % TI^TTF FtI FRFF =FT% FHF 1%FT FTFFiFT f I % fIf FftFfFF fqi^1 3TliFTF> FFRF^ ^u||«tj FTI 1f?T^ 3FF F^ % TTFT^ ^ "5^TIFT FTFT "t FF F^T 3TffFleF WRF^ ‘^wii’ch FF FFT (Order) FiFeTTFt %, 3TF: FF ^^FFT T^FT FT % F^FF F^l fFiFT FTFT F FF WF 3lfl^TF7 FFTFF^ IJFfFT FTt TIFF TFT FTT 3TffYfF> TTFTTR^ IpifFT (First Order Partial Correlation Cofficient) FTF^ t l FF ^ F^’ % TTFTF Fi^ TTFT’F f^FFT
«HIH1 •
FTFT t FF TFT FTT 3Tff9TFr TT^FR^ '^FTSF (Second Order Partial Correlation Coefficient)tl TW t F^ % TIFIF ^ FFTTR^ FTt ^ TIFT FTFT t FF^ ^ TFT (Order)
FF WTTRT%i TjOTTFr TIFF ITtFT "t^l TRTTR^ ^wn«t»l % TFT (Order) % TTR^ F FF TF^ FTFT ^P^d ^ FtFT If TTTFTTF TTFTBF^F '5'FfF (Simple Correlation Coefficient) f1 TFT FT TT^TTR^
^ Ml TFF i If FTFfTF(Zero Order Correlation Coefficient) Ml Fir^ ‘t'l FF F«F FF FTF TTFTFFF^ M 1ff1 Ml 3IR % TIFTF f1 TRTF F^ FT^ "t l
M' Ml (Xj FFT Xg) FT FTFT ^ M ^ fIf^ FT (Xg) fFF% % ITTI TFF FT FFM FT^ 3T?il f1 fFFTFTFT F^ STFIfTF FTFIFI
3Tft?TF F^TTPFF ■yrrf^ FTTFF FFTF f1 IfFTF FTFT
. (Residual Scores) % #F FfFTfFF TRFR=F MIfT tl 3TFf?TF yNl+1' f1 Ifr FF ^ lirF fFFT FTi-•FFFT
(i) FT Xj % ftfttf
di = Xi-X' 1FBT X'l = FT Xg f1 FFTFFT ^ FT Xj FT ^^Fp^ FH, f^ f^ fMIfTF ^ ^ ^F FT
TTFFT f-
<^1X'i= M^+^13 (X3-M3)F3
F^FFT MiTOf 59
(ii) X2 ^ ^rn^rf^
X2 = Xg X, -^n Tj4ct)f^ 1TH, "PtR ^Hl'=h<'J| 5TRf^'+)“dl "t- . •'ffe /
«^2 (X3-M3)X'2 = M2 + r23ag
31df^l^d yiMii'^ (Residual Scores) 'ST’^ '^N' ^■4' %n ^ 3Tm?T^ ^ tiarff^ra' 'ij'’Tf^ ^iTfT 'll ©qq^i^
q^a: quitch 'I'Jl-ll ^ '■Hl’MK^l %
■^HcT ■'R W^dl 'll 11*1^ "^cTC % 3ii’Rfi=t> ii''ii’«t) (First Order PartialCorrelation) ^ M«idi ^ ^ Pt*-ioid t-
^12 ~ ^13 ^23^12.3
V(i-4)(i-4). " •= ■^ 3 % ^ irTTM ^ 1 fT«?T ^ 2 % 371%^''12.3
■ynf^rj2 = 1 2 % li'^iiV)
Tjg = 1 ?T^ 3 ■% ITT^TT'^
^23 “ ^ 2 cT^TT 3 % ■?TPTR7^ IJOTT^
^ WH ^ rj3 2 ^23 1 ^ STCrff^ #i-
__^^32^2^2^_
^1^2) (1“ ^3^2)
^23 ~ ^21 ^31
■zrf^ ^ 1, 2, 3 % vm m x,y '^^ z if ^ ^ ■5T«|TT,'?d^ an^ -^TF^n^
^13.2
''23,1 ”
^xy ^xz ^yz
^(1-4) (1-4)y.z
^xz ^xy ^zy
(1-4)yx zx
^yz.x = ^(l-r2j(l-4)
t| w<]ii ^Jiiqi (Second Order Partial Correlation Cofficient)f%R ^ i^ ^ t-
^ _ . ^12.3 " ^14.3 ^24.312.34 1--------- ;------------- 5—^
■ Vfl-nla) (1-4,3)' ' ■= '^ 3 ^ 4 % ^ ^ ■qT'^1^8TT^2%^ ST^fw.
arff^RT ^«n'ch^12.34
wfwj^^ f^iFU. 60 SWoT
= 3 % ■R’lra ^ 1 ^I^IT 2 % STff^RT
= ■^ 3 % ^ ^ ^ 1 •5T«iT’ 4 % ^ 3igf^ 3Tf^
^12.3
^14.3
r24 3 = ^ 3 % "Sf*!!^ ^ ■?THr?f ^ tTT ^ 2 cl«n 4 % •a^’srfw 3Tft^
^14 3 ^24 3 ^ "5^^ ^lT^I«h ^'’12.3
WR ^ feft «Tt ^ (Order) % ^affVR? ^pni^ ^ ^ ^ %^iRfi«b ^prfsRlf (Next lower Order Partial Correlation Coffcients) WRTcTT tIMIV)qi «n ■R^nTI 'tl 3RT: *211^ «- = (fe — 2) ^ (nth order) % 3Tff?!^ ^yijoh M'^iii
% fW ^ ^ #n-
_ ^12.345... (fe- 1) ~ ^lfe.345.... (fe- i) ^2A.345... (A - 1))(l-r^2&.345... (fc-1))
3Tf^ WRJ^ ■JW ^ r % ^ ^ (Suffix) ^ W ^eTWT tl (Point) ^ 3TSF "^n 3TW3^ ^ ^ f 3T^f^
t '»iqf^ ^ ■^n 3T^ TT ■fpTcT "t ^ ^hiki %^-"W "tl PT^ f 1%^ % «n^ f^ni 3ToF^ 3T«M 3TSI^' WoHl 3Tff?Rr ^«liq> % "^rR(Order) ^ ^ tl ^ cI«TT ^ '^f ^ ^ RF^ R#‘ Ffm t
fg-«tOq 3Rf?T^ RFR^R^ fNfd t
'’12.34.^(l“ ''lft.3345
(A - 1)
''12.34 ''12.43 '’21.34 '’21.43-
3?!%^ RFR’^R^ IJRl^ RH RR (Order) ^TRTT t RR^ oqi<sm RRRl-Jific^ FIfT RRIT tl RFf qiKwi t 1^ RRT: <^qci RRR 3TR^ 1sf1r RTtR 3Tff?R» RFR^R^ IJRf^ R)1 RTJHT Rt RRit tl RRR R«F fgcftR RRIR 3Tff^ RFR*^ ^uifdfil' ^ Wfl f¥y RTl 3F^ ^ T^TFR^fl'
RT^ f%RT WU tl «iOq RFR^R^ . (High Order Partial CorrelationCoefficient) Rt WTT t R#T RR% WF RTRI ■^' RR R^,armr ti
40RRTr, =.62 Ft r„,'^ WTT I23 12*3F^IF^-RfR r|2 = -45, Tjg = FH-W t-
rj2 = + .45'■13 = --40r23 = +.62 •
3Pfe ^,23 ^ ^ t
'12 ~ '*13 '23' '’12.3 ” ■^(l“''l3)(l“''2^3)
=: .45 - (-.40) X (.62)V[l-(-.40)^]ll-(.62)^]
'’12.3
fe/VRT 61F^WfR
.698 = .97.7191200 ■07^ ■% ^ld<5rf ■% ^ ?T87T •STTg % %Q;
PlHqa ^I 371^ % ''R *^1^ ^ 3iqf?|oc ^ '5^1?
tT5IT «nT %r ^ J^UlJch = .68
■^5TTf ?raT 37T^ %-^NT *«»’*< ^^uu'en = .80
^TTT trar 3TI^ ■%■ t»6'H*««»^ ^4«ueh
X ■^, *nT ^ Y ^«7T 37T^ ^ Z ^
■5=rt7
= .75
era= .75r^=.80,
371^ (Z) ^ TT^ira ^ (X) f?87y ^ (Y) % ^ -w t
r._,. ^ ^FRI 3TeT:
'V=-68.
xy.z
^xy ^xz ^yzr =xy.z V(i-4)(i-4).68-.80x.75
V(l-(-80)")(l-(.75)')
= .202.08.3969
37?7: 37T^ % ^{«rra ^ ^ ^ ^ % "R^T 37ff^ ^ ^.202
tiMeieii 3Tra?^RReTT
(Consistency Requirement)c(t xi^ % oiH ■RTVR^ ^TF^rra^ 'j«ii'ch ci^ 37Tf7ra> wirra^ ir^rrar
TFcri tl ■*T?lf^T 'srra; 3TTf^ Tplf^ ^ ^ ^ TTrra ^ t ^%\sTiH'ieh ■RF^rra^ j^yn’cfi RH ■RRiR’^ ■^TF^rra^ ^uu’ch % RH arf^rar ^ "jora ^rarai
eit ^ 'TTTUR'^ ■RF^^ra^ ^«i|ch eT^ "3^ ^ "% "^tra 3Tff?ra7 TTF^T^^ ^pifaF %RPT (Opposite Signs) 1t«7W ^ 3^[TOT i\Tq^ ^ ^rarar tl ■qf^ = .90. = .60 el^n = .30 ^ era rj2 3 ^ .791, rj3 2 ^■RFt .794 F*TP r23 i -.688 "5(1^ "^l 'FT^ t ^7 ^123^"*^ ''12 t. ef«n rj32^ ’TFT Tjg ■% FFT 37f%rar f "Srafe Tgg I "F^n rgg % FTFR iF^TtF "tl 3Tff7ra> FFFra^ ^u|f^ % WqFTtt Fira ^ TR 37raft ■5^ FF RH 1.00 ^ arfirar w^ i\ t 'Ft % fff^^ ^ F^ra % fWtF ^ ti Ff^,rj2= .80, Tjg = .70, F«7T Tgg =-.20 ^ FFrj2 3 FH Fn 1.343 'Snra^ i^ FFFra^ % ■qe^ % aT^Fq 'll q^ei: "Sl^ra FFFra^ (First Order Partial Correlation) F>t F^qFl % ■% aTToRraFT "t f^. efH^ FTVTRq FFFra^ ^ H<,<rH< FTT Fra^ F^ qifty.
«R^ ^ 37Tf?rai FFFra^ ^«nV> ^ RTF 1.00 arf^rar f aTi f%i fIf ff■JIFTR % Fra^ FTI FFFFT Fra^ (Consistency Relationship) FiF^ t l FF FFFFT Fra^ -%
oqioMi FR'
^ ?Tef ifi FMt FFTfFq;-
' ^ ^12 ^13 ^23 ^ ^12 ^13 ^23 ^ ®
62 F^^rR /F/Vqf
^ #T ^ TTc? ^ ■>JTI ^ f ^ ^ ■5'iTfEFf W . yS?W77(Consistent) ^1 ^ ^ «Frt 'f ^ 31#!^(Inconsistent) tl "'Tt^ qp^m ~ -SO. r^g = .70, cT^TT = —.20 % f^-
^ ~ ~ ^13 ~ via '*' ^ ^12 '’13 ^23
= 1 - .802 _ .702- (-.20)2 + 2 X .80 X .70 x (_.20). ,= -.394
- .394 '^ ’TH ■^- 3Tfy^ ^ =1^ TFtf ^■'jn ■'TT^ % -^K^i 3T^T^ (Inconsistent) 13Tff?Rfr 'I^Hi
wm
(Limitations of Partial Correlation Coefficient)[^■^7 ^ SiiTVi't) ^wn=i> (Partial Correlation Coefficient)
rnnlcrifyd ■?ffHT3if ^ ^ itrn t- ‘
(i) 31T Tt '5’IR ^Hi^i ■5n 'Wqicii■3^ ^ ■^iT^TT ^Ttsrg f ^ 3liqj;qet>
^1 qsh1<< %.
y^dd «iOq (zero order) U’^Tfsfrf ^ % %tt y^c^i ^nn'cti^ PcldRd i\\
(ii) 3Tff^ •yrrf^ ^ TW 'STR: '^T^' W tl
(Hi) tTW^ ^'i||'«h % "^RR 31lT?Ri W^T'^«P=1 TpTf«F ^ ^Tf^-oFK’^ TPR=%T(Cause-Effect Relationship) ^TTcTT f I
(iu)^ % '9’m ^ WW^ ^ wll
(u) y'mcti ?Trf (Consistency Condition) % ^ 3T srffTRi ’TR1.00 ^ arte 3n ^Run ti
(Significane of Partial Correlation Coefficient)■^rWRR ^ ■^8%fTT % y^«Ki ^ RTRt fsfRnTt' 3Tfl7R7
ipnfcF H^'Kt f^RT '?RRT "tl 31^ cbq^jl ^=wri1(df) % RiqRy| ^ "t^l 3iiT^i«t> ^ ■^jNj ■% PiqPqa yc^lchRT ^ 1^ (df) 3fh ^ ^ ^ ti mxm ^ ■rfrt ■^fe(1 — r^)/^(Ti - 2) 'I'l 3TfT: (First,Order) 3lff7Ri ^ HH=ti ^fz ?TR
1 “ ^f2.3
fT«?T 3«F7 ■?<R % 3Tll7R) JpilVt (rjg 3) ^ BT^'hdi % 1^ y^^Ki
^ 3T33R ^ itR WcT -X^ #3-
^12.3 yN - 3
fi~''12.3t-
f^fk^ 63
^-STjqTcT ^ d/‘=N-3 ■qr ^ IR^R ^ (K-2) ^ ^ 3Tff^<jw|i«h H|’t«t> ■^-STJ’TRT % ^ ■^—
1 ” ^2.3....... k^''12.34.... k vjrnc
^12.34...k ~ K
Vl" ^f2.34...K■ynf^r % ■f^trt^TJj % tinii STffTl^ "yTT^
■RT^f^rcTT cR aif^ Pi'mU«i 'Prr % ''T*^ 'SIRSIIoRN (Fisher's Z Conversion) ^1%*rT 'sn R^rcn "f l «PTlf^ RRIR^T '5’^IT^ ^ 1^*1^ >tc‘<iqraa RFrar 1/(N — 3) F^■f Tl^nr 3lffFra» % 373^ yr^iqlaa ^ FhlT—
F^ 'SRirrc ^ k ^ ■RR ■% SiiT^iet) RF^R^^ (Partial Correlation Coefficient) % yr^iqfcio Hiiqi FITF F^TT-
(df=N-K)F«TT t =
1VN-K-3
3Tff^ RFR*^ ^ RT«f5Rn ^ Wt <1<IF<U|T IW i\ fT^I .
aqi^fui—100 yql'rq) % 3 = .30 '9TRf ^31TI =PTT "^IF 3HiT^i=h RF?R^^^«H«h RT8^^ t?
■FH-BF f N = 100 W rj2 3 = .30
3f«RT «i0<< (First Order) 3lff?Ri' TTFR^^RT ^ ^4«t)di '^iR "^r ^'is 3T3^ ^ WIT fFT ■?J5r ^ ■30^-
^12.3 'n/N' - 3
~ ^12.3t =
_ .30^100-3
V(l - 30^)2.9547.9539
= 3.097■'TfTRpHF t = 3.097 ^ im (if = 97 %\ 3R: d/ = 97 ^ ^ TW t
= 1.98 W igj = 2.63 tl mRjiPjM t = 3.097 FT ^ -ST^Rfr ti am:■?7F .01 FR ■qr tl F?#ni .01 FR "q^ FR ^-ai^qTfl %■ aWT ■qr qTFT ^ WFT t %
= .30 ^ RH .01 ??R FC TR7?fgr tl :
^.05
^12.3
aiifyicb FTFRJSRJ ^«nq»
(Semi-Partial Correlation Coefficient)ail Rich RFRRR^ '^«n«b ■% ^«siPfl«t) Iq<^q'i qw 1^R7T ^ ^ SiiTVl^ W
^ wi ^ ^ feft tM ^ (aT«mT ^ ^■) ^ ^ ^ ti ?r^‘
WfWJWhi folfipT!64
. TO ^ ^ % yikii+T cihit % irnra ^ am ^ ■qr w . w^'^ qw" vrfMmsraftrc yiyn«t>\ ■% *^«»«t> 'sn'<T "I, ^iTViqi ^«n’«t) 'll.'^q<n ■qrt ^«TT ‘^^rraci "k, w^ ijuiicb
si^-aTrfYiqr «6m*«i*^ y«ii't> (Semi-Partial Correlation Coefficient) 'll 3T^ ^W||cfj (Part Correlation Coefficient) % Wi ^ ^q»7j^ ^"l ar^^-arfRn^
^ f # 1I«TO ^ ^ fg^, xR % aT^-3TffifT5F3®lfeb ■qrt TIR:■^prf^ ?PTcT ^TTcH "tl RT ■% ■q^TR ^ «^q<n fgalq fqciM "W "tl PT^
t % ehlW* -if wr RT aiFl ^ ^nRT ^ RT (R^) % Tr^TR ^
''l(2.3)
f^RT "^RT I'l 3Tff?R) ^TF^TR^ "4 y^«Kl 3Tqrf?T^ THRfcFf oFT 'qfiv "^TR^ 1^RI "SIT^ cR TOTRT RTOT f ■% r^2 3) Xj dg ^ ®TtR TTTR ITF^TR^ "tl 3T;^-3TlT5f|et> RF^TR^■^RRT 'WTT 1^ 4 RT RTOt 4—
_ ^12 ~ ^13 • ^23
A" ^23
^ ri{2 3) ^ ^ ^ rj2 3 % ^ 4 rt4 cr ^ r4rt % ar^-aiff^Rr ^TF^HRR ^wiiV) r^2 3) % ^ 4t 4' TRT ^jl- f. 4^ ^4t 'q^ R^TT^ t' oRW^
^I^~rh ^ ^ TT^ 4 TO W 4qT TO RT
4r 4 ^ TO TO 4ft ti F4t "qro 4 a444 rt4 a^ arffro rf^trr ■jRRrf Fft wnI RR "^RFR r4 ■% 1^
frot RfRFHR^ "^RR) 4) r4 ^4 37^-3Tff4FT RFRR^ yJTlV)’ % TO 4 fTO 4t 4 14^
'‘1(2.3)
^ ^ '•i(2.3) ^ '■12.3rtot
RT «TOT
R? ~ fi2 ''U3.2) ■*' '’1(4.23) ■*■ '*1(5.234)
rfg =R^ 1 F«TT 2 4r 4r TO^fFS WR
r^3 2) = R^ 2 % TO RFTOt "qror f4 TO TO, R^ 1 FRT 3 % RtR 3Rfw
3mTR
rf(4 23) 2 F«TT 3 4r TO TOTOft TTOTR f4 'TOTO R? 1 FRT 4 3RfW -3^4%UTOR
'"us 234) ~ ^ 2, 3 FRT 4 % RTR WRlFt IITOR "4 'TOTO RT 1 FRT 5 4r RtR 3RfR^
to4F5 mRRR^-RFTTO^ ■^JRR> f4 RR! 3TT4 % RT IrTFR 4 ^4 RT T^ 4l
1.2345
TO
3.4 (Multiple Correlation Coefficient)
fsm RT % TO ,TO -^RF 1to RRtI -34 3TTft!R RT (Dependent Variable) 3TTO Pi«hq RT (Criterion Variable) to4 ■!■ F^ fro RT TO TO RTF FtFT 4 "34 TRFR RT
(Independent Variable) aTRFT ■«j4rR^ (Predictor) % TO 4 TTRtfRF fTOT RIFT 4l TRFR RT
.% TOTO RT TOfRF RT % ■'J^FTRF r4 Rf^TTl^fFFT (Precision) r4 % TTFTT^RR lIRfFi (r) % TO RT tF4T RTFt tl RT^F: f4 Rf^^rFFT 4Ni' r4 4> TO 'SRFfTO 3IT7TR (Common
3^TOT /w/vqf 65
.Variance) ^ t ^ 'fe "3^ %■ JjUii'ifi (r) ^ ^ ‘ftcIT 't'l ^
■^if^ ^ 3«iT .60 ^ -mw^ it ^ ^ ^an yi^ii^Y % ^► .36% TRm 3V^P|sa itm ^ ffe ^ •qr fH^r^ % yRHlPSiddl
i^ld<^ "Wt 'diRUi 1% ti'JllV (r) ^ "^TH ("^ ^'iiw'h ^ 3T«?^^) fold’ll srftR? ■gtdl "t^ 'j4«h«?3 ^ HR^iif^ddl -iail '?hft ^1
■?^rP5f ”'5^ 372?^ ■’^^87^^ ^nrW 1^ ■simr i ^=s|ffsfr 3«n ^ wr ^ t ^yfdy.Pd^MPtl % ^ ^ ^S7-'?fT87 3TP«?^7^ ^ *ft TJ#T ^ t ^^FfTTm^J
fnsqf% ^ •^fe % ^ .60 ^ 387T ^ arf^T^Tin' % it«t .50 it 3^Pi*>HfTt "^fe 36% 3RR^ ■^rn vjiqpti Pi^mPti 37^7^^^ %
3^T87fW ^1 ■!lf^ 3«TT 3lfi7toT ^ ^ 3 ^T^--^«T^ Tuqj flp^ frr^ % ^ 36% + 25% = 61% Wnn ^1 ^ 61% 3^75Tf^•jnTTOT % 37mR '»R ^ t PdW7f3 ^ ^ ^ 3TP*I5tW % iS7T87(Combined Correlation) 'Sfn iTPT >/!^ = .78.tl teft ^ ^ ^ "^n ^ 37pii^ % ^a? ■?^ 37^ y'^^dd (Multiple Correlation) ^TFT ‘^ITcn 'tl 373:
^-'M5«*<»'<7 lynTdi 37lf?RT (Dependent Variable) F^TT ^ '^TT 3tPj^ (Independent Variables) f^d)d (Criterion Variable) 'J^diaT^ (Predectors) %
^ '*71571 ^ qaini ^1 ’’TT^ '®T^FK ^ "dfl m«mi Fdit "^TT^ '3^«mlT^ fcu'el ^ ^<10-^ '87T '^c|«t>«i«t)l ■% UPHToF 'dTT "RH eldl ^1■^' % 3 ^ ^ 37Tp3m ^ 3871 i^H ’^^cRT ’3^’.% ^ % 3^^
3iIa'c%jiS'i (Overlapping) "Flcit "f, Puti^ 'd>Tt®T 37rf^ "37 387T wd’-d "3^ ql'q 3'Hqp»®<3 37773^ "d)! 37lf^ "^T 387T "7^3^ "3^ ■% 333f3^ 3777^*3 35^ '3T3 '3R hni
77®7^ "3^ ■’Tien "t^l sP“!iHd: ■3^'^'3F7T*3^ ^wii«n Mil'll d>fdd ^ '^TRft 'tl 37f3^I33 (Overlapping) % 37T33 '31^ F3 3.1 % 373efl3?3't 37f337 37^1 '37F ^ ^TRfTT '3T tiqidl tl
\|4d>873 'dT
itn 25% J7773n
^33^
r,3 = .50 R..«=.78r,3 = .60 r„ = 00 r:3 1 31 2
xTrif % d^dpjid ywui 33 ^(fllf^sfld «q^fi
'733^ '3^ % '3^3 37737 77F77*3^ '^ 37 37lPS3 "37 'dTT '37^ '733^ '3t '^ ti ^qd 7T?77^3^ ^713 '3>7^ '% 'f^n^ ■3^-77^7773^7 j]“n«h '3)t d^Hi ■% 'f^ IdP’i'* 7J^ ^33 yfdHld'i f3r3T 337 t f333>l
77FT33T 3|-77F7T7373 3it 3^^ 777^731 ^ 3t 3n '773r?f^ tl 3|-^F777flR7 7777737't ^ 3t3
66 3^337 Wf^mfhj
^ ^ (X) ^STT ^ ^ (Y rf^TT Z) ^ ^ WIT fw^ ^ 'H=Hd1 %~
^XY + ^XZ ^^XY ^XZ ^YZ^X.YZ■=Tfe1- ryz
^ Rxyz = ^^ ^ 00 ^ (Y) w ^^cPoT ^ (Z) ^ •^?T8^
'xy Y ■% ^I'MK'Ji
^XZ - "^ X Z ■% ffl'tlKUi ■^TFTTT^T?^ ^U[[cti
^'yz ~ Y Z % IPW’
^ '3Tf^ ^ ^“«ha1 "t l^rf^ ¥*TFTT3T)’’
37Tf?T?f ^ 1 ^ ’d^n drt 2, 3, 4\...3dfd-3TSFf "if^ diTdT 3T^ yM^l dHdT fldd d|-'dfd^d^ -dd dd^^dd "^Jd tdHdd •iii^*ii—
= ^1^2 + ^13 - ^^12 ^13 %
1-4dT^-dT^ 3nf^ dT d^ Y d8d 4drrd d^' d^ Xj, Xg-.^ ^ f dd'dd^
R,,1.23‘ \
tW
4, + 4, -'2'-;,,.RvY.X1X2- 1-4.^1*21ddddd fd^dd ^ PT^/I % d^-dFTWd '^dldT (Multiple Correlation Coefficient) dd
■ R TT^dT^ ^ tdT^ fl R % dfd-.d^ ddiR (Suffix) ■4' ^ dd dT 3?^STTfdd dT dd dddl t ddf^ did % 3^ d7 37^
3TTr9?d dT d7l 1 ddT■4'd^ did% ^ f I ^ -4' arffda .dT X t dd% R^jj^g ■4' sTrT^m dr d^ 3i^
1 %dT ddT tl d^--?T?TWd ■^ojldT dfl Wd f¥d 311^
ddfg4U|-d^ r^2= -58, /-jg = .42 ddT 7-23 = -35 dt R^ 23 RT^ W dJ^f^l ^-dld t ' ..
drf drl •■^^dT?^ f I ^ R ^Rd^ d^' d^ 2. 3. 4 d 5 ^ ^ ddT tl dT^-dT^ 3Tlfdd dT dfl
' f1.2345
^3 = -42dT ■Qd^ dd dT ^ d dT cftd d^-<ri5M^-d ^pifdT did dR^ dd '^Jd t—
R - 4 + 4 - ^^12 ^13 '*23
1-4 -.
rj2= .58, '23
.58^ + .42^ - 2 X .58 X .42 x .35i-.SS^V
.3364+ .1764-.17052\ 1-.1225
■34228 \ .3785 -
= /3900^
= .625
HiT<&^<*i}(4 /d/ddf 67d+dd7
'jj,un'ch cBi’ siH^
(Interpretation of Multiple Correlation Coefficient)^pnf^ oqi<sm ^ WT -dcJdi t^l
oqt<s4i "t■sncft ■'TT^ ^ yKs^-%T-T|'3#: aRTcTT f <^qcl 37lf9[cT rT^nRelationship) % ■*TH ^ sididi fi ^
^<541 q4*^ei ^dlc^eh qi'JllrH'h
^ *+'. ‘- ^ it w ti ^wfT i?2 % T^' ^TTychdT'tl ^^«u=h % ^ 3T8?fcl R2 cr) ^-'f^ykch ■’prfcK (Coefficient of MultipleDetermination), t' 3Tlf?TfT 'W. «qd-q ,'^' % (Commonvariance) ^ '% ■^’ qdidi f 100 ■3^qpi'>?5 3RTX^ ^ ^TR yfdsfra i■Ein<T it 'm\ %\ 5R: ■?lf^ R = .75 R2 = .5625 ^1 ^ W X?^ f % snTaR ^ R«IT ■^' cp '3V^ldcc5 ■5XTXR cR RH- 56.25 VildJ^id t’l
5^ ^ XTMXR XT^-XT^'*^ )j,w|i’<ti ^ oznxgTT •46 «<R f % R^-xr6'’H^-^ rtI ^
•^d-q ■% it^ XITXF^ (Joint^-XT?XT^^^ W '^iX^ % X37 ’xt XW-?^ %\ cHlPti
^cfidl ^ ■fXXf^ ■^-XTfXTT^^it it
Wd-q
(Remaining Variance) 3X^■srxnvTl-R^ 'it '^-3TfR?lfXcB ipu'cb (Coefficient of Multiple Non-determination) f
qdidl "I 1% -aTTf^ ^ tVdHi 'SrXR’^ 34^Hid
oTRXR '4' XTT^nX^ XT?XX^^^ ipiT^' % 3XRRTcf ^ -qx' r^ X- fe2 ^ 10 %
K “4 y^f^id RiX^ tl '^-'3Tf4?^fx=F ii'JH'41 (K)^ 3?fix5q^ ^ -qr X^ tl ^-3Tf4^fe
(Coefficient of Alienation) % XPXqtSii f^x!^ iii t. -a ^ ^RHx it R2 + K? ^ ym 1.0 it^ ti
■5R 1TR Xt4^ it 3TTfam
cTgn x^TrRi % xrrmxR xt^xttst^i -ijuTf^’ % ttr (+ ax^rcn _ xr- wtr f^) ^im\ tl 3m; Rj 23 ^ 1TH x^4^ it rj2 XT«TT Tjg % iTR itm\ ^-XmXOW %XJR xw #n ^-XmXTR"'*! RTT axfecR RFf cR WXT ^ f ^ r23 = 0. ^ tl ’SIX^: r23 = 0 ttt RX ^-xmXP^ ^ Rj + r^3 ^ W t f^' 41 ftt Rt rj2XTsqr r^3 % %TX Rj 23 ^ 3TftRmR RH WT ^^^1
3rff^ ^ CTSTT 37t^ X^^rm % R^ZT
.■^-XTFXTR^'ipTt^ % XXR^ t 41 t^ZR 11% (i) XRcF? XTtxqX XmXT*^^(^23^ ^ ■^-xmxoR^ U’JTToF ^ RH R^fTT t xRrim r4 % 'qxxqx xmxHR^(^23^ ^ ''^- ^“XmXTR^ ^M’«b RH ^TRI t XT2?T (ii) 3nf^ RX tqd-q Rt %xmxF^^ % RFfi % «r^ “qx ^-xmxxR^ ip^rrsF rtt rr q«di ti 3m; ■^-■ ■yTRt^ RR XR 3TfiRKR tmr t ^ 37Tf?m RX XXcFSf ^ XIBXTR^ 3Xfu^ (High) it ^3Rf^ XRcfm Rt ^ xmXTR^ 3Xrqm 41^ (Low) Ff FX^ 3iHqi'^icHCh ■fxRfcRT (Exceptional Cases) ■^xfl tl xmml t rI 3H<)c*d Rf®m ftqtf ■% fq^Jd ■qfoqm "sx^ r>x ttl ti 4 tltl l^qn tr^ ^«til
If xnx41 1 F' ■qfx^dxR^ xft^’ xw tl xr^tliXTTXtrft 1
XliqKUl XTfXr^cRI ^^Olicbl ^ R^-'Xl?Xrt«RI ■’jqRoF RT THTTR
R2 ^1.23^-XnaJTT ^12 ''is '’23 1.23
1 0 .2500.2976.3906.6944
.5 0 .502 0.5 .4 .553 0 .6 -.63.54 0 .8 .83.5
68 J«qd< fiUit
5 .5 . .3 0 .3400 .2619 .2500
• .2778
.58
.51 .6 .5 .3 .47 .5 .3 .6 .508 .5 .3 .8 .539 .5 .6 0 .6100
.4405
.3906
.3611
.78/10 .5 .6 .4 .66
.6311 .5 .6 .612 .5 .6' .8 .6013 .5 .3 - .4 .5476
.5476
.2619
.5476
.7414 .5 - .3 .4 .7415 , .5 -.3 - .4 .51 .16 - .5 -.3 - .4 .74
^ f-(i) ^ ^ 'tl
(ii) 3Tff^ ^ %. TTWj ■?F)l^ <s<(odl •'t'l '
(Hi) ■'TO'R ^J^iViT %, ‘'R ^ RR '^TZcTT "ti
(iv) ^u||'ch ^ iTH 37lf?R cT87T % RWT ^'i'mkui ■% RRTarfM^ ■2IT tl ...
3.5 (Multiple Regression Equation)efts'll
<rqa-5« *ft fqt^a tstt ^c^ai 5'H'til^ ^ ■?Wll TRTR RTPRI (Two Variable
Problem) 133’ ^ RBPRn 3iT ‘j4«h«R^ IRnT RRFTI ■^' ^ ^ STTTsR ^ ^
■% %T1 "STcftWR ^hIcix.^! 371 ^Tf 'll 'f^Rfl 3TTf^ "3^ 33 ^ ^3?F3
■3^ 3it RBFRT '^4«ti8R 3R^ % y^«K1 ^713; 31^ 3Rft HcfbPFR ■RRt37T'3 37T ^(General Form) 13R33 b13T f—
X2•^b,3,2X3 + KXi=bj
Xj = ‘STff^RT 3T Xj 3T RH
X2 = f3cRT 31: X2 3^ ?fTcr RR
12.3
3BT
Xg = ■?3cF3 3^ Xg 3^ w 3R
^12 3 ~ ^1 ^ 1^8R % tRT3: Xg 3^ "Mr W 3R
613 g= Xj % -^SR % Xg 37l 33T 3r(
K = ft8Rf37
33t^ % 13Rr3 37l <jM<lew 3^~33l3333
3if Xg 33T Xg 31 3131137 ?R ^ 31 •311% 5flTf9R 31 Xj % 313337tl fl«Tlf37 (Constant) K 33 -^RT 3T113701
Hif&f'=hl<i 33ftRf 69TEWol
^ t % '«j#g?r8?rr x^ Xj %q<iq< 1^1 -iH^leKi '«Hl<=ti<«i fej2 3 ^13 2 (Coefficients) Xg Xg ^
Xj ^ #> WI Xg ^«IT Xg ^ % "RPTf ^ ^ "RR (Weights) tl 31^:6j2 3 ^ ^ Mflicii '!' % X^ "4 (Unit) ^ RT Xj % RH I^RPTt
itJj\ Xg ^ iiRR ^ wm ^ 1^ -w^i 31^ ^^13 2 ^ ^Xg trgr 1«FT^ "^fe ^ ■’R Xj ■% RH "4 rod'll '^[fe ‘SWt -^iqlV Xg "Sf^ RRTRT^ ^RTT ^1 ^ ^ ^ 377%^ ^u(f«h (Partial Regression Coefficients)fl 3M0=kl ■^-'sratWT Rh1'+<'J| Pqxiflria HiKii’chT (deviated Scores) 'W*7T Hii^ Hi‘<it't)l (Stan
dard Scores) % ^ Pi*dqfl 1?raT
itz
f-^12.3 ^2 ^13,2 ^3
^1 = P ^2 Pl3,2 ^312,3
•iM^qd y^eW '^tzi (P) ^-Jii'chT q[^ Hi'iq> siiRfiqi ‘^cftWPT ^ynt^ (StandardPartial Regression Weights) "tl ‘rpW' ^Pny. "t ^^iiyiiebT % y^i=w 1^ ■3fT^ ■f l ^-^uii'cbl .(6-Coefficients) ^ '^FHT(p-coefficients) Wsi^dl ^ R^nft "tl cT^tt ■ft^RTeF ^ del’ll
cfr ■§■- . '
^1^^12.3=-^P 12.3S2
= — P13,2 13.2• ®3
^12 ~ ^3 ^23Pl2.3 1-4ft - ^13 - ^2 ^23 ./'“■ 1-4
K = Mj - (6j2 3 M21 6j3 2 Mg)dM<l<w 15^’ ■^' TT^cHl 'R%fnSR fl ■^-|]'JH’«*i')’ ^«7T RMRIT ^
(R) ^ TOTT -PTH ^ ^ ^ ^ t-
■^1.23 “ VP12.3 '*12 + P13.2 ^13
^ %tJ 6j2 3 ^ P12.3 ^ ^ P2 ^ c787l 6^3 2 ^ Pi3 2 ^ ^^3
^ pg ^ %\ ydlyiMH ^sin WtpT ^ ?cq f^TR^cf ^'^Tl^-
•X, -= 62X2 + 63X3 + K
■^1 = 63 ^2 + ^3 ^3
^1 = P222 ■*■ P3 ^3
^2 “ P2
ft - ^12 ~ ^13 ^23' ^2 1 ^21-^23
K=Mj-(62M2-r63M3)
Tiq, .63 = ^P3_S3
Y ^13 '”12 ^2‘SP3 = '1-4
1 ■
70.
^1.23 yf^2 ^12 + Pa ^13
31lf^ ^ Y Xj ^ Xg TI^ffRI I^RfT ■^n^, cR ycilHMHi
^hI^iWIi' C^^^T ■^' Pll-lqa 'Jll^^MI •
Y=.6iXi + 62X2 + K
■ y = b^x^ + b^X2
“ Pl^l (^2^2
6.= *^®1
n _ ^yxi ~
1-4K=M^-(6jMj + f72M2)
^«TT1S2
^yx2 ^yxi ^xiXjP2 =cfen
1-4
^ml«kl qfSia yalni]^ ^wii=bl cl^lT 'FT^^ 1P?T ^ % 'SRTtWR yHl+J,u| ^8?T n^: ^ TRH t, 37^ -^T^rn^
% tl y^d 1^57%^ ■SH^dd TFT^TT W ■!■ c7«7I
3T3?7l7H ^FRf ^ "57^ tl Mld'cfiTFn feft ^ FI^' ^
. ^ ^ fl "3^ T^ 37PT WT ^ ^ RT^I ^ ^ FT^
^Sfqd 1^ TIcftWT^ ■% •% XFP7 TRftWT^1^ T7T W^-'^ fl ■^rf^ Xg ^ ^an f X2 Bfw7%cT'1^ RT^-^.
<i'i'^ ^IT^ ^ sRqfan ft '^7%f f I q|,ydl4^!^ 'SifSRT 37ft "7^*7^ ft'TFf^ftl
"aqig^’Ji—ffr Rf Xp Xg ^7^ Xg i7*;2T^ R *^i'i=h fq'qci'i 25.6 R 5.8, 47.2 ^ 7.3 W 16.9 R 3.4 ft = .47, rj3 = .38 W r23=:.24 ft.cR R7 2 R 3 ^ WTO
RT 1 ^ ■SP«R ^ WfTWT Wft^nTJT W dSIl -^-WW^ ^ WTT ^1
^Rt3^ % Xg 34 ^ X3 ^ 16 37^ ff, Xj R7 % yikll^ft ^ '^aR ft ^‘l
f . - 'r,2 = .47 rj3-.38
r23--24R7 Tim 2 ^ R7 WT 3 ^ WTO
Mi = 25.6 M2 = 47.2
(jj = 5.8Og = 7.3 . ■
Mg = 16.9 . CT3 =-3.4f RT- 1 % ■<^^aR ^ feTtj; wtWTH Wt^TT'^T
• \
Pit-iqfl ffrt-
X,^b,X, + b,X, + K
.ft TjqfEFf 7M tWWT ^ WRT fFT -TJ^' f ^ Rlfft-
^12 " ^13 ^23^2 ” ^ P2
^3 = f P3*3
WT P2 = 1-4 ^13 7’i2 ^23W1 .P3 =
1-4K = Mi-(62 Mg+ 63 Mg)Tran
d^rt< wJhH'=t>l<i 71
HHfhPUIR .47 - .47 X .24 ^ .3572 l-.24x.24 ”.9424
sm: P2 = = .379
5.8^2 = X .379 = .30
7.3.38 - .38 X .24 .2888l-.24x.24 ".9424P3 = = .306
5.8= — X .306 = .52 2 3.4
■ K = 25:6 - (.30 X 47.2 + .52 x 16.9 ) = 25.6-(14.16+ 8.79)= 25.6-22.95
= 2.65
Xi = .30X2+ .52X3+ 2.65
^ ^ t- u^1.23 ~ 7^2^12 + P3^13
Rj 23 = V-3'79 X'47 + .306 x ;38
= V-17813 + .11628
== V!2944l = .543
Xg = 34 X3 = 16 ^ Xj ^ ■'J^sicT ITR W ^Xg = 34 ?«Tr X3 = 16
X j = (.30 X 34) + (.52 X 16) + 2.65 = 10.2 + 8.32 + 2.65= 21.17
3R: Xg = 34 TT«n Xg = 16 ^ 31^ % Xj ^ ^ 21 SRT ^ ^ ^
SR:.
/
tl
3.6 3if^ch ^ (Multiple Regression forMore than Two Independent Variables)
^ ^ 3TfiR7 >qa-3t IqwiiRci (Extend)^=bdl t'l ^ <|win RfeR ■?! ^TRft ^ fR
wri ^ ^HoIh'IHI % ■^tel .^UlfeJiT ^ WPRTtl WcT •^fVifdci(Doolittle Method) 3T8|^ (Aitkin’s Method) ^ 31^ ^ tl ^ tt■foff^nt ^ (Steps) Chi'll 'Jilkn tl STR^Rf % 'gRI tl, ^ 3Tf^
qi tqd'O) -q
ydlHMHd ¥4)qi<.wi % <j«ii«hl ^ t; i^rf^ fsiftprl
72
, ^ ^ ^ ti ^ ^ -sri'UrftwH ■^, qRr>=td ^=Rr^ % ?7F^V jtf^wt?*?. — 1) ^d-?i ■^. % ^ ■^-'STcfhTWT
yHl=h<«i wiFT ^ srnff^ ti
Xi = 62^2 + ^3^3 ... + 6,A. + K
Xg + .....■.... + P^—X^ + KXi=P2^X2+P3 Si■m
S2 . Ss Sm
t. Pg-^ lyi ^ P^ ^ •4’ P2 ^ ^ ^ p cT«n p ^m r^/n ^
^ t -R®# (R) ^,wn Rtr ^ ^ t-12.34 13.245
p lm.2345
VP2^12 + P3^13
t«id-5i ■^' ^ '^pff^ (R) ^®7T ^ qRqifijtddl(Accuracy) '^TRfl ■!■, ’'R^ WtWT^ % f^ciH t«Jd'^ ■^’ ‘^^TT '^^ Id-dKyil^^
■5R^ 'tl eFi)^ 3?®M 31^ 'adT^'H'l 'HHld5<'Jl 'dl'sA ■'7^ ^jnTfefi id®n
MRMiFSiddI ■^’ 37«T^ 37r£RT ^ t TR W\ ^ fl -Rs!# R■33dT f Rr ^ ■?^?F5r Rd^RWT ■^' fiRnRld RT jpilch
RrfMf mw^ (Forwardsolution), 'i^'iinl ^*7H (Backward Solution) cT®TT WlHiit'f^d (S^pwise Solution)
, 'i^l 3T5FTTR1 ■RRT^TH R ^ MdlncHd 'RRrRTcT Rj^lT ■^TRTT "f 'aiajRti '^4mih1
FSRT "^incn f I tilniil'^d HdlncHd fq^c^qu] (Stepwise ■f Rrt^ ■'3^ % til Midi R ^RnRid ^PFn^TPT .'^ HrqctJ rImm R
^r?:% ^ WlRWT ^dleh'OT ‘4’ W 37^ RNPcdd WF t R> yc^di R
arRT^fPT ^wiiq) TIM ^ TT%I ^ Tl^ R ^iMidl'^d TPffWR Pq^fc^qyi TJ^TR^lMid
RqldH ■'3^®7^ (Best Predictor) ^'imm R’ RdlcR '^4=^iH«hT ■^,.... .....d®7T ^ TRTR
R rWh m ^qldH tiRnRid qi<4i TFcfr'TWT TPTMT'JT ijUdl 't’l TJc^
tilnM Rt TIM ^-R^R«l'^7 IP^TM R % ^yiH Rt TfM IFTM ^ R
^ ■RT®f°MT ^ RtI^ Tlf4Rd % TM“?^S7TiT (F to enter Test) % ^ Rm w R^dl 'i'i ■y^TM R 'RT®Rt "glck f M TifflWTd fq^criq^i R- 3T^f?M
RRRfdd ^ ^ ^ fdyRd .^cJdlRl TM®IH R Tf®m "RNHRp-nRid f cR Rtrc ^ f Rfmt qV^n 'ft
'HlMld)'f)d ^TRl^ '4 yr^di rImm RT 'R^ ■% rFiRR Rft "^73^ "I cTSTT 3TM?®MRT R^ RT
37RT®fRr qVi'^M ■^’ oR) IFSI ^ "tl rIrrI^ ydlMCHd fq^c^'PI R R5R?ftcFTT (Tolerance) RH RrRR snRRT R^Wjyt ^ f I R?Rifl)cid] (Tolerance) qiwq
tional Accuracy) RRT^ RR^ RR RcF "^rtr ■3R^ RcflRRRR fR?^R^ '^' R^HRid ^ (Variable entered) % R^ R^ RFR^R^ % R^ % (Complement) 3T«7fct 1 - ^ tl STcR^ 37^ 37®(Rr TMT RM^tefT (^ .001) Rl^
■RT Rt RdtRWT fq!?<^qui RpRf^ RTtt RT: Rf^RTStRilRT ^leRI (Rounding Errors) RRtRl • RcflRRRR fR?^R^ % Rtt^TRl RT ^RTR R^ Rt ’3TfRR7 Rr.RIRRT TMI fi ^ r1 1r^ RT Rt
RMTfteTRT % RRM 3T?7R| ttt RiT 37®! t 1% R? Rt ■'^ t RpRfeFR (Extered) Rt RTF .R#3R (Linear Combination) t ^RT RT Rt TJcffRTO fR?^RR t RpROdd. RTTt ^ Rt^
^1.23 ■Pm^lm •+i ■
RRIRF f RR?-!^ qi<<4) Rt' Rt f^RRRR Regression Analysis) RR^: 37TFTT^ "RRIRR RR tt R3>’ RRTR
•3R R^' ^ ^ RIRT
RTf RtRRd^Rf Rt^rqd'q •
# TiFn Rr4 ti nR^aSdi (Computa- tl ■^RcPR RT Rt WR?ftRRT RTRP^ BR Rt R®71
3Wrf< wVs^‘fil<i 73
(Additional Information) ^ tl ^IhRI^ci WfWR ^
-^fwIcH Icrflj
itz (Doolittle Method of Multiple Regression)
■^' lT8Tf% ■^’ ■^Rt’^rn^T CT^IT (i'J|l’4) ^^ T33^ WT Wt -ER^ t^R.13:^. "^Rrired (M. H. Doolittle) -^R 1878 ’4' fel m
^ t . .^X. + KXi=Pj-^X2+P3i
^ $2 S3X3+... + P; ■j
■^J
Xi =fc2^2 + .V3'^-®/j + ^% '5KT qt^a: p 'Rpff "S^k qqPid "«t><*ii ‘^tcTT "t % Xj
^ Xi '*TR1 % 3T^ "ijHdH "^l Tjflncl-'R ■'TB^ ^<ddl ^ ^ i^tr .
(Method of Least Squares) "SI^ 'PTB .^<aTq ^MHd (Linear Simultaneous
Equations)
3187^
Wt'TWT
P2 +^23(^3+ ''24p4+-+^2jP;= ^12 ^23P2 + P3 + ^34P4'^-'^''3A=''i3
^2/2 + ^3iP3t"4)p4+- + P;=^ly;/■'
HR "t t^4)<4vl^t ^tdt "t l <4HleRl*^lT
H^1=R<u| (Normal Equation) t HTHRT '4' HTHTH^ ■^' HcftWR HHNRR PlHcid #ft-
^ HIHFT^IHM yirMq>di CRT TFSRT ^ "t! fqfl-q •’
Si SiP2—^•X2 +P3 — X3 + P4 Si X^+KX4 +p5 ---Xg + P ;X =1S5S3 S4S2
X J — fc^Xg + 1^3X3 + ^4X4 + ^>5X5 + k
1H ■ft«7fcT HTR (Normal Equations) Pi*^qa
P2 ^23^3 ''24P4 ^25p5 ” ^^12
^23p2 '*' P3 '*' ^34p4 ’*' ^35p5 " ^13 , ^24^2 ^34^3 ^4 ''45t^5 ” ^
^25P2'*' ^35P3'^^45P4 + P5 = ^15
31HHT
14-
tIR RT HtRP HR ^TR 'll 'HH %HR ^ RT "f cR P HHI "% fRH ^ HtRdT RRf^T^ RT ■f’l ■% rH'e?<^ <gws '^’ qf^ld HHT p^ ■?Ht yq>K ^ HHtHHHf
HTR M f I HH5 Hfe ■?^cTH7 ^ HI p HRI HTI ^ ^ HHlHHHt'IJHR^ °R HR HHIHR HHHT TFR "ti 1H HHHt Hit HHH HTcftH
HHIhh^ (First Degree Equations) HiT HHTHR Hfl iHfHHT "I, HH^ '^fRfeR % ^
yfdMlRd Mh 37f^ tl W MM HFH^^ HhMrt (Symmetry) Hfl
nif&ct)l<i MHHf74 dwHcr<
rj2 = ^21, Tjg = Tgj, ..., '?fT*T <i<Ji«t)< 'iwiii ^tT4 ^ f'l
■3^IOT«f ^ (Xj) TTSTT ^ ^^cF5T ^ (Xg, Xg, X^^ Xg) ^ '
TIT^ Pd-didd! ^8TT '^pfsFf ^ W^ft 2 ■^' IRp 1%^ W tl
BTHlft 2■qfir ^ ^ (N = 600)
tiQti«««*«4 ^^wnW (r)Hineii 1«rsr?PTTTtZTOTTVariableM s Xi X* Xa X4 X5
X,120.30 24.39 .61 .42 .24 .53
X246.20 6.56 .57.61 .39 .23
X332.50 7.34 .42 .39 .36 .55
X410.40 2.60 .23 .36 .38.24
72.50 Xg16.47 .53 .57 .55 .38
T0^ -q? Pft ^Ml«h<uT ^-T^gf 37lfSRT ^ W?T ■f¥iT=I IPTf^’ (r’s) % ^WTPT
P2 + .39 Pg + .23 P4 + .57 Pg = .61
P3 + .36P4 + .55Pg = .42
P4 + .38 Pg = .24
.39P2 +
.23 P2 + .36 Pg +(^1^ 444i<t»<U|).57 p2 + .55 Pg + .38 P4.+ . pg = .53
^ bNt ■f ^ «il'*iqr«ici % ar^Ffdqrmt ti <r^Pid W? ■^' ^ wit 3 ^ ^ ^ ^ ^ 1%^ wn tpjjy^ ,qi64>M''i ■^’ Pifea 3 % ^tidi4>H ^ qwt^r^’^ 'tl Mr^«h (Cycle) ^ 3?5rRl Tlftl (p) e^iqi 'qm "tl 3T5IT5T ^.iRtnTqit qn ?pq qq qqr ^ciai "t
fq «Hl«t><.wiT .qjt qj
^PriH 3T5nq Tiftr qq qR ■qpq q^ ^ qnqr ®i5nq^ aqr
^ 3??Rq^qqqrqum^^fq^qTif^ qq qrq ^ w ■^'
qqTR ^ qqt 3T5nq qfW (p’s) % qR w ^ fi wit ■jqRjf qftTsqR
■qqfqqqi qfi iq^qq? qq -^idniR ^Kufl 3 ii<i«w Iqfv qq qq?f^i Wt 31^ ^rlqi^ qjt wit 4 '4' i^gq ^ tft ti wft 4 ■^‘ ■'feqt qrt A, B, C, ....3Trf^ ar^'qqr wrf qfl Xp Xg... arif^ ^ t^qr qqr ti 3?pqqw7,1^z ^ #Tq w i, WTT qft aqqfe qrfq % wn qqi ti qf^ qqi wt %
■q^ q^ % qR qjt "3^ q^ q«Tr 3q w? % «4)dits^ qrt T^q» ■qrq qi qrtt^qr ■^'^ qqr ti ^ [AXg] ^ qf^ A qqi wi Xg ^ qj^ qi^ qR ^ ti 5 '4' qtzi TpjpFjf ^ WTT qrrif ^ qq? qR4 6 ■4‘ qqtqqqq ■jqfqjf q ftqqqr qft qmqr qit ir^et qqr ti q^: fqfq-^ q|-qqtqqqq,1q7^qq qiT^ qrr wq wtt qq^ wft 4,5 qqi 6 ^ ^q wst IqTT qt aiqqft ^ ^ f¥q ■4'q^ ^ffqrqf qrt arnt-Sregd -I^ qqr ti
«t><iiq «a<t»<
wTh^^l*4 fhf^<4t 75
WhT^
TTfel' (Operation) (Solution)(Row)
3I2n7 «h1'»)<'J| p2 + -39 ^3 + .23 + .57 Pg = .611
^nleh^ui
1 - .39 ■fm ^
2 3 ^4 ^ ,8479 ^ «7m ^
,39 P2+p3+ -36P4+ -55 h = A2 - .39p2- .1521 p3- .0897p^- .2223 pj^ =-.2379
0 + .8479 p3 +'.2703 p^ + .3277 p^ = .1821 Pg ^ .3188 p4 + -3865 % = .2148
2
3
4o
^tfl^ IVi<gl■'Tf^ 1 ^ - .23 ipn ^
5 ^ - .2703 ^ 3pn ^ 6. 7 ^e^l 8 ^ ##9 ^ ,8609 ^ ^
,23 p2 + .36 Pg + p^ + .38 pg = .24 - .23p2 - .0897p3 - '.0529p4 - .1311 pg = - .1403
- .2703 pg - .0862 p^ - .1045 pg = - .0581 ■ 0 + 0 - .8609 p^ - .1444 pg = - .0416
P^ + .1677 pg = .0483
6789
10
1 ^ - .57 ^ .'jpn ^^ 5 ^ - .3277 ^ ^
10 ^ - :i444 ^ ^ ^ 11, 12, 13 (Ten 14 ^ ^ 15 ^ .5242 ^ ^
.57 p2 + .55 pg + .38 p4 + Ps = .53 - .57p2 - .2223 pg - .1311 p^ - .3249pg =- .3477
- .v3277 pg - .1045 p4 - .1267 pg = - .0704 - .1444 p4 -.0242 pg =-.0070
0 + 0 +0 + .5242 pg = .1049 Pg = .2001
1112
13141516
^ 10 6g = .2001 Vi\ P4+ (. 1677 X,2001) = .0483 3m: p^ = .0147
pg + (.3188 X .0147) + (.3865 x ,2001) = .2148 .am: pg = .1328
.17
^ 5 #P4= .0147 ?i«n. pg = .2001 Jfif
1 -H pg= .1328, p4= .0147 fJ®lT Pg = .2001 51^
18
P2 + (.39 X .1328) + (.23 X .0147) +(.57 X .2001) = .61,
3?cT: p2 = .4408
. 19
4
(Multiple-Regression by the Doolittle Method)
vftm fdWM (Column)
XiX4 X5 LX* X^(Row) (Operation)
X2 % ^*1 ^pif^A ^ - [^\X2] =-1.00 ^ *rm
2.80.57 .61.39 .23A 1.00\
-.'^►7 -2.80-.61-1.00 -.89 -.23B
■^.Xg %^ A IB Xg] = - .39 ^
+ 2.720 -1,002
.4200.3600 .5500C 1.0000-.1521 - .2379- 0.897 - .2223D
76 9‘^c7<
ualnuH’i^ C D ^ ^E .8479^E^-[EX3]= - .8479 ^ ■4TTT - 1.0000
.2703-.3188
.3277 .1821 + 1.628 - 1.920F -.3865 -.2148
X4 % •
^ A ^ [B XJ = - .23 ^ ■3^ ^ E ^ [F XJ = .3188
^ G. H ^«^t I ^ ^ ■4^ J ^ - [j XJ ^ »Tm
G 1.0000- .0529- .0862
.3800 .2400 - .1403
+ 2.210 - .6440H - 1311
I -.1045 -.0581 -.5190+1.0470J .8609 .1444 .0416
- .0483K -1.0000 -.1677 -1.2162
^ Xg %A ^ [BXg] = - .57
■^E^[FX5]=-.3865 ^TFTT ; ^ [KXg] =-.1677 ^ TFT.
L, M, N O ^ ^ P ^ - [PXg] = - .5243 ^ "m
L 1.0000 .5300 +3.0300 - .3477 -1.5960
- .6292 -.1756 + .6292
M - .3249. N -.1266 - .0704
0 - .0070-.0242P .5243 .1049
Q - 1.000 - .2001 -1.2001
5^ ipiTrajf ^ TinpTT
(Solution of the Beta Coefficients)
4lci ^wuch TIUHT ^5T4.
Po -[QXJ =.2001
-[KXJ + Pg [KXg]P4= .0483 + .2001 (- .1677) = .0147
P3 -[FXJ + Pg[FX5] + P,[FXJ= .2148 + .2001 (- .3865) + .0147 (-.3188) = .1328
P2 -tBXi] + P5[BX5] + P,[BX,] + p3iBX3] _= .6100 + .2001 (- .57) + .0147 (- .23) + .1328 (- .39) = .4408
6f^deb ^ OUHl
(Solution of the multiple Correlation, the Regression Coefficients and the Constant)
Pi Pi ^'li Sj/Si bi M M,b,^li Si i
X2 .4408 .61 .2689 75.71726.56 3.7180 1.6389 46.20
X3 .1328 .42 .0558 7.34 3.3229 .4413 32.50 14.3423
X4 .0147 .24 .0035 2.60 9.3808 .1379 1.434210.40
X5 .2001 .53 .1061 16.47 1.4809 .2963 72.50 21.4818lb- r,. = .4043 112.9755
^^1.2345 “ ^ Pi ^li “ -4043
= /4043 = .6358
M, = 120.30 K = M,-IM.6.
= 120.30- 112.9755 = 7.3245
1
5^ = 24.39 R1.2345
wfmffh /g/W 77jWf7<
(Regression Equation)
Xi-62^2+63X3-h6,X, + 65X5+K
Xj = 1.6389 X2 •+ .4413 X3 + . 1379 X4 + .2963 X5 + 7.3245
(Steps)(Doolittle Method) 41h<^ 3hjtdQi fVi4 ■^il^
^ fci<ai ^ t"—(0 f^i Xg
3T8# [AXg] ■^' 1.00 f^i ^ iTHf -qtn I %^\
(u) B A % ^ - [AXg] 3^«lfd - 1.00 W] f^r
(iu) C Xg % '?fT5l 'S^qHji'^ (Remaining Variables) % Tjqfap"4 "t ^ ^ 3?gf7T^
Ag % 3r^ ^ ^ ti ^ % Xg [CXg] ■^‘ l.OO-^1 w ■^' "^n^" Xg % 'HT'q 31^ Tjnf^' ^
2 TdWI -^f f^l
{iv) TTf^ D ■4' A % ■*TPff ^ [BXg] 3T«rfd - .39 'irni f^l
(u) E ’4' e :d«TT d ^ ^ f^i
(ui) ^,F E % iTRf ^ - [EXg] 3r«?fcl - .8479 ^ ^ %^l
F % iTHlf ^FT ^ (arfSfm ^<Twr % tth ^^ ^1^*7 ) ar^rm TfTlST % TTR araf^T [F] % smoR ti
(uii) G X^ % TTI^l STcffin^ ■^’ % W Tg^% ^ ■4 ^ t' ■^TBl 3Tcif^ Xg ^ Xg %a^Pdftdd 31^ ^ tl W ^ X^ ^ 3Taifd [GXJ 1.00 'f^i ^■^' 1^ ■qiTf trgf X4 % RTef 3Fq -gt' % 'JiqfsRl' ^ ‘^bi X
- %a\i(utii) H ■^' A %1TT^’ ^ [BXJ ai'qfd - .23 '^ott f^i
(ia:) I E % iTT^' ^ [FXJ 3Tqfd - .3188 ^ f^l-
(:c) J qf^ G, qf^ H i ^ f^f
(xi) ^ K ■^■ ■^ J % -RHl' ^ - [JXJ 3T8lf3 - .8609 m ^7% 1^1
■3tNt-^ ^ Tjf^ K % RHt w ^ arf^rm ^fiwr % rh arsihr [KS] % ^rtsr
^23
tl(xii) ■qf^ L Xg ■% 'RT*T 3Tgl7T^ f^R^I «pfff^ Tgg, Tgg "d^
r45 % ifR ^ t' IRfeq; d?! 3Td1^ ^ t dTdRf X2, Xg d*^ X4% 3?fdftdd 3RT f I fR q1^ % Xg dT^ 7d»l 3Tdfd [LXg] 1.00 f^l
fR "4 TPft dT^ T^ Xg 4> THd 3Rd ^4 % TTFRtd^ ^ dW 2 ^d®T
4 t^l . '{xiii) df^ M 4 df^ A % dT^’ [BXg] 3Tdfd - .57 4 ^ dR% t^l
ixiv) df^ N 4 df^ E % dTdf d^ [FX^] 31dfd - .3865 4 ^ f^l
78 3®d(77 f^ffifVf
(xu) 0 ^ J' % -ciHf ^ [KXg] -. 1677 ^ -jpn f^l
(.rw) P ■^' L, ■^Tf^ M, ^ N 0 ^ ^ %#l(.vt;«) ^ Q ■4' P % TTRf -^ _ [pXg] 3T«rfcT - .5243 ^
'trf^ Q % ^ ■?itn 3Tf%ir T?twT % STSlfrT [Q2] % ^TraT*Tntl
(.wrii) fTR ^ ^ Tpp^‘ ^ WTT ^-
h=-mi]-
P4 = -rKXi] + P5lKX5l.P3—[FXJ + (3,[FXJ + p,lFXJ
P2 = - [BXi] + 65 [bX,] + p, [BXJ + P3 IBX3I
n# ^ 1|H ^TOR tl(.vis:) PtR ■^' -^ R2 cTsn R ^ RR W ^-
^^1.2345 “ P2 ^'12/^ P3 ^'13 ■'■ P4 ^’i4 ■'■ P5 ''15
^1.2345'
<4*4)e«^«i ^f filler % RRwhrt
ML2345(.vv) PlR "^3^ "^tsi ■JRfcfjt tTen HM'+I fcf■City'll ■% RH ^ ^ '^_^
\ = 63= P3^ *4 = 34-^ A = P5^'«4®2 -S3 H
(xxi) "PlR TRI^RR «6iHqi K ^ RR
K = Ml - [62 Mg + 63 M3 + + 63 M3] .
^ ^ ’TeftWR ^ RR 5TR % e^ri-q Wt'T'FR RRI^RR(Regremion Equation) «<cnai ^ fcT^ ^ ««t)dl 't’l t ‘SRhlRR
IRt^R’iT ■pTRoR WR iWt- -
Xj = 1.6389 Xg + .4413 X3 + .1379 X^ + .2963 X5 +'7.3245■'R ^nvii'chl ^ ''5^«R i^Rn
t f^ra% Xg, X3, X_i w X5 ^ m uTRTf^ m ii\[?! ■^) %
f 1^ srfe^ ■^‘ % %T3; Rt f^RgtF (Extend) %R ^ tl ^-w**^ ^ ^rr^l 1R;1R%.'7F3^ "^RBr (cycles) f) "RiRT
^ % «l<|6|<. f I Vlr^ ^ % RT2? ^pif^ ^Tlf^ ^ WSl ^ t ^2n ^ 3Tf^ -qf^ % Sim - l.oo ^ TIM cRJ ^^RcTTf i ^ <t4rg^ui -di X^ ^ 3ITf?R cISTT Xg, X3, X^ ^ X^ ^ ^ tr^
t. FRg sTif^ ^ qrt wn«I11 1%Rt ^ snf^ TteftWR ■^TT R^kH^ ^RWT ^ ^ ^ MT«i -rnmnii
Tc^fR 79
XSignificance of Multiple Correlation)^«n«h (R) ^ ife (Standard Error) ^ "t-
1-R^-1
N = yfd«<J?I ^ 371^
m =.««nT^'ju|i«f> % 1^Tt^ 3RRT^ TTM ^ "f l
^ ti:qi-a7^ (F-Ratio) ^ ^ ^ ^ ^*nw
ft t-««t)CI
R2 N - ni -1F =1-R2 ■
W . R = ^ 1TH
N = 'Sff^ 3TT^
m = '^’ ^ «(sMi
^ lTT«fsfKn df = m cTan N - m-1 tl ^-3T^W ^fcTT^ f1^ qR^iPyi'd (R) ^ iih ^ ^ f^ t -^\ ^ N = 200W m ~ 2 % ^ R = .45 ^ ^
m
.452 200-2-1F =
1 - .452 ■ 2
= 25.01(//■= 2 ^ 197 % .05 .01 ^ ITH 3.04 fT^Tl 4.71 t 3^cT:
■qroir«ld /= 25.01 ^ -tlH .01 TcR h R.= .45 ^ tTH .01 tl
(R) elft •■^n^fSFTffT % % ItrTi; ■qftf^T^ 7 •^‘ Tfl ■?TT8^^ ■^■■g’nfsFTf ^ ^’t tor w ti ^ irn^ t N K % TTpft % ;o5cT^n .01 "qr ^ 3i(qjf<4<?) tR ^ tiN = 200tT8TTm = 2%fRilR^iTR^^^ .172 ^ .212 "q? ^ .05 ^ .01 wt■qr #ni
3.7 <?teT ^iVeiil WsfeRlT (Significance of Beta Coefficient)
'SPTpft^ ucftqwT (B’s) ^4«hdi % '^-'qft^ ^ 1%^ qn ti^ IfTf^ ^ ^ ^-arjqm ((-Ratio) ■sFt fmi Mm ^ -t ^t-
Pl2:34..... mt =®^Pl2.34..... m
^iPJI80
■5I?T ^ (Standard Error of Beta) ^ Wn fiR ^ tl wmif v[^'
1-R?1.2345.....m^Pl2.34,
^ (l “ ,.)
dM<l<W ^ ^ MR'lfuid ^-373«7IcT ^ ^«f4rai d/‘= (N-m) ^ tl
^ ^l^-??5^T®SFSr ^ ^eiii
(Comparison of Two Multiple Correlation Coefficients)««ta5i HR'ifwia '^-
IJOTfs^ % 1^ t^fTOT ■^TRTT tl "at "t ^ITcft t~
p, (R;-R^)(N-mi-l)(1-Rf)(mi-m2)
■5T?f Rj =
Rg =
= arf^
m2 = ^ distil
w '^-373^ ^ ^ fHvfn (mj - m^) Tmr (N - mj - 1) -g^rTRif (df) m
'J'jjiqi rod'll wan«6«1W
mj
13nrn tldiif^ich TRir Tprrajt tf iT*3rtT
(Relation between Partial and Multiple Correlation CoefHcients)^ 3uT?i«h cT^IT- ^T^<RT ^
"sn t-
1-R? - (1 ~ ^12)(1 ”'1I2) •1.23
- (1 “ ^12)(^ “ ^13.2)(l “ ^1^23 )(1 “ '15.234 )
3lff?R» ^ "^tZl ^wiiehl «Ft
1-R?1.2345
^ 1^ ^ % IRI ^ ■3n4ie*rdl t—
'il.345.....m “ P12.345..... m ■ P21.345..... m
'12.345..... 'll ” ^
SnR'icft rRIT
(Assumptions of Partial and Multiple Correlation CoefHcients)3llT^I«h ?I«TT ^W|l«t>l ^ ■’Fpn ^mKW| TpJlfef^ % ■5RI t,
5«nrm. 3Tf^rai tl^lT 3Tf^7W t % t ■^T*ft
■^TRTtnst cRtT tfl 3Tcl: 3TffS!TcFr
^ t % 'HW ■^’Jpnp^-STT^ ‘jwilsh ^ '*TPRn3Tf ■'pCI ^ it t^l
ni • ^3?«T^ 2.345 1.345.
H6T^
81
cR? ^ 3Tft?T^i|«(teh ^ Hi'<<ai37f o|^ I^TR '5^
f^oi ti«t>cii't^—
(i) . rqm«i (Univariate Distributions)(N.P.C.) % "tl
«ihRT
(ii)7T?f ^ il
(Hi) ^-qsftlq Pqa^wiT (Bi-Variate Distributions) % 1^ ^RftWH (Regression) ^ (Linear) %\ '
STftTRFT cI«1T % SRI (Canonical Correlation)^ ^TRT it #ni ^ ST^ anf?!?! (Dependent Variables) W ^(Independent Variable) % ^5«*q'<I ilRT fqjjt^(Canonical Correlation Analysis) ^ TRin ^srcH tl ^ ^
. fq;f<Aq«j q<^n: ^ilPiqici 4iSfl*q*<I ttiiVci^qOj ^ IqHnoc iiq>j< 'fl ^'i^Piqxn•He^*q'<? fqjfci^qwi ^ T?tiHT 31?^ qfici "t f^I% 1^ it 11^ 1%^ '^ncll "tl '
^ra* (Student Activity)1. TOt it^ t?
2, ST^ arrf^ ^pif^ ^ 3T«f we ^srflf^i
3. ‘^wnq»l ^ '5^ ^ "^i
82 3^5717 /q/q*#/”
3.8 ^Emf?r (Summary)
■5n^ ^ ^ TiT^ ^ ^ ti ^ irajR % ■^' ■^‘ m<wR+ 'w ^srrar t^T^^T dcqi^n ■a«?T Tt^mR, w srif^ ■^' ^ ti
• ■3m ^ ct«4’ ^ -risstt^^ ^ ^ sT^mr ^sif •4‘ cr«TT ^«zt ■4' -TTR^rN ^ wsm ^ ^ eft "ZIF ^ M t -f^ -3^ tTSzff ■^' tl
• ■% RWT ^W||<^» ^ "^n ■’R "^eR M<^f^«ifliai ^ ^ ^ HR'ifwia 'RF ^iH«h ^ 'fT<*'di 'll
• ^ ^ '5RINRF ^j«ii«h ■d*TT 37ft^m> ^pnf=P ^ ’’RFRT?en ti ■REif^ -m: srif^ -^ijn^mn itb ^ -sim ^t arfl^ ■jpJTf^ ■qn ^yR>JT ■*tb srf^ «it titrttistiai "tl
• STtRsIcI 'cl*!! 31^ tqdd "% iT*m 'jpnf^ ^ iTH 3TTfald ^^871 "^^eRr ^ % ■RTWT ■RF^F^ ^rkT % tIH (+ 3l«mT - W7R M i^) if stRF) ^ ti
• ■RFW 3Tf%m) 'WO'sj ■^’ % IfR ^ ■foTRntTel (Extend) I^jRT ^^T'^ai f I 3TRm7 "^R ^-■RFR^SRT ^«ii'«t) ^ '^pjFT ^ "^neft■f iieilM'iHi IFft^^rmr % ■^FFTt STsmT ^-i^'i|i’<?j*l ^ ■RFFFT’FRT ■^Tleft ■ft '^- fwfd 'SRftWF ^41^<'J| % ’pTFFf ^
% %R ^rdHid 3TsmT i^iP^iH Wi ^ 'sNtn ■srmr ti• " 3TTt?l^ ‘jwiiVi ^ ^luidi "RFtF^^ % FRI "^neft IRf^
3Tff?F) cf^Ti ^-^FywRT % 37m?imr f't^ ■?Fft ■RWRF f]uii<+i arm?^■iTFitnaif ^ ^
3T^2irH-WT (Exercise Questions)1. ^ trftRmi ^ ^ ^ 1rrtr ^ ^ftf^i(.2. 'J]’J|f'f> Id'^'sHI "^l
oqusHi ■^ORiy.i
^f?Xg + K X, 6 alR K ^ PR2 ^
6. '9eftWF ^ ^ «tilpji»<l
3. ^4. ■^-'StftWF sTTmiT ^ enm5. Xi = fe ^2 +^13.212,3
Tl^ (Reference Books)1. Wmf alR 'jmtn-a/TRR^ r^ ^2. <iiT<S4«tn<i S77 ^7R/
3. 3t!^ ■H*Aif^M-/^V/c7'=/*f li-s^d, RRT^F ST? ^7R/
•4. yif^chcii f^^ui— *f//5-7 /WR)% Rii^n R^F^TT?/
5. "W^, liiliH Hf^4>i(l'tl
R/I^4>?«V /Rfwf 83
3TSirra-2
TrfrfERTr fc^diUI
(Sampling Distribution)
. /•//W^^ (Structure) / //.
11.
1.1 (Objectives)1.2 ywiqdi (introduction)1.3 (Finite Population)1.4- (^Distribution)1.5 Xg-t^TcTTPI 3lk 3^1^ (Xg'Distribution and their Uses)-1.6 . (Degrees of Freedom in Chi-Square Test)1.7 ^ (Hjq)othesis of Equal Distribution)1.8 WTFT f^d<u| qft (Hypothesis of Normal Distribution)1.9 ftd<M| ^ MR«birmi (Hypothesis of Independent Distribution)
1.10 (Summary)• ^ 3<»qi'U-'5rFf (Exercise Questions)• 1J«1 (Reference Books)
1.1 (Objectives)
■5H % 31KPPT % fensff ■2TN, #1-• mRRf! yMRd. ^ ^'i
• sfk Xg-fqcK^i ^ ^1
• 3RT yqiH^
1.2 (Introduction)
^ it t w 3ir<i yfd^MwnRd % yy-^c^' ^ cPnqj ^nciT Mfos^tW % '
(Distributions of Statistics), yfd'^^R (Sampling Distributions) it f, ^^ «eiqai
84 3^537
■fl ^ ^hP'* ^ 3^I^»n ■% 3^^ WMyffT^viT ^ ^TRT yPcKi^l'Ji ith! ^ 1^cr^ ti m: •^rf^ feft ■?FTfe ^ f^-f'T^
yP^'vnl «Fn(■^ TTt^HI'l)
^ ti ij^ ^ ^
itn "Sf^rW^ f^r1i<JlcSMM WoT
”5yT^
^ f’ldl <id^ y^«h "Sri^^ % 1^ P^^hI ■'R '^‘rf clftW RH
'3^ 11H
A
(■“^^ ■r^hi'i) utrt p3ia^
WR ^ f«I5r it t ^^TT 3TT^ ^ ^ . ■^’ oqqR^ f^RFT R^kTI "tl. !lf?T^?f^ Rpit fR t^d<*^ ^ Hfn«iq»i fqtnuj (Sampling
Distribution) fl ,y(aqqi, fqa<wi ^ ^*iPd«n 'STf^ ^
R^RTT "t^l RPTT I^RTl RRfe ^ "t Iqi*^ ""R yiv-ii’eh junsfi: 1,2, 3, 4, 5,6. 7, 8. R 9 ft ^ RHfe ^ RWTRH 5.0 ^«T[ RR^ 2.58 tl it Ul'^^l*(1, 2, 3, 4, 5, 6. 7, 8> 9) ^ ^ W^' 3TRR?f -sn^' cR 36 Mm 3lf^CSrfesiTPH '^) tl m 36 'srf^ 7!8n R1W7H fn=mcp ■^-
it aqf^ ar# ^ ^ f^ -^n'3'\I6^''I
wlaq^f ^ -inch RSZTRH
ylnq^f Mpiqyl VPd^vl ■R®J*TrT RSEfiTRhmIhR
TURra? Hi»<1(«»» Mi'oicnuiKiicn
1. 2 1.5 3.02, 4 3.7, 5.0 5,7 6.0
1.3 2.0 2,5 3.5 3. 8 5.5 5, 8 6.5
1,4 2.5 6:o2,6 4.0 3.9 5,9 7.03.01. 5 2, 7 4.5 4,5 4.5 6,7 6.5
1. 6 3.5 2.-8 5.0 5.04,6 6,8 7.01,7 4.0 2, 9 5.5 4,7 5.6 6,9 7.5
1,8 4.5 3.4 3.5 4,8 6.0 7,8 7.5
1,9 5.0 3, 5 4.0 4,9 6.5 7, 9 8.0
^ 2.3 2.5 3,6 '4.5 5, 6 5.5 8,9 8.5’
1 i FT yPd^VlT ^ TJTRT RKRTT^’ m 3?T^ ■fgcR'n ■30^f^TRl^RR fqd<,^ 'StrRT
RTTtrft-2
3TT^ 3TT^R®nTR *r»wpr •ItqHn
8.5 1 6.0 3 3.5 3\
8.0 1 5.5 3.04 27.5 2 5.0 4 2.5 2y
7.0 2 4.5 4 2.0 1 '
36.5 4.0 3 1.5 1
FI ^ ^ R^^ptpT xPd-qqi 1^WT (Sampling Distribution of Mean) FT^ grff^ ifTTRift 3 i TR^ t cT^IT FT^ ^ Mi \ i
tl •<reqrf< 85
. tiK«n~3
TRZIHH W yfrRnR f«^d<U! (Sampling Distribution of Mean)
7.5- 8.5 6.0-7.04.5- 5.5 3.0-4.01.5- 2.5
481284
K = 36
12
10
■t:4
2
2.0 3.5 5.0 6.5 8.0
1 .
■% yRrciMH t^“(0 ^ ‘^Rf^Tcf 1^cT^*JT
(Symmetrical Distribution) “t 3TF|f^ "f ^ aTT^frnn
^ tfI t, (ii) w ^ iiKm 5.0 t ti ^ ^ ar^ryRi'^i^W , Hiich fq-etciH, ^pTT^ ^ yfa'qH't [qa^.^! ^
^1 ai^HHIir^'+i % 'gRI yiqciT Sf^tTH WTT^ ^ yla'A^^t-fl ^
qi^c( '4' yfcit^^flMj fqo<wiT ^ (Characteristics) ^ Wl
3R?RT^gjT RKmR -HRfe % R«2RTR ^ ^ t cTaTT (ii]lltaq^H fqiK'Ji ^ TflRTR? yiPHcii fqfi<''i (NPC) % "STM 'll yfo'^^'l fqcit'^
^ Tfhn (Central Limit Theorem) % ^ 3I^' "Mr %, ,mPi'+idi '^;5i 3Tc?F^ 'ti r^ztith % %Ti ■Rb=n ■9^ ^
f^i ¥'*<[[ ^1
g?r RKmpT cmr Tn?r^CT (itTFmH arraJiT <n) '% wi^ "*1^ tifrtc^^'TT fq(i<yi, n^" T^TPT"% ^qt'^l W.SI 5)^ "RT, Mifqqxii fq(i^«i
% '?)(Tr '!' Rimchi Mt-MHH p (TSTT HI'icb f^'eli^ c / 4n ^ q^iqf
l
smK (n)% ^ ^ ^
86 wf^ii¥f^ /wfy^.
» ♦♦ ♦ ♦sVn 2sn 3sn-3sVn -2sVn -sVn
f^sT 2. ^
ntfRv^fsT % yRft<^H n4ci<u| ■% Hirieh PqtKri'i ^ Hinqi (Standard
Error) ti a^^t: TqwmB ^ ^ o/V^ % ^ tl ’^TH^ ^ SE 3!87^
% ^ «rtr<i;U ^ -^r^cTm (Suffix) ^ ^ ti sTcr*
ifttn ■% ^ ^ Her^«[ t"! xRrqqi ^ 371^ n% ynfe ■% ■f^RR’JT ^ "R^fH ^ 'll ^wfe Slkii’^ fqni'n fepr^ft ^yefTT ^ ^ Mpmsff % STRTK ^ q»^ ^ uRiqq'i f^d(^ H*TPT Mlf^qiai '^3r % 37^^■STRTT ^1 ^ cl«^ 3 ^^“*ni Isalq, fqo^^l % Hiiq) fqqtni 37^yfir^U ^ ^H^Tfe hiH^ (q-q^i t’ ^ ^ Hi'tef) fq^d^lTRfe ^ ^ WT*PT <j ^ sT^iTH -yf^ %
(s), ^ ^ t, ^ i' "Rfcl^ mf^ 3TSfT (Part) ^ fTTf^
yfrf^f % ^wyfi Iq-q<n'i TET^ ■% tTHef) [qq^ ^ ^ ^ ^ ^i^^iq-ii '^hfi't"!71®^ ^ «q»cii ^ yfa^sd Ht-iqi f^qci'i ^ ^'i't> fqqfri'i =h<^ 3iT^i(Underestimate) "t^l yfci^f % ni-iqi fqq'ji'i (s) % nidq! fq^ei-i (a) ^ ^qTdnST^RH (Best Estimate) '^TFPT^ '7^ "triMi^l «q>dt ■§—
nc = s.
^ Rf^ ^ srmTR (n) RRfRT ^ CaPI: 30 37f^) t cIR Vrt/(n-l) ^ 'RH
. ^RR -Q^r % R7KT ^■^tl^f^SO'^RTR^RPT 1.017, n. % 50 R7 1.010, n ^ 100 ^ RT 1.005, ;i 500 ^ RT 1.001 RRT n % 1000 R^ 1.0005 ^1 n, % ^ RT
. RRfe % RRRT fETR^ % WT Rt Rf^ % RHRT fRR^ RJT tort Mr ■jz ^ RR)n "t RT^ yfd'^Jtl R)T 3nR)R n 'Sfel '?hn "t clR c % RH RR^R ’[fe’jpf "5^ qqlt^TRn RT s^ RRioT
?R ^]nl(n-l) RJT RH RRRJ^ fRR ^ a % aT^RH R^t RRR^t RRlf^ RJ^I SIR: n %
^tel ^ RT R7 Rt *inft> ^ "4 RRfe % RPJRj [qTicii RB T73RI Rlf^ 37RRT R^ RfRR?fIrrbr rr rIrr?! rtrirt) r^RRR7 IRRBR % 375^ R^flfVR RR ^RT qil^l RRf^ % RTRRi’
RT RR»R1 f"—375RB ^ % RR^ cfRiRT
<j«q(i7 «/J«w«u?q ^/VRT 87
I(X - M) \ n-1
3?amT <y =
%
i
■to 3. ■fM'^ ^ ^ to Tiftoff % to •jifton to^iX "Slftof % yi‘<H'^, M ^ 'cf^TT 71 ■'afcto 3n^R f I % HM=b
(o) % ^fsTH 'siftof % tTH^' tom (s) m ’stor ^ "q^ ^ ^^ ftom ^ ^-^dl f-
Wf^T^ /^tof88 7^9f7T
f<^n<a flFHch ■^RniT (Standard Errors of Various Statistics)
Statistic Symbol for SE Standard Error Formula
CTSEj^^ G„.1. VJT
1.253(1^ ^md2.
.707(7a3. Hiiefi fq'qd'i SE„ -^TT a<3 a
__ 7^
.7867(7
■Jii
4.
5. 3n^
SEq ^ Oq 4nVwSE^ ^
PQ6. !><f(Km SE^^a,,- V N.7. 3i^HI(!
V n
18. ^IwUsh*^ ’pifth- . SE^ ^ yjn-l
l-r^9. 'jwi'i't><ri-'3TT^ «eu«<'<^^n=b SE^
y/n-l
^ wgew ?T (XJenominatcrs^ if n fOTT ^TT (n - J) 7^^ WJf^ ^
o #r WFf ITT 17^^ ^ i^^mw (s) w irshr ^ ^ ^ 77^/• rq(K«i HHeh fq^cii, yPci^fKi, ST^HId
3Trf^ % ^ai "t ^ ^ yfa'q«H'i ^ HPT^3T8|fcT fd=hl HM^h W "^iT^ %■ f^-f^ WK^Il 4 yt^ci fen W fl
"t 1^ MnctJ fe?it nT^rfe % ■ferm ■*TFT^ ife felt ^rfek irennH. 'r^^, ^ fera fef % fe^ ^feldi^ileicii ^ 't‘1 Hidet) jjfdni % ■§ f^ '^fet Tmfe fq-qd'i^iidd!37«ifn % TWi^nnt (i«n ^t?n ti««t»ai "t 1% MidetJ fq-q^ri'i % cl^TT yRi'^d % 371^ % Hi'i«h ’^^ncft "tl
*77^w
1.3 ^rfrtMd (Finite Population)
Hi’i«ti "^fet % fej yRuj) 4 qf^ld STnftfe ‘TRfert (Infinite Populations)hRi'^ iT % ^1 TFlfe "^Fn "RfeTf '^tZ\ Btfll f cR ^ Hi-lch
cTS^fTT 89
3T5*TH ^ f ■SR ^ ^ t ^ ^ "STfR■3f^ ^ ^1 "irafiT TfRlf^ Iq^thl’ 'SrfcK^f 3JR: ^ ^eRT ^ ^ f ^®7F
^ M^Vllt*^^ 313^'VRf f^WT (Exist) ’it R#' T??ft f I ^fliMshl % 1V^fl 'SlfrR^t ■^ '!ITR hR^ih) «(nn=hl ’’Tt ’it 'dPJ^ "sn "^RkTI ■§■ "f^FRn ST’tt 'J1-h’it -gaiT ti ^-HPaij. aiftRrm fterfNt' ■^' 4 ■4’ 3i^ 1^ ^ ttprt ^ iratn 1^[cbifl ebldHI^ ■% 1^7^ 'STT y«*dl 't "SR hRRici cT^ llfcR^ ’l^lfR
fR yiTtS4<=hl4R<‘('°Ft ■^RPT "5^ ^ STR^^RRT "^tcft "f I q«^o: 'iRfHd "^RfelTt ’T^ '5lt%^?Tf %
■sn’RT '5^ '^7^
v/<7'?<(f /qcT<W ,
■5ffe
HH4) ^5^^ R % 31TRtR dl<r=b % 31R>R31RRT ^ ^ ’it TJ’Uf^ ^ tl 31RTH N RJt ^ftpR 31R7R n "% 'STf^ %■JT«RH Rft ■RPR) ife ■RrT ^ W R^t sn IRTcft t-
N-71 -a
■SIB? <T ■RRfe % ■i^ ■RPR) ■fRRPR
4M<I<W ■^ ■^ 1R^ ^ ■f^ ■SR N = n SlRfcI RWjyf HRfe ■Rlt "SlRR^f % ■^ ■^ fR Cj^
■RiT ■RTR ■^JR ■^ ■SIRH "tl 'SR N 'R)t ^jeFTf ■^ n ■R^ 'SteT ■gt^T 'RR ^(N - n)/(N - 1) RR ■RH ePI’lR
1 ^ R<1R< alRT "I ^ ■^ 6dNI ■SIT "RRIRT 'll rRrPTR: Rf 'R^ ■STTRT "I “Rt STlRfRtT■RfRRTlf ■% ■%t3; ■ftRI f I
1.4 (t-Distribution)
■R?lfR ^itRf (Central Limit Theorem) % ^ Ri hRis=!iT % ‘SrfRRRR fRcR’^ R)t IlffR ■Rit "Sn ^RRT t ■RT^ ■^ RfRR^ % ^ ■^^.■RtRT ■R^ % ai^RR ut^RtR
df = 00 .df = 20 ^df = 10
R)fRRR *in)Ril {df n) %■ %t{ ^-Rid<.«i
■Nr 4
[qd<^J| ■^ ■RTRPR Rlf^RRT RRT,% ^ ^l^hK RTPIT R^RT ^r^ci ■^ slcil ^tl 1908 ■^ ^■■Rf^ (William Sealy Gossett) 'RTRT ■% RUfR R^ 'SrfcRJflf ■% R’RRPlt Rvt Irwi ITTRPR ■^^ ■^
“BtcIT % ■fR)^ ■^-■^ 'SricR^f RR 31TR)R 'Sfel ■^T^ ■SR^ ■f ■^-‘^ TlfRRRR fqd<.«i WTPR [dm«i ^ ^ ■gZRI ■SORT t RRT "R? 1ROT aif^ ■gRJtpR (Peaked) ^ sipn tl ^ '91^' ■^' 'SlfRRRR tRRlIR
(Kurtosis) R^ sntft ^ fsra^ ^ aif^ qsuli (Leptokurtic) ^tRT RTRT f I Rf^ ^
(M - p)/sVn RR tRcRR tRR tR>RT ^T^ ^ IrRRst (i-Distribution) R^ tl "RFIR)t RRkIT
tl^Re RIR«T
90 RpRRT /RfVRf
sTVi-f (Dublin) IWrl (Guiness Brewery) 4 'iDia?f %^ ^ ^ ^ 7I(f ^ ^ «ft «^>4Tj|(l Hm ^ itfu|^ imyehiP^m ^ ^ 8|ti ^ff^ ■# (Student) % ^ (Pseudonym) ^ y^rfvid3T^ -q^r (The Probable Error of a Mean) ^ ^ IR^ j ^ 1926 ^ 3TR. f9>yR (R. A Fisher) ^ % SRI t f^?R^ 3R3(T % ^IRT ^ 94^^ /-■f^cR'n ^ ^tni ^ t Iqo<«i (Student's /-Distribution) % ^ ^ ^R1 WT ^1
^ f^cR’i? 1?^ Pqq^yi (Continuous Distribution) t ^ ?:q ^ + oo*^ tl ^ t = 0 % ■RRf^ (Symmetrical) t ^ sfrf^Rp ^ %
srftTfteRT «bt'+>t ^ rR? ‘^iRT!^ ‘STlI^Wcn BlcfT ^1 lqa^«i ^ 31l«6K "% STRiR nRT ^ fqf^5l arraJRf % mRi«^5{|T % /-IcRRHI t^-f^ ■f’l % 3TIeFR
9iP(«6ai ^ % 'i'ji'0«6 srrar ^n?n ^ cwr n %^ ^ ^ tl P ^ ^ % -SFRq ,
^-f^RR'tft ^ 'SR^ 3RR*ig %\ nRRfi'* 5 ^ ^ ^ ^-fqa<.«i ‘HROlttii4«tidi % 1^ foff^ (df) "^R / % RH1^ "tl ^ewRI (df) ■*2;^ eeh*il«h1 'f tRRRfft RR! 3TR1 ^ rI tl TR f<BI WIRT RRIrI t RWRTTRf % ^ df(n ~ 1)i^h
nfa'^<ii /q(T<w
n ^ RTT^ RT f—IRcTRR RTT ^ RTRT^ ^ RT RF RTRI^
Tif^ % "gRT ^-IRcRR % RIl^ RT 37^ FRRhit R>t ^IrT Rjt R^l STT^f^ RRRRTf % ^ 4’
■% ■?RB RT RH FR^ 3TfUR> FR^RT RTRT RTTFT "tlRlfRRRTT ■^RTTRIRTrf
1.5 X2-f^m«t aftr ^«ehi (X^-Distribution and their Uses)
RTR ‘R^TRR RTT RRIr lRiRT RTT TFT tTR 3Reio*fl ti4®h (Available Data) RIRTRTRi R^>RT (Quantitativ Nature) RR ^ t RRT ^ RR (At least) RIRR % 3RlfKRr TRR (Interval Level) R1RIRT RRT fIrt "tl RT^ ST^RWT R>Rf ’pTIRR* R^Ir (Qualitative Nature)% Wt^ FR^ ^ fI ^ RTf^ W (Nominal Level) Rm Rif^TR 'RR(Ordinal Level) R1 RTRT RRI fIcTT "tl RT^-R)^ Rfll^la'Ji-R RTTTRI ^ SRiftcI "RR R^ RI^^ IFfUR R1 RftRf^ R»tA RTRT R^ %\ R^ HF RTRai ^ “Hlltld RT sblRdRT RRi^* (Subjects) RJI fRURlR iRRlt fqRiioi "^pi 3TIRR R7 RRF (Categories) fRRRR 1VrT WTT ^ RRI tRfRR RR? '4' 3TT^ RTR^^qf Rf) ^'ohi R)t,R^ Rit% STl^ftl^ jITIR R»t Wft ^*1 snfiia IRT R^ RRtRTi RTI fw R^ RIFT Wn f RT ^tRRT RR RT-dfifen RRT (InherentOrder) Rt ^ tl ^ RR^' ■4’ RTF! "RT RRJRT t % RlfRR RRT RTfRcT RT FRW RRR) RTR ^
^ ^ RT srfRRT RRf (At least two or more than two Categories) ^ 37i^f^' % FR ^ ^ f I ^ RTI 3TRRT ^mifRR>-3nf5TR7- TRt, 3TRRT 3TRRT RTI^ R^RTR, SqRRIRfURR 37RRT oqfRRtR RRTR 3TTfg ^ 3TTRR RT ^ RRI IrRRR R^ 1rRT (count) Rt I"! FRI R^ ^ RKlRdi) RtI <M^fa«b Rvit % RfR Rleq-sni, 37RRT RIRT^fRT rRR^ % FPR^ ■4' RR. 37Rcn ^ 3TRRT ^RRRTR ^RRT RTRRT RTIr 3TT^ % 3RRR R^ f^RR RRf ^RT ««hai ^1 RqT ^ Rf^Rf^.^srlRR^ Rfl Rt) FR RrF R>t 37I^[rrT (Frequencies) R»F^
RRT f R%RI5R FlR*^ Rj^ ^1 SR^fR^ % ^ RRcT^T fR RR>R % «hW1 RH tR?^RRt ^ RTR: RiT^-R^ Rfl^
(Chi-Square Test) RR RrIr '^RT RTtTT "tl Rnf-RfiRIR STcRR RFrR’pf STSTTR^iRr RTR^TRjIr
RRidT
RT ««hai
91oW7<
(Non-Parametric Statistical Technique) «^qn: (Discrete Categories)(Frequencies) % 'WqfRl^ (Most Appropriate)
q^dlqci H^'W (Widely Used) ''TtleiRT "ti ITRlf^T^ fqijiMl %H6Tq ^8^1 ^L|qVi o[T^ 1^-'^ Hcq^l TfPTRt yif^fli !
T^-''TTt^ ■% «*i5j<rq tql'hn. ■^ffcTT tl
W\i (x,) qiwq "4 (Greek Language) 3T8;r (Latter) 'll 1^cR^(Chi-Sqnare Distribution) 1875 (Helmert) Ml cTKIT M TT^ 1900■4 (Karl Pearson) yRiMim T^mi 3Tf^ ''77 =bt^-'^
(Chi'-Square'Test) ^ cT*noin^^Tft^ sh<h1m M’ W PqMqi ■gT^cT M',
% <ii«ialq fq'^qi 1Vqi ‘STT T?T 'tl '^iT^-'^ qrl^'ci^^d:
^qeiln^a [qd<.«i (Observed Distribution) ^ "5^ ■gftoRc'CRT % 3TTMiT ‘'RMRqilVid Iqa<.wr (Hypothetical Distribution) fMcR’^T (Theoritical
Distribution) 37?!^ xr^ifVicj fqd<wi (Expected Distribution) Ml "I, M ^RriT fi fR '3^ M "^Tar ■^rrar "t ct^ MR«t>RrHd fqd<.«i ■% fMfMR M ■R«rt ^TT^i^rqt M tie^Pd(Agreement) 3787^ (Divergence) faRT MIrt cT^ "t^l 3iqcilRdd iqd<^ ^87Tfqd^.'Ji "aft Sii'^Puh'I % ^P'ledlP^ 3RR TnM^iRr % «f>l del'll ’^Sfrat ■fl 3RT:arr^-'SFf trI^ % 3ri w 1^ t ^ ^ 1^ "W 3Rf#f^ teon RftaJvw % 37R7R RT fm M Wc^lRld fM^R^q ^ ^TlMar ^ 7I, f^ f Sjmi TR w ^ IJ^nTRR M "SR gR<+i(rMHI ^ eql<j>Tcl 3787g|T SRql'jifd aiRTf "t f^TR% STRIR tR yrqiRid [qd^^i ^^R
fqiqi nar ■§I ■jqTFRR Mi Rfq f^Rt RTRlf^iar Hp<qdd % RfcT.^sud lOO rMRMI% -gRiqM M 254 tM m^. 30 M r?rRt, 12 arRriw, I5 M ^trfr^ ctsti is M3i<H6Hfd oq'^d Rill Rfq Hp<'h<rM'il RF't 5TRfe M aft ^ifdf^qiq RTat aMf M RRPT RR 'MfqdRd t’ •sRifcT Riat yldf^qi aMF % %q sn^fRar rrr t'l |r wRedc^ii stwr rt Rrat aat M20'20 RRRa '-diP^ Ml RT^ "I tai R( ^q^rilRh^ taaiRT aft sFIRT: 25, 30, 12, 15a 18 f 'JiqRti y^iRid farpRi aft sn^faaf arq?!; 20, 20, 20, 20 a 20 'ti rtr af f i^ aai sraeitf^ lacnar aft aar arqilVra faaTRi aft aa^faat M strr afaaaa (SamplingError) % anra "t 3TaaT aRaa M strr fi fRt aaiR M afa arjRanaiat aprai ar^ar ar RT afaaM ■M ara Raar rr M faafta f', aa at 37avitfaKT 3713% taaRR aft Riara laaRR aft RftaieRaT R7 aTiaifta RWif^ia taaRR M ar{M> M^aai ■^Rn % atat taarMt M ^fVMlPdd strr Rfaaaa Mr arRa Rilaar ^ Rarar t ^raar aRaa M 3rr ti RRt RftfRTfaaf M arr^-aM rMst^ an Rata laTai ar Ratar ti rrs f far an^-ad RMajiTT Mr R%a strr aft RiMaRn an rM^
■fi irM an^-aM (Chi-Square) aiR Mr 'j^nfar aft Raar aft arat f, faRan “Tja ■Rnaan f-
^ Mr arRa,
tt^aartarRt
arrar
an^-aM, =fe
a^ /q taRtt aM aft sraaftf^ STl^fa (Observed Frequency) ■! aai aR aM aft yrqiP?id aTl^fa (Expected Frequency) t RM 2 ^ tMf Rat anf Mr (fo^O^^fe "9^ ^ Mta aft ^Rid arTai "f 1 anf-aM Mr "^a M rt^ "t fRan rh sTaaftfara aar yr^iRid sTi^faat Mr 3rr Mr anf an R^aRaa RcaifVia STiRfaat M ^i^nidl an atn atar ■f'l sTaafttara aai Rcaifara sriRtaat % 37^aM Mr ?tM RT 37af3 3jqcitRbn aar Rcarfria ^Ti^faat Mr rrh atM rt an^-aM an rh aja ^ anai "t aafar
92 a^cgna
■ ^ -m ^ ^ ^ 11^ ■^-■^^ T3^ ^ tUTc^t^, (+ve) ^ f'l sm: iTR ^ 3?a(^ ■?Jp!T ^ arfe # ^
■§■1 "4 cT^ yrHif^w 3?lo[1%^ft' aRTT % (f^ - ^ ITTf^
^tcTT t ^ 3T^ ^ f^' (Direction of Difference) ^ 'll
■^T^-'^ ■'TftnfR^T ^rrf-'^ ■^n^fcpcn ort tqa<.wi(Sampling Distribution of Chi-Square) % 37M^ ''R (In Terms ofProbability) fW>*n «n?n "tl ^ yRi-qMi ^<^ci -^qdlf^^ fqa^wi %
{df&) ■'R f^*ft '^Rn "tl '51^ «FT^-^ ■% ■foRRR "'R 311^■'7^ "tl ■g^RTRlf % ■'R ■^-■^ ^ yfaqq'i ^pnctR? fMMq
(Positively Skewed) "'R^ ^^iViT {df) ^ RR «n?n R? RTRPT f^cRR Rfl37k U^ (Tend) ci'icii "t^l fsrfRR ^qdKl) (d/s) RT fcia<«il % ^RT«ta RRf RT RR|-RTf % ITR -qkkT^ 8 k R^ ft Rf^ mR'IPuM RR^-R^ (x^) ^ RR ’TRRRrR ■gRRmf {dfs) RRT RifecT RTsIRkR RR RT RRRT RR k RTR ^ t RR W RR|-R^ 3f?RI«tR? RJFT ■RRT ■f I 3RTT8JR5' RR^-R^ RTT 3T8f ■!■ 'STRRH^R fRRRR RRT RcRlfinR fRRTRT % RIR R»t fRRcn RRIRRRT ^ RfRRRR % R7RR RRl RT
J^^RRR /q<j^
R^oi
^ RR 37RR\ftRl IrR^R RRI RRTifRR 1RRRR "% R'RRR»aiRfRRIRR: RRfe k IrRRR
RRT RR Rft^RRT R^t, tRR^ 3TFRR R7! VrqiHlia 1RcRRiRiRI ^ RRhIT
Ri^ yrqipfict fRRRR % 37^^ RPR RH RRiRT % RRT RT, "TRIrTR fRRT RT RR7RT^RR/IRTRI ^1 iqHdfl RfR nRHiPwici Rik-R^ (x^)
■gRRkif (dfs) RRT RffeR RTKfRTRT RR R7 RTRift RR k 3TfRRT ^ t ■RTRI "t^l Rn|-R^ RiT 37^ ^ 3IRcfl1^>R RRT yr^il^ici fRRRR % RIR ^fW'^lfqa fRRRT RTl ■RRtRR^T kI RfRRRR Rfe % R>T7RT 'TRtRjR R^' fRiRT
ynP’RRRR RTR
RTTf-RTff RT^ RT«fR> R)fToR mTkT
■f T^R 37Rcft^R RRT ycHiPfia fqd^'Jl %RT R^oTRtR 3RR^ R ^ RtI RtfeTR^tR RftRT^RRI (H^) R^ tRT^ IrIRT RTT iRRRR RrRT^TR IRRTR % R^ RTRT RT RRTRT "t RRT RftRTcRRT RTt, STTRR R7ydJiPvid fRRRR ^RIT RRT RT, 37?RlRrR (Reject) RTT
■fl RkRTTRR: k«RiRT
tl^ Rt iirqif^ld IrRTR iRT^ft Rt Rft^cRRT % 37TRR RT cIrTT pT>qi RTT
37R#%cT l^m«i k Rit RT ^T^cft ^ R7^ RTRTTRRT:
37T^ftT 1rRRR ^RR RT^ 'tl ^ Rfl«P(rMRTt( "f-
RRkTT "t RRT a««tj1 '5^ 37)’ % 37TRR RT yrqiHfid
(0 RRPT ^ ^ f^RkrT ^ ^ Rft (ii) ■RIRT^ ■^R fkRftcT R^ Rft
(Hi) tqn-q ^ (qaRd R»1 Rfr
RRTR RR "k RRT ^ihM RR k PqdP<d R>t yRqicRdl^ 37RRT RR fRRRR
R^l RkRiRRRT3Tf RTT 37*ft^ RF ^73R FtRT % RRl 3<q<nlf«t>d tRRTR R?! ^*qf-RR Mp.qj'rMdl % 375^ n^dRd RTRT RTT RR7RT 3I7RRT R^l "SIRTR Rf^ RftkRtR k «RT^-Ri qjtdq k dHfftMrtll 3iiqind Rfl^Rr RT kRT^JcTT R^t^PR (Test of Goodness of Fit) 37RRT ITF^Tfif Rf^^PiT (Agreement Test) f RT^ 3Tqeilf^d lRcRR/37T^fR^' (f^s) RRT ycHiPdd lRR7^/37T^fRRf (f^s) %
RTP7T «*in (fit) ^ R^ RTRT (Goodness) RT*! BIR.RRdT ^1 ^qn-q RR 'k fqdRd R>t Rklft
qjvH'U
q>VHdi
% ■fl ?R SR>K
RftRTRRRI ft-RTRf IRcRR ^ ^ 'M tl RIR 1^ R^ RTR ^-RtIrR. RTfRR, ST^RkR a7RRT 3TlHlPd+ Rt R7 Rfe ■RTRI t RR RT^ 37T^ fRRTR RJI fg-RT^ 3TRRT fg-RT 3?T^ fRRRR RT^ tl
3TR R^ Rt % independent)^ 3TRRT WFJK fR*k (Dependent)^ Rt"k STT^kr^ % fqdP<d
HWi\ tqd-q
Mp^qicrH'ii ”5^ RTR RPRt t RR yrqitVfd ,37T^fR IrRFT '^Rlf Rt ^ iqd-q RR
HiTisn'=ti)<i f^fVRp 93RwRfR
% 3TTVR ^ "Sfrcn tl ^3Tcfeft%tI Ig-’^TTlf f^d<wi ^ ■% 'Mdol Hr^iHfid fqd<wi % ilpn^ t •3T8T^ ■^■| ^ ■SR?R ^ WH'^I (Test of
Independence) ?tc?T f ^ ■^* ^ ^T£?r ^psRf (Association) ^ ^ ^ '^iTfll
% Tiw 3Tr^ fctdi^il ^ ■^-■^ ^rfNnn ^ ■fsF^ ■sn 'H«hdi c<^d:^TF^-'^ ^4)^1 TRiTtRfC ^ fqdRd ^ sRcdWil ^ f^'WK
(Extension) 11^ ^ fPJ^' % 1^ 3IT^ m W t fI«qT 1^
STT^IR ■^’ itm.%. ^
^ ^Rdjcrwd) ^ "f 1 fT®?] t5rai3^ %% irfd "TO^ ■% 31T^ fqdi'Jil "g^ % ^ ^^^hdi fl
ImM (Sex Difference) ^ ^ i|H^, 'm %, ii% % 1^i^-iTTHf 3Trgf% f4d(^i ^d-^ fqd^yi ^ Hi<«tJ<rM^i ^ ^ ^J^dl 'll ^ ^
cod’ll "% 1^ MRePwid ^ cfl?^ '?tcn "t "fe % f^cR^ ^ 3P^
t ■qftTTfoTcI ^ t ^ ■?P3^' % f^dlWf 3RR■^tl
if/(7^</'r /^cf<u/
"RR ■feRT ^TRIl "tI dc^-ejiq^ ^doi
5RI ’T^
1.6 M<1*^ui ■$ ■^WfVT (Degrees of Freedom in Chi-Square Test)
otl«|ld f4d<'j| % ^ iRgd WebT % 1^ ■^©n oRf (Categories) ^ dom% ^t^it't PAdA ■'if^ % 4)4! % ^ "A sijAiAd "^igi "^pg^
(Arbitrary) 3ng1%qf '^Ritft I'l ,% At# ^ "t, 37igf% f4d<wi gsfi-TTTAf
(One-way) rT«ft fg-Rpff (Two-way) ^ #1 T33;wff fqd<u( A’ Aicfd
t ^ ^ •icAlHd "4^ "RR 1wf^ ^ (Categories) # ^nf^Rid 3T(gi%At ^ AbT37igf% (N) #t?n "tl ^ 3TTgf%4t (N) K^gf Af 3ngt% f^cir^ «1'1ihi Tnn ¥tcn't era gbi (N)'^ twe gg A>qd-(K—J.)^Af "A i^srgRR an^fAAT airafei ^ ^ Ararat "t srf^ ^ ^ ■sngf^ ^ ^ TrarR # ^rarat^ ^ KraAf airgf^' ^ AW N^rarrar #t Wl STd': TrajR ^ fwW A gefdj^it (df) ^ (K —1)% 6i0“i< WAt #, wAt ^ «<S4) ^1 "WA HbA) aiqcilfAid fqd<'J| W 400 A^ira % 3^1^ AT ATA AA? W TTRAt 1 % 3igAR fAdRd f Tra'A^ AT ■g^WAf(d/) AA AH 5 - 1 = 4WaT AAfWf STgAAMArat Wt 400 AH aW AaW flT %AH AR AA? W AAAl^l 3TTgi% TT3^ A»t ^ W TraWt f I aN#' a4 Aft 3AgfA Wt g>H AW #? 37^^ T^ ^1 A^ AT AF «7H T13AT-Wat 1% aW Al# a4 Aft (d/) W AR^ TTAA AAf Aft TT^W A^’ Wat ahi ti
TTTTAft—5
400 'snif A5T 4lchlA1MlRd 3TTff^
AtAleqrra TTAJTA ArfApATraiTA tJChid ■f^ TIATTA.^HTTraiTA
70 4006080 90100
tlAWf 3TAAI IIaT sngfA IaATA "4 3TigfA4T aIaaAI AAT of® TA’Wt W f^dRd FWt f I FA AAAT % fAcRA #■ A ^A^ SngfAAf AH aW ATA -Mm TThW A AfAAAf % aW ^ AT?! ?A. ■# STRtfWl AAA ^ tl AfA 3AgfA fAATA’t aWaW' A?t Tl^ (r) AAT TA»Tl' Ait TR^ (c) WW t AA WlA Af^' AAT TTRAt' ^ angf^' % AWf Aft f^ go; %AH (r - 1) (C - 1) AAft^' Aft (Cells)
94
■4' ^ ^ ^ (r + c - 1) U^tM' ^ 3TT^|f%^' ^ W T^nT f^(fM
«f><’il ^1*11 'fe hT^<<1 cM 37T^ftT^ ^ 3iqcnllVa 3TT^^3T^ sm: ^ WT ^ 4' ^<4dlijlT (df)'^ Ttm (r-1) (c- 1)%r hI^'hI ^ c ^ W<SHi ■f I l«t)«l fg-iTFff Siqciir^ci fqfi<ui ^
cR^ ■%
t, ^q^iq<
cT^TT "^iHif^eti-^TTf^T^ '^■fg-'RFff fqa<«i % y<+,')tqj1 x(
500 'HK‘41 164 ■% f^Rd cR tgRPlf fqaT'Ji % (df) ^ ^iNshi
(3 -1)(5 - 1) = 8 #ft.
•mnfr-a■sri^rR^-Tranr rTSrr lTRTf^T^-3TTf&^ ^RR % 500 oqfcw^) W STI^f^ f^fi'<«i*
qincch "^fTT
3R?InT 1^ 4T^ 3TRrrT
304^ 25 20 30 120•15
■R«m 40 45 30 40 18025
•f^ 40'30 50 40 40 200
90100 120 110 50080
«wlT^ 'ST^^I'lchnf ■’if^RRTt cf^TT ^ *^'11*1 ^15 ^qci 8 "3^1 kil
qiqi^l 31T^i% T1§ ■^RFcfT 7 ychi'll ^ ''li^ «a’1l "^t^l
■?T^ 4 -zrF K^TH jmi ^ -etW ^ -d^n ^ w ^ wi Wn■=T^’ «nrn ti •'./
TRt^TJT % #EiR (Steps of Chi-Square Test)
TCt^ ^ (Chi-Square Test) TF^ yf^qi (Process) ^ flH ^lynT (Steps) '4" ■^n "^4bdl f-
(i) 3TT^f^ t^cRTJT % ■^FeRI '^' HRqirH'ii IhmI’JI cfR^TTI
(ii) 3Tl^f% 1^?ROT MRehc^di ^ 31FIR 'Tt ycqiRid Sfl^qT (fj ■qrt W «IR% yr4lRlld3Ti^fd fqa<«i ctRdll ' •
(Hi) "RffeT^fh? HRehfpH'ii (Hq), 3iqeil[^d fqa^wi ^r^iRid fsTcR^ 3FrR f, ■Rrqfji ■^rwi
(iv) • 7<R "^R^t
(ij) tjqdRiT (df) ^ ^ ■^R^l
(vi) qiTtsn TTr^fsiRTT tT^ e*qRq^ ^«kiRi) (df) % ^ <:iiMl IIH tH?! ^R=ni
(vii) Siyi'Rbd ^ tFPTT ^R^—
^ (f0~fefX^ =
fe(viii) yR^iRid (x^) ^ TTT^f^Rn ^<aeh<,, ^RcucTHdi ^ «ql'5»fd '^1 Sl^qltjiid %
Wq ^1
(ix) '?!3^ mR«6^’11 % STMR '’R ^ 'MR'4>trM’it
^ 3T«1^ fRRl ^R^l ■ ’ ■
HiTis^'^l^ 95d-c-ciai
^ 37Fr ^ ^ 1^ ■q? im ^rc=iT<sf^rt ^ <sM<l°Kt ^ ^ ^ ^JS^qrfy wReRr^'ii % f^f^i % 'h'Iwh
'SrFf: ^fWxnftqj Hrioi ■^TTcTT ‘qftfiirq fqn<yi 'qft HRchcrHii 4l<.«hcrH'ii
■q^T # ■qm ^ 3mn-3T^ f i
1,7 ’^TOH f^diui MR«*i?^dl (Hypothesis of Equal Distribution)
■qr^-'qr^ 3T5??%iMcbdf wcT ■q>T^ "qii "t 1^ qqr ^nf ,^ wq ^ 4^1 %2n ■3n t 37q^ ^ -qTFR ^ ^ f^-f^ fl W 7^
qjT^m ^RTTcjci^l 3icitnintia f^cRtiT (Observed Distribution) ■qjrf-cTf 'qft^ % 'gHT =h<'^ f^qi
^T^kTT tl TRB n^d^ui ^ mRcH'ctMHI ^ yjRraidl sRcticTH'ii (Hypothesis of Equal Probability) ’it 1 T^ nRchr^'ii % 3T^ fsff^ orrff (Categories)
gn^PdqT f I ^ TT^' ■jf f 1^ ■4’ ^i-^Pd’qT ■gqf -^itir P^idPidcfft ■qfrqiRRT qf! RRt 5
■% '^’ ’it STT^frPTf "^RR ^ IqdRo "^t^ ^Pey., 1^ ■jtRrpr ^ % ^jRR,
W^: %n ■^tcTT f, cR STJ^MWcTI '^it ^iqcilPhO 311^1% foRRR % "^RTR fqaRd "St^ "qf!
- MRqivMdi ^ "qilOT q^t sTrq^qqjqr t^s^Rf qrif-q^ qit^ % iro SRRtf^ qqr■ yrMiRm [qa<yil % 31^ "qf! 'W^iqrqT qft Rtq qft qUclt "tl qqff^ 3TT^Rt^ % whm ^
RidRd qft MR'tKrM'ii qit qitq qv^it t, §^Riq. 'Sjfqq^ % yrMiHrm Iqa^'Ji % q^
(Categories) qft yrqtRid ^i<jPti<iT (f^) "^RR iHt Ri % iRq, '^R (N) qit q^ qit WoMI (K) ’RT. "f I SR: RTR fq^RT qft qftqfRRl"4' ■jrRTfTiq fqrRR % ‘^’tt-q^ qft ■jRqri^ sTTfRiqt' (f^ q^T'RR N/K.% qqqr ■^Rr 'll TrqrfRq fqqrR (Expected Distribution) qft qJT^ % fqf^ q^ % fR^ /q q 4 3Rp; 3^qfq^(fo -/p w qq%, qi qq t qqr /■^■^ ’RT ^ ti q’it qnf ^ ^ 3™ w^ (fo-O^^fe^ ^ ^ qrr^-^ (x^) qq hr ^ t f^Rrqft ^^fq^q df=K~-iw: qqf-q^ % TTT^fqi qft qTRit qffeq qrfifqqn rp qr IqqfRq qft qr ^ctial "ti twr fq^TRi qft qftqjRRT % qft^ TFjyt ■srfqrqr 3q7t qqiFRif qft qFrqqr qft rt 't’l
-500 qqr qfqq^ ■4' 30o- r^ q 200 r^Rf^ Mfi qqr qqfeR5%-R^fqRit’ % fqqqn qft qqR Rq fqqRq qRT qr qqiqT ^?
RR: q^ qRqiRRT ^Rr q>\al.
'‘g) RR. qiT^
qqTsTnr
fqa<,«i qft qRqiRRT i", ^THfciy, ye^ifVld fq^TM ^ RS% q R5fq>^ qft•^—qqlPfaq^fw RqR qT%q;i soo to t, ■^' ft q^ qft q^nf^ 250-250 '^i
RhR
3iqdtp4)d qqr ycqiRld fq^TR % 3TRT qft ■^«iqRT qq qtt^ qqf-q^ (Chi-Square Test) %qi RT TtqRT tl qil Tt ^ qRq^RRT ^ Pqd<uit‘ ■^‘ 3TqR q^‘ tr3R:-?pi qftqTRRT
(Null Hypothesis) qft R^qr^it '^" Rpq qq fR^q Rqr^ f-
lio- • 4■qrsfqRi RR = .01
^qqfR((i/')=K-i = 2-1 = 1 .
tTRvft qR, (i/= 1 % i^POl RR Tt x^= 6.64
wfm^ /qf^96 <i^a<
^ yr^lfVId fc<d<uil’ ^ ^ tjui^ ^ '^311^1^ WT ^ TTRoft 3 '4' 31^ tl
-HKyn—7
^ ■^ihht ■JTte
grf 3l3[HricF?T ■f^Mr«4i(fp - e,fSn^fri
fc(fo-Vf - f ‘o S fe
^0
300 250 50 2500 10cT^f^F)^ 200 250 50 2500 10
^ (/o-/e)‘<+4T=1) - X^=20N = 500 31^:
fe
Wfqi nR^rwui 20 ’7H (£/= 1 % .01 ’HT 'HKW|) "iTR 6.643R: .01 ■'R 'tl 3R: ■’Tft^TRRT, 3iqcil(^a 'cT^n ycqiHfiii Iqa<«iT "4
^ t, ^ .01 ■'77 f^C-fd f^qi "^JT tishdi tl hR^jiIha: ^hm fqd<wi ^ HRchtTHii ^ ^ Pi<w37R7■^RTT tl 3R: '3TT '^RRTT t WhRc t 'cT^ dsf^qT aTT^f^Plf ITRR <<^let)K ■^’
^=hfl1 tl
1.8 'HWPT f^m«i ^ MRch^-ii (Hypothesis of Normal Distribution)
't "JIR: 375^^M«t>fi[ '^TRl 3Tra?^Ri^ t Rh'41 ’’R 3IT’R
7TF1P7 "yiRrefrdi (N.P.C)'^ 373^ fqnRa ■^qlehK ■f^RTl ’TT^FdT t’-ar?!^ i{ 'HTRRl
5nRRRT 31^^ f^cR^ HRq)C^il
RR t 3iqcilRhd rqd<.w| tl nR^Rid
'MH chl
■Rrr^ ’t ^ ^ tl ITT ■SlH %1 ^ TTmT^% 3TTVR ^ "t^R ■’it yrniRid lqd<wi (Expected Distribution) ^
(Observed Distribution) "t '^-'^ ’’TtST'^ ■% 'gRl ■^R% 1^ THdidlRH TTT^ 'tlfll t "cT^ rR^RRT, 3iqcilRhd Rid^^l cWf TTWRl fqd<.'J| ^ hR'+ivS'II %
3TmR "iR "tlK ^jriflRld fqd<.w| t- 37RR '^Ttf t, ^ Rt^td 'efR "^STM t■f^rPTOT %
yqiKi'di. t ■?RTcT "^RdT t 'TTR?^ ■RRl ■^n TT«6~dl.tl RimOh HR^Rid3iq<rilfq)d lqd<'J| ^ TTIRPT
^ RH 3TRT«f^ t^cTT t ^iH^qd 'yR^cr'H'tl ^l«t)K ■^- '%RT '^STTcTT t 'SRTRRR t '?Rlcf ■eFRdl t "f^ aHqdlRbd lqd<'J| ^ TTRTRT fqd<'J| ■% 37^^ RHT ■Hdidi tl RIHM RldT^i ^ ■’TR^FRRT ^ RTRFZT yRqqidi HRctic^di (Hypothesis of Normal Probability) ^ t l ?R■t dld74 t TTRfe t yiKii’q) TTTRRT yifqchdi cR> 'Sfft TRf "t IddRd 11 TTFTT^ 1%1'cRU'l ^ 'TR^TRRT
TR tt -nqial t "R®!. RRdi ^ ^TR stjfHd RR (Ordinal Level) 3TR^ STRtRcT ^(Interval Level) "'R TTRrf^ '^'l fR
t'-(0Ril nRshcryii-'^ ’iRRr^RiqT t 'TFRR tl 'Hqidl
■RR RR^ q^if'd<l (Class Intervals) 3iicjRiql 'F’71 oqqR^ "tit t 3TRR1 (U) RR RRSfi 3TRtRr rr rt Rt 5biHd 'Rt! (Categories)Rt 3TT^j%Rf f "tit t.i hcci! RRTTT rtI ■'TirR^TfR
Rl t qqf^ yq>K 'Rfl "^Rr^fd RR "^RlFTRl WRI ■RTI 37tRTR^ % "SRI ^ "i^RR "ilsf (Grades) % "RRI 3TT^fR iRcRVf tl "^R RIrI tl RRftRfRRf t ■RT^-'crf Rt^ "RTT RRIR 'IRRRR Rt RRPRJT "Ril "RTR 'Rw. 'f%RT ■RIRT tl Rh^l 37RefttRiR "fRclRR Rt
T
3eRiYT R/ifefq^ /r/RrT 97
■ ^ ^TT^TI tl ^ ^ W(. ^ "Srf^ ^ ^ 1^ huK yc^lPvirl .3TT^ iTt^TTR W TTH^ ^ ttH 3i<^dlfa~d an^ l^rj % TT«TT ilPI^
% «!^i«K -cjifey. cT«TT fqa<wi ^ TIf)1% '^IHM yiP^efTcn ■^. % 3^3?^ -cilfgi^l
nfh-^M’i f^ai»i WTP^ fqmwi ^ % 1^ ^IMM ^sfT ^
HR«bc^*ii % 3?^^ yrqiRfic^ 31l<|f^ 1^?R^ iTIcT % ^m^iti
^ ^ ^ wn ^ ^ ^ ti "^nsf^ ■*?n ^Hifq«t>ai fq<Rwi "^Ft HRshcrHii Wla«t) ^ «f^l% 3RT8f^
^IRRI fq^K.'Ji MRehc^ii ^ ^ 5jlash 'tl
^TRPT (qmwi ^ flujof ufg^^ 3?p^ 'S^TfT'nf % 'gRr ^
Mn.«htr*H'ii WIPIRR Rft^TRHT cT^TT
% ?RT
h3^iFTR‘-400 ^ ^ TTg>-ijf^ ^ -3^ ^ % 31TVR ■qr ^ ^rPm ^pif ^
■^fel Wl RTT ■^r^’ "q? fqa^’JI ^14R «ifq«t>ai fqa<wi % RHI ^TT R^kTT t?
WHRI
8520 200 80 15
■^R—(0 ■'R 3iqdint)a fqn<«i
iJlrqiPfR iq?i<«i 'R^SRT 400 % IR !4fd<?f ^ tiiHi-q % 3T^?^(q»iero “«i)X<^ IRTlf^ f^RROT |^l 441(4)' oqiqeif<q) XW^arf WFT^
yifqehdl (NPC) % fqwK 4)^ - 3Z '^ + 3Z 44r fqt^jd RHT ^n XT4RTT f, V^fdy, 6Z 4)t ‘fR ^ 4jt 474 471^ Rmf ■4’ 4fe^ 47 4r^ 4T4 ■^‘ 1.2Z ^ '^l ?4 44? 44 t4W7 44T
4if44>4T 447 fsrf^ 4^ 4)? yrqiH'm 3n^f44T (f^ f4R % ai^RR '??4tl
1447’4 4)t HRq>crH'll % 3TT4R 47 4^TTlMpq
744P4
77141-4 ^
4R ife 4Jfcf?4 ^ 47?
f
gi^iiy "^fe 47?. ^ 47?
Ii i I I-3.0 s -1.8 s -.6 s + .6 s + 1.8 s + 3.0 s
H22.57%
46.41% -----H4
(4 M50%3.59% 23.84% 45.14% 23.84% 3.59%N ♦K H4 ♦K ♦14.36 95.36 180.56 . 95.36 14.36
ftT4 5. 400 75T4? 4>f 7TT4FT yiRl+dl 44^ % 373417 4?4 «mff 4^ 144144
/
98 3^4ff7 wlwi^fhj f^fim ;;
' (ii) ^ ^ ■gr^ hR^cthu 1^ f^diui(TS?T ^IRFT ■^, ^ IqnRo yrHinjia Iqa<^Ji 37^ ■! 3787f<I^
■ ^ai-^i = fei
(lit) ^ = .05 CT8TT .01 d/'= K- 1 = 5 - 1 = 4, 3Tcr: d/= 4 ^ f^ .05 Tr*7T.01 ■'77 ^73^ ■’K,
X^= .05 (df= 4) = 9.488 nsm x^Ol (df= 4) = 13.277 (iv) f^d<uif ^ ^
W^-8 : -
■gjlf-^rf ^ ttuhT
v/ot^-v-v /q^O/
^jThprX^ ■f^pf ffrff«T*TR7
fo 15.00 80.00 200.00 20.00 40085.00
fo 14.36 95.36 180.56 95.36 14.36 . 400
fo'fo 15.36 10:360.64 19.44 5.64 •
235.93 377.910.41 107.33 31.81
31.81235.93 107.33377.91..41 7.9495.36 95.36 14.36 = 2.22
fe 14.36 = .03
180.56 = 2.09= 2.47 = r.l3
■ x" - 2 —^—fe
.= .03 + 2.47 + 2.09+ 1.13 + 2.22 = 7.94
■ (u) vifm\m x^ = 7.94 ^ ^ ,05 ^ .01 ^ %q;■TTR^ 1TH 9.488 ^ 13.277 ^ ^ t, 3Trf: W ^ %\ !JRT % smiR tR ^
71 %37^ Wnl'iq^fi 37T ^f^dl % 3iq(nlf^a f^?97T oF^ ^iMi-q yifqq>ai Iqa<«i.%
37^ 17FI 7n tl '
Hi<q)crH’ii WlqiK ^77!^ "^TFr 'TT^kTT
1,9 f^<ui MRcbc«4-n
(Hypothesis of Independent Distribution)
wn*q f^cTT®! ^qa-qai (Hj^othesis of Independence) ’it^ t'l aT^TTnmuTl ^ ■5TPRT t ^ ^ ^ tn^ ir^ ^
(Independent) ^ (Unrelated) f "fTl 37?^ "SFl "3^% ^ ^ 77^ tl ^ ^ f 371^ tItTT t 3727fcU^73^ (Cells) t s^qcilf^d tt?t tl TT 37a[^fri%?l 37T^f^
mTw
. ^a-q fxT^ mJj] oTTcn tl ■^nt-'^ «raidi t ^TT |qfl<«i TI^TT McMiHfia fqdi'JI ^
3cqn<
/K.
37^ ■% «t)KW| ^ 37^?^ I tqa'^at % 3TWR
l^d^ui % UcfW (Cells) ^ yp^lP^ld 3Tf^f^ ^ % %tt -3^ y«hW % '^<!^
37T^ ■#T ct^TT ■qt^ 37T^ '#7 ■^?T% W^ H'JU'+d ^ ^ 37T^ '^iW (N)f I 37rT: ^ ■4' ^ ^7^ t tT^TI ^ TR^fe # fm yc^inild 371^-
y/fl^«y’t fqai'Ji
’Tfe^»xCj4 =
N ■
•57m r. = i5ff trf^ ^ 37I^f^’ ^
Cj = j^ ^ 3n^f^' -^T
N = 3n^f^' ^ -^iWW ^ -^rnl TT^t^' ^ ■f^ ur^rrt^ ^ ^ ^ f 1 377^ ■^'
yr^flfVld 377^f^ (f^'s) ^ "STT^: '♦il'<5'til (Brackets) .'^ %73^ "tl
■H?7t yetjl'^l ■% "f^ yr4iRfid 377^ftT^ (4 ®^ ^ ^ .y<7Vi
^57:% ■5Fn^-'^ ■'PW ^ ^ f I ^ 371^ r^dlui i( qTdd4l' ¥t '^77^ rf7877 ^c "t '577’^ % %7^-^etdl^ ti<S!4i, df= (r— 1) (c- 1) fl ^77^^ ^ -q^ (r - l)(c - 1) % -f^ ^ tl ■7772^^ TTH
MR«h(rM'ii f^dT.'Jl ^ Hr<.«t)<rHii ^ 3i<:«n^fd 'flqRh 37W*P^ tql«f>(dyR-qiqcK 'tl STRt^ 377^^ l^fTT^ jjd’li % ^ ^ ^ ^
^ HR«bcrHii (Hypothesis of Proportional Frequencies) *7t ^ ^ tl ^ ■f ^-577f ■qftimq Wd7 t ^ 'f¥«7!7 "TTi^ % 377^ Rldtui
tt 3751777711 37«7m ■qtti ^HitiWiRiqicii sR«t><rqil •% yrqiRrd Ruk'^i *^ t
mR+^hi W7: 37^ Htf ■577777 tl 'Wd'^ R(d<ui ^ HR«ti(rq'ii ■% ^qf!^ f^iRj
377t 3<ilg<'jil' -^W qfl ■577 Ttt tl
ypd^ivt % ^7f^ qft^ ■qft^Tq 77577 yimPd^-sTTf^
■7777 ■% f^ 3HH|Rt !qcn«i RimI^^K 5771 ^ tiHq> qt^-nR^iH % «i*iiPji«n-377f87^ ^
tit 37k ■k^T! ^ f ?
^iTcn
RjTTT'H q>t yRd5<rqdl tl §«lRnl^ «*+i3HiRiq>ai qt
w7^
‘ ■TTTt^ mRuIIM■^[tfkr M\tm itnft
30, 415 5■3^ ^0 3n» ^ 6
afkm STTo 10 4010 155 •
fqT=T ITT® ^o W 6 3010104
^ • 100 •30 203515
■^—(i) eWlfd* ^IddlRhd fqci<.«i qt 5^ t^^d'-swi yRdicrqu (Hypothesis of Independence) % 377VR "qr qt HCMiRid t^777^ "k dil-tl t, ycqiRm Rici<.«i q><'ii eViilyc^iRnd 377^^7^ qt 'q^qq7 hk^iI 4 "t ■qff ’it ti
100 3^CT<
tiKyH—9
rf^ an^frnilf (fg*s) ^ ^iothi
mRuiim
HTmf^ra>-3Trf5raT •ff^nr ^ruft ST^rfW ' ^
30x15 30x35 30x30 30x20= 4.5W® 3?^ ^<TT = 10.5 = 9.0 = 6.0 30100 100 100 100
40x15 40x35 40x30 40x20afrtitT w® arr® m = 14.0= 6.0 = 12.0 = 8.0 40100 100 100 100{
30x15 30x35 30x30 30x20W 3TT® m = 4.5 = 10.5 = 9.0 = 6.0 30100 100 100 100
15 35 30 20 100
(ii) % igKT hR'^ktH'II "t "aiqtriK^ci fqm^i cF^«qa-?i fq^KUi try'll % 31T^ "'R Vc^nP^ra cpt 3RR 'f ” 3T8?fcI^
. ^0 ■ /oi = 4i = 0
(Hi) RTSif^ RR = .05 W .01 df= (r - l)(c - 1) = (3 - 1)(4 - 1) = 6, 3^: df= 6
X^= .05 (df= 6) = 12.592 x^= -01 (df= 6) = 16.812
■qr-
WTolt—10
^ RUHT
aratqrt^ifi3TT^ 31T^ (Pp-fe)^Pp-feRRR/
LePo Pe
.*1 .1 6 4.5 1,5 2.25 0.501 2 15 10.5 4.5 • 20.25
16.001.93
1 3 5 9.0 4.0 1.782.01 4 4 6.0 4.00 0.67
2 1 ‘ 5 6.0 1.0 1.00.16.00
0.172 2 10 14.0 4.0 1.14
12.02 3 15 3.0 9.00 0.752 4 10 8.0 2.0 4.00 0.50
■4.5 ' 0.253 4 0.5 0.061■ 0.02
0.113 2 10 . . 10.5 0.5 0.253 3 10 9.0 1.0 1.003 4 6 6.0 0.0 0.00 0.00
1}- = 7.63u1013»WfR
(u) ^R'lpuid ^Flf-Tsnf ^ TTH 7.63 t ^ .05 ^ .01 m (12.59216.812) ^1,31^1:
4R«t><r^ ^ 'wleftR ^ ««t>oi 'll 3Icf: ■5f»?T tfeB'dl "t '9^W-hR«ih cl^lT wnf^T^-'STTfjRJ W Wd*^ 'll
ufay^in /^cTTW
f^^ircF^inT (Student Activity)1, "sfiT^-^ ti^KiKi ^ 3Tn ^ w^.t7 sRin^t
2. Xg-t^d^ sfR dM^lPrdl dd ^tf^l
3. ftd<.wj «pt ^RcurM-ll
1.10 WUl'vi (Summary)
• ^ ^ w t ■dSTT w ’srfd^ w Wid yrd<j;M'd>t TT^TddT ^ Hi'd^ 3i3_*iH tri'iNi '^ndl %\ % fqa<,«i(Distributione of Statistics), Tlid^dd (qo^wi (Sampling Distributions) ^ f, dd IdidT ■^ndT t'l
• ”51^ ‘Tjwjrnd ^ 3fzj|Tj ^ T5I "I dd (Available.Data) dTdlrdd? Tl^fd(Quantitativ Nature) dH b)c1T ^ did ^ did (At least) dm % SRlRdT
fsrfW102 d«d<f7
(Interval Level) ''RiTrTT Tftn tl ^ 3T3?mH ^ 3TffH(Qualitative Nature) % 'Smci'*! t’l '^’ ^ (NominalLevel) 3187^ Shin Cl (Ordinal Level) "^TM ^^cTT 'tl
® ^ (x) 4 liNr »TNT (Greek Language) ^ (Latter) 'tl r=<d<u|(Chi-Square Distribution) 1875 ^ (Helmert) ^ Mt cl«TT
1900 (Karl Pearson) §««t)i yfanis'i 'waoi ^ tC?SIPIR TTxl^ (Chi-Square Test) iq=fii^ci f^T^i✓
• "^SinJT HR^rwia ^iT^-'^ oR oRTf-'^ % MicTq<H'i Iqa<.«i(Sampling Distribution of Chi-Square) "% •'3TWR «ifq«ha{ % ^ (In Terms ofProbability) I^PTT ‘^nclT "tl ^ idci<«i 4>qcri ^qcilPtia fqa^wi % %
-g^cimf (dfs) m ^Rcn ti• * ST^^tlH^Tclf '^^ ^TTcT '^rT^ ^ "t wf! ■^' 37^Rt%cl
WiR fqaRci «q1qiK f^PIT ^TT yqiai "t ^ TOTX ^ 't'l
Distribution) ai^#%cT .f^cR^ (Observed Distribution) ^ ''TTtOT %■gRT IPRl t^PTT ««hai "tl whm fqcK«i ^ HRq»crS'ii '^PTR Hifqqjoi gil ‘RfteRvW (H3T>othe8i8 of Equal Probability) ^ t'l
• fqa<wi g)t ^Rshcry'll WTP^ xiiq«t)aj gfl hR^xthu (Hypothesis of NormalProbability) ^ ‘tl dldPl 1^ TPlfe yiKiiqi ‘HPTPg TJlfq^cii ^5) gTFRiaRa t'l ^Hi-q fqci<wi ^ kR^crH-ii clW «q>cfl "f ^ra '^H<+ eRR shRld
(Ordinal Level) 3T«^gT 3PrllTcT (Interval Level) U'+Pcrld ^'l
d (qcic.wi (Expected
ar^IRT-TIVH (Exercise Questions)
1. '^m % 3P?pfd an^ ^ B^TPif dn ^'i2. ■^PTPT fq"d<.<i| dft HRqxTH’ii gn "^if^PcT «0Pdi<l
3. MPiPHd WP^i aiFT ^ t?
4. itm anq ^ 200 ■5^^ 100 Hp^diqi ^ % ai^^3^“Hpecniai1i % fddT'd ^ ^PTR ^ RidP<d iTPTT ’an y+dl f I
TO (Reference Books)1. 'HffeRitq f^vgf ^ 'SPitn-«;m ^ q%^7P7/
2. -HiPsqqilM fqfqqi—^rqq liHHH 57? «f>RI
3. aiFR ^*HRid-/q/c7q*f q’TPdT? 57? ^HWI• V
4. i(iR<q>ai t^cRR—*77/^ fH^h, f^’^U v/^cj)^/d/
5. ?^-qE w*^q, 'T/<)«/ q/%^77?/
■ \
Wf^Rfhl 103<J«fO<
^ ’sn^r(Test of Hypothesis)
, (Structure)
2.1 '3^9^ (Objectives)2.2 X'Wiqii (Introduction)2.3 ^ (Essential Elements of Hypothesis Testing)2.4 l^inr (Standard Error)2.5 TFTR ^ (Utility of The-Standard Error Concept)2.6 (t-Test)2.7 . TTuqiTHi'% 3RR ^.
(Significance of Difference between Related Means)2.8 % 31^ ^
(Significance of Difference between other Statistics)2.9 ^ (Assumptions of t-Test)
2.10 ^-TRTToi (Analysis of Covariance)2.11 yf*41
(Computational Process of Analysis of Covariance)2.12 lq;?c^qui •4' ar^Pff^cj
(Assumptions underlying Analysis of Covariance)2.13 Mil’ll (Summary)
• (Exercise Questions) . - ■• (Reference Books) •
2.1 (Objectives)
• ^ ai^snwfT
w7<w«/)?<!/ f^(^<ii104
• iTP^icnT?; «fPT^ "^ i
• W-'SraT^ 1^7^^ Apt yrW>«Hi
2.2 MWK^-ii (Introduction)
rH'*t)4'0| (Statistical inference) 'SraTR ^ ^ oqqgK(hypothesis testing) cl^rr ^i^hm (estimation) I 3TRPT ^PTH-hihiWI
■% ^ ^ tcilqiK 3i'wlq)K "t 'Jiqfqi ST^iTH eh^qi Hi4i'«t>'l % *j<rql■% t l
feft wm % TTNf^' % % ■snt•f f+ql'n^i fqqKMd *HIHlVl’ % 'tl. oqiHlH-qi.
■>3;^ tuTo^qilq 4n.«t5<rH'ii Hi'qdi ■^1 §^[(ny. <+6^1^
37fy^iKt, iNHlPd'^ ^2^1 ^sft^ % yr^lqi % oZff^ Hi-qdi^^f %31MT ■’R Pi^fq "^n;^ ■f’l ^ <^qci 31^*TH "t'l '3^% 3p5T yfa'^^fl
■%■ 3TTtrR ■'TT 5^ ^ Pi=hiQi^ 'll HRqurM'ii
% 3TWm W<\ ^ ^ ■5THT ■gifftri<i^=hl isius-l
2.3 cl^
(Essential Elements of Hypothesis Testing)
Piled d^iV'ld ■^STPT^ % f^, Mpqi'rH'll "'T^^ ^ -diley.
I =Pfff^ Pis*^4 Pi'blci’^ "t; ■Srat^T ^^I’PsHchlq H=&ld ■^’ 'gWf
dil'jdl o^cR«n % 3T:?Ffcf 3i<riQid ‘4’
t "m ■pT^ ^sTPTT f 1 ^Eifer ^ P^ iTpn ■^nm ti ^ ^^ <ir^c^ (Prosecutor) TfcTF tl "t’, ^ 11^■QoFr sP=b<rH'ii ^ ■'7^^ qi^di t, Hq ^ Tf^cTT^ W ■H=hdi t; % WPl
oqPi^l Pi<T^ tl dir44 t iV 4^1 P'S =h sP.qxTH'ii ^ t^ra^TPT t Hj ti4idi^
^ t % ■5q1^ ^ tl ^ 11^ ^ ^ ^ ^ cT^n ■^m■-■pT^^ffeT ^^ ^T^?87T ?iTT Hq Pl41q Hpchc^dl 3^^ tl ‘t’l 3iq.icid Pl^fq ^iTffl t % ’^TlMt
Hq %.3ratcT t, tt cTF W Hpcbc^HI 3T?tl^ Hj' oqpKi % tHt ttt . hR^ic^h 'wlchK qi<di tl '
ftsrft ^ tl 3TT%?T^ Mpcfi'c-H^I tr 315?TR ^ tl t :
(1) oilPw Prtf^ t (Hq t), 3T^i^ Pi^l'H ■'iitl.t (Hq ^ +ctTqiK ^rr^l t)i
m: ^ PpPt wn%\(2) ^Pkl ittl^ t (Hq'^7?^ t),' ■'TT^ 3l<Al(nd tltl deudl t (Hq^FTI 37^tt^ "^iTtl t)l
3RT: TT^ ft^ ttRT Trar tr(3) oqP+d tl^ t (Hq 3R7c^ t), Si'Mcid "3^?^ tltl H^ntl t (Hq ^
■SFntl t)l ^ ItopT 'W tl
(4) oC(f^ ^ t-.(HQ 3T^ t), ^ 3T^I^ P^ wl t (Hq ^■'Wl<+)lt‘'ERcfl t),
3m: 13;^ TreicT 1^ ■w tl
1053cWd<
HRffxrsii ^ ^3^ (1) Tr«?T (3) ^ wt t ^ (2) (4) -^nawn^jf‘ P^ i\ ^ ^ (2) TTOWTT ■^‘ Hq ^ ^^4iKwtl Hr<4i<rMHI, Hg f^IW ^ t. ^Tffeq^ ‘-^J^Hr<«t>crH*ii' (null hypothesis) «FFT '5n?n "tl Hj MR«h(rM'ii ^ HR«t»<rH»ii’ (alternativehypothesis) '^>?T 'SiraT 'll nR^rn’ii 37^ 'JiqRh ^ -WI ■!■ 'Type I Error' «t>6C1fll %\ (4) TRTSn^ ^RebC^II Hq ^ MCial "W "tl HR«h<rH'1l ^ tql«t>K«t)Vii '*ioiRh 3RT?^ ^ 'Type II Error' «t)6ciifli ^1 ’^fT^ '4 4—
=bK Chi’ll
Pv^Hq tql'ya Hq
Hq TflW t ^-froRl I-T^fTR "Ft "5^
Hq aro?^ i II-W7 ^ fFN
■JJUft 'W^ ■?T WT ^ Pr^ ■^, ftFT 4t, •?TTra 44 4^-^' 4 ^ t f^RSR-qf^nm4'it^,q?!Flmt aftrl ll -srarKq4-gWq^ti
MR«hVMii-M^l<i:i«i (qRi (H3T)othe8is Testing Procedure)—qr^TF qi qFqqr "^raq)!qiqfqr qr/a^r qq?. 4? qrfqqrqi fqqTqr 4 "t, irffeTqftq qft^rqqr <t»5wio1 "ti qfq
qRqr^Hi qq ^rqq % qrqfq? 4 # ^ t, ■qiqfqr qrr 'pt^ qrprt qr ir qrqfq44 ■^rqq ^PTF qf 44t qRqr^qqr qrt ‘qiqRilq qRqjpqqr* (Parametric hypothesis) qiF4 'ti
(Simplehypothesis) qiF^ ^l qtq qiqfqrt 4? fqr4i' q4 PT^ R fqiqT qqi 41 ^ ‘pqferT qftqj^qqr* (composite hypothesis) ^f4 f I
fq4 4n^ P7^ %4 qnq 41 -34 ‘
fqfq qr fqqq f^rpqiT qqfn qF iqqR'q q>T4 4r 1^ Rh^i qn414) aq^ qftqJFPn q4ti 14)41q^FcTToT
qftqrrqqr 4? q4^ qrr 'F^^q. qqip«7q bIp pq 4 qqit. qiPiq■^t4 "t 1^fi4) (qmwj FP ■’j4 R(q<»i % PIP pfI ■ph4 "ti pqiFpq^qpq, f4 pf ^ f1 14) qqr ■qpTPpq pq 4 fqqfpT t Ipp^ wt fq^^q s t. q^ f4 •3p4) PT«q qq 15^ pfI wf ti 441 qqr 4, FP ppq-PT^ 4) q4 4 qRqjpqpT ^qrr Bpqq q4^ q^pn qr^i f41 qqjR. f4 pf pfi f1 14) 14^41 ppq qq PT«q 15 FPT qPN IPPeR 4 t PPl BP PPP 4r IqFPF q4 % pR 4 ^ 41 ^ F fIi 441 q?n 4 FP fqFpq, % qR 4 qR+^qni qq q4^ qR4 (BqiF^4. fpt4 qftqrFqqr f1^ 14) PPq qq fqFPq qPTPFT fppqq PT^ 15 FPT qpiq fqpRF 4 "t) I 41 qrqq qil pffeiqvlq qRqjpqqi ^ ^ Ifpi fit
1pp4 4 I^TPTPF ?nF qrpTT4 b41 qqnr 'qr ppp ^pt 14) qRqjpqqr 4 fqqr ppi "ti pr: fp 44 "tq-qR 4 ©qqsK
qF PPfl
4 ppq % pR 414741qiwq
tlpqTFT
l^ip qRqTpqpT qq qR^ ptpit fIft b4 ‘^jpt qftPi^RT’ (null hypothesis) ptfR f I^TP^ p%FT^ Hq fIft h qR+rmi’ qq q4^ ^ % I^'fp '4qif^ qRq)Wn’ (alternative hypothesis) 41 pi4l ‘4q>14qq) qRqTFqFT*4 pRtft^ IqPT qiTFi qftqTpqpT q4^ ^ pf^ qqRq qjR qRqTFPFi fpt 4«Ff?qq7 qf^TppFT PFiFT 't'l qjiq pRqTpPFT PT RqTfvP^ qRqTcPFT ^ Wlq>K pttR ^ 144q qtFq4 4 'ptf pIfp^p
(statistic) % apiIR PT IqTPI PUFT "tl IP PfFpRq pR pR^-pIcK^Ip (test statistic) PTfR, "tl pR^-qfFqRp. PF PTF 1pp4) 14n^ qjR pRqrRTFT aT^plqqr pr 41 ptfI R, ‘pfwjR Rp’ (critical region) pTFeTlFl f I pR^-PlFp^fP pq PF PB f^TPR) 14n^ qRpTFPFT «q1«t)K pR PTRll f ‘*qlq)i4 Rq’ (acceptance region) PTFT pr PPTFT tl
106 FPqFT nif<sfi<t>l<i Iq^Rf
W t # ^ ^ ^ ti ■3?^ ^ ■4' 200.W^ ^ ^ 1^ 200 'f3feif 3Tf^-ftf^ t TT«n ^ ^1 % 3t=^
*Tfl’ TT^ fiRt)41 ^ fqa^'Jt H^IHI-^ ■!■ ^8?T 3PTT3 Ri'eHrH 4 f I iHMfdl *1^1^Pli^5nT ^ IPIrT crx'ii ^150^ ^ 200 ^
^ ’STfcit t cit IciVqT ■53^ t' op^T "^irrat ^ cit oqiniO y<if^ "t afir (viebWd "t^l ^ «<2Mi 'Jii'qA % "Slt^ 25 PietJj^ "Sn^ "t fI«IT [dl^qt
fnt ^ fi "^if^ fe^rqf 196 - 204 ^ ^ ■4’ ■qrat ■sritft t ^ ^ f^nr^ror '4' itht ^ t196 ^ ■^n 204 3?f^^ ^ t ^ T^ ^ ^ w •^nm t sflr ^
^ ^ ^ ti
'4, HR«h<rM'1i ^ ^ T^ "f 1^ ^ ^ fsf^T^ oFT T?T^(H) 200 i ^«n Hr<!4)^rMHI t 'RTKq (\i) 200 % ^t-
Hq : ^1 = 200, Hj : n ^ 2001^ ^ ^ i?#=if SRI '^‘ ■^' ^ ■5IT^ ^ ^ int^T % 3RtR
^ '^m ^RHT t (Hq) -^ft : (Ht)■§tnt : |ij — ^2^'
. MR4.trMHi ^ fmfiTn ^ ^ amw ^ t.-Hg ^■’Rt^ RT^f^RIT RR (level of significance) ^ ‘^TRIT, 1^ RRTHI?T: yiti^Td5%, 1% RI«teT RR 3?lfs ^ ^ T^' ^^RRI ■sfmr f I 5% ■zn o = .05 RRta ^ 3T5| ■?!? ■!■ f^ FR I 31^ "^fe (Type I Error), (Hg ■! FRg wle^K f^RIT ■f), Fff yihetidl .05 "^TF^ f I 3PT TI^ "4, RPTFI 'lR=hvH'ii 5% RT^f^iFI
: 100 4 5 4 ^ pfM ^ -ssRft -sn^ft ti^ fll4et»ai RR ■’R SR^l^iR ^R^. 4 100 4'4 6 hjhciT 4 iRdicrMHI- ept Siwl«t»K
^ ^ # ■STFI tl ^ FtI^ 1®^ (a = .01) Rr4^ RR VX ■RfRT WI t ^'icifl Piufq "4^ WlRan 100 4 4 1 RTr4 4 4t ^IRlt 4t RFt^^ 4F.'(critical region) 45" RF
4, Rfs o = .05, X = 200 FRI 0 = 4, Ft FR RfRFF’TFT .Fit SR^tFIR f44 Ffs X, RRTR fFFFR 47 1.96 FR 4 37f4F> 4 f4F%F ^ 4l
hR^IXTI'II ^
FF Hj 4r fFF4FRcFFI
•^'iiq FR4RR FF^flFT
i FtRR ’’R l4tFIR i^FI 'FIFI RTRPFF
4x1.96 = 7.84
Hg 'SRFtFFT
Hg ^fIftr
^ -;---- , Hg 3RqT=hK f4 "FlFt "t FfF4 ^
F^ X<192.16 or X> 257.84
192.16 < X < 207.84FfF
STF ^ t 2 Fft WF FRFT:
z< - 1.96 or 2:> 1.961f-yfas^U (Test criterion FT
test statistic) FF '’^FF f44F f4^ 4r IFcRFI FT 4^ FT^IFiFtfFFFF t44 FfFF RF 4 FFtF Fg r47, FF ^^FIF FRFTI f4^ 4 RTRHd: FIFF ^ FIIFFIFTfFFFF f: t, F FFT X^ I FfFF hRwiih 4^ % F4y^ FF fIfF FlfFFIFT 1fFFF FI fFFfftF StFI
, FIr4 ftcTT 4l FFRRF^F^, FfS 4tt aiT^ 47 TlfFF^ F>t FTRFFfi «shc1®F 4 Ft f4^ 4? FRIFFT IFFFF FF FFFtF FfFF F^ FtFTI
107F«F(R
2.4 mhH (Standard Error)
(statistic) % [qa^ui (sampling distribution) ^RFT (standarddeviation) "3^ ^ WT #n tl WI W '51131 t '^TF31eRR ■271 % cfjRW fq-cKun^ncirtl *^l4n1 ■f) .■f%5|7t^ "mu 31% %T -3% tn% % 3%% (^d<u| ^ ■3TT^ ^ ■5tF "ITTWT ^.yfa-qm [qa<wi «n5ciiai FH 1^3? "^TOl ^ ^ 100 %-'57f3% "33% '*71^ ■f%T^, "Ft 100■*71% % %! "SR '511%, t%Fn 37%% fqa<'J| "^Tl^ FTl yfa-q^^-i ‘^FT’F "FlFRl^l '5ffF%-'*n%371 TTl^ W1 37T TTl^q (p) 37FcTmi tl 37f%rf "iTl^f 371 WT t333B ‘tHKT 371 '51313 (standard error of mean) 37FR[31 tl ■*71«7 % '51313 f333 371 3733-13^ 3«3 F% 37TF1 t,333, X % ■f3f33 TJ^ ^ 3;^-'^ "% t; ■^. X, 331 p %7 3^ ^l3F 373R 373 tl.■3l%1313^ 3t^ 3I X V 371 '3f3F 37^313 'FtFl tl yffl^^d 371 37137R "3^ ^33111,■31^ 37T 3313 f333 373 F^ 3frai t, ^ f1 3f^ 371 37137R -FFRl '31311, '[¥33 yrd^vff %7 31%
37lV37lf337 33H Fit '^nt t '’7f%ni3W?3, ‘Fi^ ^ 3fF% '31*3 '333 '31*3 371 37^313 ■?%ntl 37^ ■?!% ■4, fFF137R yfas^fl "cT^ 37137K '5lfF%f pci'll't 37f3^ ■f3?333i3 "Fit tl
337R f3f33 3fF%f % 3313 133%' 371 3313 133^ ‘'51313 133% 371 3313 1333* (standard error of the standard deviations) 37F^nFl tl - 31^ '3ife73fi3 '3lt, 'tt 3«T^, ■3F3*3^-'5’337, ■Tjq3*3^-'5’337 37lt3 %> 3t 3fF333 f^cK^i 3*3 '51313 1333 '^FF %t '31 337t tl l373i 3fF%3 371 '511F33F 13F% 373f%f713 3Ft3^ l3Fi3F137i 3!l F3^;^ t—
(1) 1^37 3lF333 f3F33 333 13373't 'FFFI t '% "FF 133313 '31 37Tv3pTF7i
(2) %-333-f-t 3fF%3 37% W 37137H % 3lF% %T 31331 3fF33F f3F33 %3F Fi■3% tl .
itz
(3) 133^ '333 ^ '31 -371337 '5rfF%3 ^377 '5rf%3F IfF^F 'FFl '33731 tl "Fl^F % ■511F33FIFFT. '. 'Ft H5Tci‘^yf 1%FF1 'FFl 3<r^<a «t)<.di <j|%d 'FIfII 30 'FI 37fVF7 '5371% % 371F7K % yfas^iT 31% F)T 13F3F 37^3133: '5r313P7 fIfI tl FFT F^ FTF 'F^ 3l F^ f% Ffi ^7^773 t % dM^dd F7*TF ^ FFH't 3l 37F fIfI t 3RfF7 tJR 333, "t 31f% 1% 'Ft t.
F71 l3Ft3 '5I313rF 'FI "TTi yfdd^if F71 371F7K "Fit^ 3333l % 3 Fi 3F 'FIF ■% tl 3TF: 3F F7F1 31 '33770 t % 'Fit T^Fl -333' f ^
n(n > 30) 371F7K ■% 3*11 3*3'F 3lF% 1% '31^. Til IfIfF '31f%! % '31% 'FTl fF7!3F- 33T3FF FIfI FFl FFF77 31*3 ^133 tr aiJUFF: F3FT fIfT. F31 '33371 3313 1333. WPT % 3313 1%H3 %'5lfF% 37TF7K % ^ 313 tt 37 W 3^ % 33F7 FiFll
■% t. 331 IF 1%3F13lf -t f1 31F371 Pi fed 'Fit t, %
T^F7ct)K«i 'Fit F% %1^3'%F7 ypt'^S'f '3%1 't 3
31 1F33 % F3F1 % •% tl
2.5 ironT ^ 334) Pm(Utility of The Standard Error Concept)
PlHfdf^ '371% 't dlT^sHdild a7*3FFi't 3313 1F33 371 'F^ 'FFtF t—
(1) T7F7 ^ -FFI HP^trydl 371 -Ft^ 377t't 3313 1F33 '037 FFFIfI 7137% tl
MRdxrMHI '371^ 5% ■31%F1 'TTR (o = .05) 37 13731 'FTTll tl % STFcDIftF F31 yrdlPfld fRumhITTTFFTF:
Fc3F7 Fri^FFT^F l3/%^108
■sm % 1.96 ^ ^ 3T^ ^ t ^ ^ ^1chK ^ ^ ■311# t. ■f^'ti 3FT 71S#
#■ % ^ TTpn wr tiarm #?fT t f# ^ tlCMIHI
■^, 37^ ^ *l5Tq‘^«{ ^H§ll '3n?TT "f I 3Rrt yfdti^H %f?i% aT^feTtf#^ ^sTi yr4iRid ^RuiiMl' '^f wi % 1.96 ^ # srr "I■# ^ 3T^ 0# ^«foF "^T# ■*TpfT ■^5ff?n "t^l 37^ '^, 37^ "SlfTra^ % ^j-c-qiq-q^iT ■% «t)KW| # ti«i>aitl 1% ■TfR (a = .01) m ^77711 ■# 3fR 37^ 3Rm % 2.58 # ar^RT# # "3^ '7TT5^^ H5r=i'^'jf TTPTl '311^ ^1 nR'hurn'll ■#) 71# «<flehK f^RT '31171! ■§ '31#^ 37^WT t^?R % #^ ^ (3 S.E.) 37^ ■% wr ^ cfh 37fM^37^ #t yiRicbdl 0.27% # ## t, #^71 % ^ -311^ f ttt«t ± 3o/99.37% ^ #1 <S'*di ■§■! 3a "S'?! '4' TFTll ■!■ ‘31#’ n ^>l#t #1 ©qq^K 5% "TH 1% '^Tl^f^Kll■77R ^ ;gT8t^ -q#^ mn it
d'cqiqq
(2) "SRiq "f^^R ^TRTm yfaqsff "# ■^ifqs'q^'il^di qaml 'tt URN f^'di 37fM^■#N, ^Rdtq*^ Iqa<'J| ^rqiRld ■f^TTNT ‘STPIT # 37^ #N, 3771:
/ -I \
##l "JRN f^liR 0^00H (reciprocal) ——- , yfd^sfl #t Iq;fq«id1qai "ttI 'TJWTl #t "Rq #711 'f I . \S.E./
■qRiq# ■# '«Vsqi ■% q«l«^ ■% '^[^-"T^Tlt fl ■qRt ^^*7^ ^ ^ #711 "I Tit 'SlftN# "RT 37N>R RT ‘^pll RHl "## "t^l
(3) "SRN 3# VRm #R37f #t ■fRlfftn R# ^ -TTfRR #7Tt t f3Ff% 37^#! 'SITWR HiMi# % Rt ■qrqiR ## "tl "^F HRiqqd "^TlNl #t fq^lqoi % ePTNl "RR^ '#711 "t 3itai^Hidci: ■qTTRP! "f^TlNT #711 "tl fH fq^jlnni % RTN1-*
68.27% #cT^ % R«T RR ■R«7 ± a ■# fq^K ■# 37R7f71#^ CqT STR Rtf # ■qfd^#3i Tm % RqfR ± S.E. % aTR^r #777) 1 urtt,
RKq ± 2a % a7R#l 95.45% ’q# RT 'RT#?! #7n fl
± 3a ^ 37R#1 99.73% "q# RT RT#^ #7n tl
■qT%21 ± 1.96a % a7R#l 95% q# RT RTlt?! '#711 11
RHT ± 2.58a % aTRtn 99%-q# RT R7lt?T #7n f tfR 3!tNN t 'Srfd-dqq Rt^R TIR R«SR7n qfty# Rt'1^ F# TRI# ^ 37H7Rf i^
(i) ^ RT "yfcraRl (Sampling of Attributes)
(ii) ^ RT yldq«4d (Sampling of Variables) ^FRRR "qlTN#
(Hi) RT yfaq^d—aTTRTT % TlfcN#
I. ’jqft RT PffIraTR (Sampling of Attributes)
, -3# % itRtrr "t. -yfcTwr # ^ iit qq t fR# -3^.% #^ r q t# '^ vm forr #7n tl FRFTqr^^^N, ‘■3R7t‘ ^ yPaq^'i 't F^RT ^ R7T # # ^TRTTT t Ir d^ld 1#^ FTFRT t■R d'SqH I yRiqq'i t lq># "qq RT •^Hni * qc'ii' (event) R * HqVi (trial) t ■!## -3^ RT #R ‘-TTR^' (success) FR -3^ 7J07 RT t ■#R ‘377TR^' (failure) RFT R7n tl ■RFcTTnart womart rt ■qfTT^iRi ■fRiNT; "fl^Tq 'snfqRTn fqq?t ■#^ % ■rtnt,"Rt "R*^ (u) = np, ‘sthni (variance) (a^) = npq FR RTN fqqdd (a) = "#711 tl
■^q yr^ "t 'n' ■q^qraft ■% N 'sfqq# ti f3iqt "fR# arqqieiTTT "Rt 'snfqRTn q = 1 -p ti ■qqicTTTTaft rt uRq fqq^nd yj^PQ ti
■3TTi<R :
t qR ^ qt qqRFvlTTTT
^qzqr Rt TETRcTTlT ^Rt TTlf^RTIT 'p' FR
fk^RT 109
1 .
^ 1000 ^ ’T^l 550 ^ 'STHTTI '3<5HiPia ^ «(*«•<<' ■pK?! % if ^3TTI
BH— "^ir p 37T^ ^
ai3*nPTcT «(K‘^Kfli np = 1000 ^ = "T - 500
f^-qei'i = ^j^PQ = 1 “ X -i X 1000 = 15.81 V2 2Hl'l«h
«ikMk<ii ^i^HiPid qK««Kcii ^ 31^ TTH^ ^ ripT -^pr (15.81 X 3 = 47.43 < 50) tl
■^' 3T^ = 550 - 500 = 50.
37^: WTFT Pr^ ^ ^ ^TTOI |37I, ^ tIHt ^7^1
"^^TT ^ "t ^siN) 'MH«1 '^, f^iA IIPt^ ’T^ 3T?5nf?I^ ^ wm H ■^l W ■'?? ^grtit % p^ ^«TT p^ ^ 37^
(Pj-Pg), Pi^P2 ^ dt-qiq-di^' % q>K0|: 31^ =fP (association of attributes) % a7«7^n aricft
■f ai^k 3i|«hK % ^ipT^ ^^1
il'*11 ^«hOI 5i
%\ pj, ^«7T P2 ^ 3RR % -sm fetr ^ -Tj^ pFq f-
Pl^l . P2Q2S.E •1-2 ^2ni
■*?f^ 37^ (P1-P2) < S.E. (5% WpFm ^ ^). eft ^ ^ 37^7 % ^ 3lftK9f ^ ■qr 'TT^TFTT ^ >*11^111 <^11^ ^ 37^ ^77^ % <i'cqtqqi) % chK^I 't^l
^TTR (^ yf^4Vf) (Sampling of Variables-Large samples)
«l'^eridl aTTTTJ^raT-'^*■^-^<r4t % yH<tJ ti<SHirH«») 7T«^ "t 31?^^ "*7^ ‘
3n^, aTPT^ 3TTf^l ^ cT»if % -jaf. -qf^ ^ -q^' ^ ^ -^f ^ 17^ ^q^f-■^n TT^ietTI , o^I^ql % 17^ 'TT^ 37^2777^ •4‘
^7§T '^TRIT sqf^ f 3T«raT <^^ii PiHf^d ■*Tm ‘^TPl 5«»)i5*il
««t>ai
^ ^ ^u<d)4l<i<l' mn fi
(1) ttPr:^ -irn ^ -m^r, im 37if^ ^ TRTP^f^ m ^ 3731TH■ .
(2) aiddUd 77871 yciJlP^Id 77871 ^ ^ lip ^ ■^‘ 37^ ^ ^ 77^% ^x^jqqi) % q>KW| ^77^ '
(3) 37^*77^ lqj?q«n1<i<ii "qn ^77771
^ 77«IT ^ Wlri^^HT 37^
(Difference between Small and Large Samples)®rt 7787T ^ wRiqq'll f'lp(^cl ^ ^77^ '^Isi t, "qT^ TTTHT^
^ t Pp 77f^ 3if^ ^ 377^ 30 ^ eft "3^ ^ "ypT^ ^ '371^7 '^if^l qfTT^ ^
t««i3ai
: ^’<sMRii«q1 "4 "^T? ■*77277
110
«iT<sq^lq TTirilf % ^ Wj 3lfff^VlT ITT^' ^"tl irfcRTif % oqqsK TTp^dll^ I' fH 'SRjTt "t—
(1) tii’nsqehlq ^ xlaqq-i I^IrR^ d<^HMd: 'SraRFT
(2) ■srirof-'^ % .% ■q^' % 3Tf^ I^Rre f ^«it ar^qn % w? fsraq % ■qr■5bW tor ^ wm ti ..
yfd<vi ^ ■qrqf ^ "q^^ ‘wr Iwt' tiR^m ^ ‘srqhti ^ ■qr ^ yPd^j^iT ^ wfewN 5Tqf % iRiq ■^ft fq^d ^ Tt f i Tqrw ^ imnT f^v*T(Standard Error of Mean) •
■qi^j 'qq "SRiq (sampling errors) 'qrq %\ yrd-d-m ^f^^FT "f ■qft "^nTn ^ a^ray'^di ■siFBq^ ^ <iHqVi 1^ nfcT^ ‘^■^Fni ^4' fq%tT fi ^mzT ^ ■smrq ^ ^3TB>R % 3BB yrd<^viT ^ ■qT«T qi^ qq^i qrwr % wr qft wn ^ ^ ^ wr t-
(3T) ^ qqrq P^^trin w^-
Standard Error of Mean =
sRftxrHii ^
tl '
HIHI«h 375*^■f fqpT% 3RFfcT .«rtqociini
Standard Deviation of the Population^Sample Size
S.E-^^ N
■f^nq^— S.E.^ = ^qi*27 qq ■qqrq fq-qci-i (Standard Error-of Mean)Op = i?q!I ^ wq rq^ci-i (Standard Deviation of Population)N=% q^ qft (Number of items in the sample)
('^) qq wq fqqtn'i w q cit q|qq?f qq qqrq [q^cii ■^q^r qnq qq.qqN fq^"ilRT %qT ■^TT 'qqqn "ti %ft "4 tiqTf^n 3ijhh % qq "TT^ftqq (Bessel's correction)
V 5' "tl 3B: ^ W WT Wr t-
• or, .
%qT qmn ti qq ^rsntqq
Vn_ g SampleVn
qfq ..qfqqjrf qq snqqr q^ t ^ Vn 3^ V^-1 q^ 3br q^‘ ^ ti
4«^i5<«ij4, qfq qf^q^ ^ 100 ^qrfqi ^ VlOO = 10 qq? -^loo - 1 = 995 ^W, q^ 400 fqq^
^ q^, V400 = 20 q«q ^400 ~ 1 = 19.97 ^1 ^ wr wr-fq^ qft wn Iq^tq srr
q^' q^i q^iT ■4qq qt qqtq -sn -qqrqi t-
a SamplearqqT Vn^
Standard Deviation 6f the SampleStandard Error of Mean =
^Sample Size
a SampleS.E. j =
qfq qsq qf^ % qqrq fqqqpr w H, ^ qr^q % qqrq f^qq qft wn % ■^qq fqq^q % q% qfr qrqteqr ^ qqqt qi^i
srqsq, Vn
"3^^^ 111
yfdBEviT % ^ nnnr(Standard Error of the Difference between the Means of two Samples)
^ ^ -sf ^ -q^' ^ Tnssn N21.■50^' eft iTT«if ■4‘ ^ 3RR tl a^^efC ^ ^ 31cRR % t ^37=21 ^1 -IIF W ^ yPeKi^i? % RTKqt' ^ 37^ ^ IpqFT t^^TR ^^■tl ^ ^ qf<i^?lT % tTTWjt' ^ 3T=tR OTTT % #T ^ ^ 37f^'^t (Xj - Xg > 3 S.E.Xi - Xg) ^ -3^ -RTsJqr 2it ttph -snen t afk "3^ 37^ % # -r^'
tl, 37^871 3RR ^ 37^ % ITHT ^1131 t 3^7 F^ 'RF t % yPcKJ^lT ^Ft TTO f ^ 137T, tl WIHMd: 37=eR ^ 5% ’qi 1% ITTst^ "737 '^7 mt^ ’^neH tl
5% 7n«fqren TTR ■q7 ^ 3TR7 WI % 1.96 37*7^ 1% ■7n«fqun 7eT7 "^7 37=7R.W7 ^?7R% 2.58 ^ sTftRT Ftm t eft 3^ T7F'Tei>iul 17RT "snen tl
^ yfds^iT % RRzft % 37=37 ^ yniH t^^TR ^ R^ft ■^HR^RTt % 37T87R 'R7, fRH "qt 77FF73I ^ 1!TeT -51737 t-
(i) -57^ TTRtl ^ 1737R ^'>SR (0^^^) W
- S.E. xj _ xg =:a^pop
R7t7 ^
r 1 l^ 1 137«7^ opopJ—\ NgIN, Ngj
N, W Ng yPel-^^^iT 37t sFR^: 1^-77^ ^ -STfeTf^tTc^ ^ fl
(ii) ^ 77=777 377 ITRTR 1333R ?773 3 Ft- '
3f3 t^-yPcKVin ^ 37^-37R=7 77=7I7f f %t =7t Ft eft -3=7^ -m^Tf p 37=777 377 -STRn
f3^ Ft37-. ' • ’
VN, NgOj 3«77 Og 373?7: ^33 3«77 -1^3 ITfTT^T^ % -57373 f33R=7t‘ 377 3f77ftf^ 377t tl
(Hi) ^ ^ yPcKiiiT 33 ^^373 37?73-37m ^ 773!7t' t ^.-^7 -337 ^ 377377 t77F77T3=3 tt eft 7J3 33 373t3 %37 ^ t-
^ + £2.11 Ni Ng
(tu) 373 -l^ ^7f^ 37«T ^ ^ 57f33?if % 77^33 37«7 ^3ft377cittcr57i3=T^377iratn -57737 t-
.. ,37337. VS.E.f+ S.E.2S.E.X, -X2 ”
_2r-^x-^ Ni. Ng
. S.E. X1-X2 -
NgNgS.E. X1-X2 =
«i«f«i>ni 3it 31177 (Test of Significance)P 378i3t 377 37377 77ffel3it3 37R 7778f^ t 37337 HFt, 3F 7773 377^ % '^ 377 37^
1^^t-
pop X NiCNi + Ng)' /^P^PV-NiCNi + Ng)II
D-DAT =-----S.E.X1"X2T = Test of significanceD = Difference between two sample means (not taking into account + and
- signs) . , . -DA = Hypothetical difference which is taken equal to zero.
112 HlJ<3‘i<f>l<4
III. xlfl ^ TrfrT?^) (Sampling ofVariables-Small Samples)
1?;^ w t, ^ yld'^iriT'm o^q^K ■fi■^•■STf?!^ '3T^ THTFIT f’? 3^: -5-I'M ^qei< "ST^ '^sn^ tl■jrni: ITM ^*i'q>T Pi^fq '^’ f^r^TF fl ■^. yuiPtJfT^srr^—fq<^'t>Hch< aii^fq^iH cT^ o^iqeil<ct> fq^iiql ■^' yfd«^vl <HH=h "Sn^: 37^
3noFR 30 fi ITT^Kuiciqi, '^FTTU TRN fqqci'i 37^ t cf«7T%■ "sriTN fqqfri'i 1%^ ■^5n?n 't'l ^.¥Ri<?f 't, iptn fqqtni 'ttito % "srp?
fqqoii fq^ci f^TR ««tiai 't'l
37T^ % aRl«\5ff ^RTTF % HlH=b^ 37^^11 TRP7I s«r<J. ■^.(W.S. Gosset), (Student) % d<3rcg,^ 1908 ^«5iPq'^i 'SrRr^R^ 1^7177^aqii^i-i f^rsTf stt, ^ ^T^FcTtTTj^ '^yTfn^ ^t 37^'fl“d (ratio) f^ 7' 3787crT ‘u<i ^ t'(Student's t) cT«7T yc^«h f^cTRT ^ i-fqa<«i (i-distribution) “FTfl ■^jTRTT ■§■, alciHt'^d1^ «7Tl i-fsRRiT W ■'TFTcn m STT^nftd t % ^ TTOI. f^nFT^' fm RH t, y^lHM 37«TgTSh^hiici: «^ihi-4 |qd<'J! t\ •■?^ ^ iTPT^ ■^. Tity^Rf ^ "TTinj yj ^TT^''37=rR nRctxrM'ii ■'TTT^ ^TT;^ % f^. ^ R^kTT t 'yit' TTRI Iqqcri'i ’SM '=7 ^t .?Tf^ Iqd^*^ ■^'.•fqdA' %yT''nyT ■!, 'SRTRRTcn 37fy^ fqqci’i f (U ^T J ’g^), qdiql "rRi^fy ^T{OffflT % 1^ yr ^dll f-Hinqi (i-statisfic) '5^7 TRiR W 1^FRT yidT■f ^^ilHi'^-fqq<,»i-2 (normal deviate-s), 37yfcl_"4 TTRI RT^ q'diq)<, «^qd 37RR y? t % ^ 3TRR X % at^MlRdd W7 "4 WF ^TTn t, ysny a % RH^ ^ 37?m w ft ■
vRsfJC-v-f/ ^ WW
(X-m)VnX-n■ o/Vn
Where : X = yfad^l ^>7 RRy (Mean of the sample)|j, = ^ TTR7I, f^R74 "4 TtRt^ f^n^T 'W ■§■, y7T (Mean of the parent
population from which sample has been drawn)S = uRt^ °FT W^-Iqqcii (Standard deviation of the sample)
t =s/Vn S
a = RRn ■yRR R^qcid (Standard deviation of the population)
N = yRi^l ■% 4757T (Number of observations in the sample)
, t-fqcK.ui 4t RcT^, y^iiqiK 3iitjiid ^ rfyi 407aid'tr'4 fddl’^il 4 <4qci IcRT 4t 37RR bRtt "f 1% ^-tydi^i 4 Iqq^'JRiieiai 37fy^ ■?7?ft‘ i-y^-3T'74
414 (tails) Rt 37fyy7 4k4 Flcfl "I 3^7 47^ 4 37fy^ 4Rfl ti Rtr 4 f-f4?RVT cf^t rttrrf fycTTTJF 44 yry-RTy ^ ^4 f I f4 u = 0 cTyr o = 1 ■Rn ti
Normal distribution Distribution
^ 3T?i2T^ ■gfTT^ ^ rI®T ^ Hi'-Mdi "fHt i-lqrK.'Ji "O;^ eW^. 3T^ t l ■^FSW ^lci-9q 3T?T (degree of freedom or u) ^ t ^ '51%^ % 3Tf^ 13^ ■
■'R W<T ^0^1 (N - 1) ■% '^<ra< ^1 ■^-■^ ylas^I 3TT^ ■§■, WFTP^
% Rchii ■^nn ■^srl^ sltr '^t^.N sprt % ^ t-"^ wrpt oraT" % 37^??^
WT tl 3RT ■^■4. -ST^ N 37f^ ^ ^-^ncTT t, S. O % -sn^n t. ^ ^ 3TT^
TTTni % 3TR7R % t eft S, a ^ 37T t eT^TT f W 3 ^ *ft 3RR
■=T^’ Wl; - .
^ ^w^rnt ■% 3T=fR c^ ITT^eFcTT % (Test of Significance of Difference between Two Means) 3(5HiiirH«ti ferf^’ % 'gRI ■^sneTT f f^- '77*7^7! % 5RT7
% ^ yfc<r!?f,^fc^Hi'il 3iqQi'lI^a 37^ ■^' 'TT^ftn^ 37T^ efft yilqobdi ^ %7 ^ '^^ yiRl<?rc(l^ ^ t cR 3T^c#FkT 37^ ^ ■¥n«f^ tTHT W t ^«7I ^ ^STfclT t % % T7KqiTHf
^ 37^'t’l "^77^ ST^^ftf^ 37^ % 4<'qViq^l 377^ ^ yifqqiai 3lffc7ef> 'ftTft ■!■ cR 3iqcilf^d
37^ ^ -TRt^mr 3TRT ^ RRT t TR^Rf % ■i7K7T7Rf 37R7 t l T7Kri7Hf %^ ^4'»>di % t RWn ^ W "57^ 31^^11 77^^ tt t tfeH tlt % tt ■*7W!TT7Rf % ■^, 37R7.^ 77T«ta ^37^ TJfq t, tl MR+erHHI ^ ^ ’TTrar^T •^' 37STR 14
■^‘ TTf^ 1^ RT ^ tl RT3^’ ^ ■^rc7R4 ttR tt R^RTpff %. 3RT7 Ml7't>'<rH’ii % % 1rq; Rf 37R?R^ t % HRq>erHH %• TTeR tt^ oft t RWRhI' %
■ 37^ Ri7 yldqqi t^cRR (Sampling Distribution) ^TTcT ttl % 37R7 R7T yid^qd IRtR?^
37^
t ■93^ nRoherndi, ^ Ri^HidT % RtRqi«tq
q^d: ^3^ HRohCTHdi %• 77eR tt^ Rt ‘ftRtt t tt wnfWqT "t Rt yfdS^'iT % % 37RtRiT tRcR'n tl l^fdqq'l ^j-c-qiq-q't % chK^l 177 "SIRR % 37R7 RRRRR7, '3TR7 Rt'^IRep RR]^ 37^ i[ TTRTt tl ■TTffeTft^' t tW t % tt RWTRlRf ■#> 37^7 % yPel^^H
R>7 R^Hid R3!^ % R7TR7 (N.P.C.) ti WT^ ^77^ tteTT t RR% yfd'^^'lT % Rf yRiqqd Iqdl^l (t-
ttRT tl tt fwfdql t tt R^hmI % 3TRT % yfdqqd fqd<,«l RR RRRT
RR?
Distribution) % T7R7RtRR^d, tTtZRTpt % 3TR!T Hidcb (Standard Error of Difference between TwoMeans) R>¥t t, fRH 'TJRT^^TR tteft t-
SEj3 37«7Rr (Tp = -^CTm, +<^m^
RIFT <7M| RRT ‘^Mj, R7R7T: URR cIRl fgdlq RVqHidT % Hid«t) '/JR-mT tl
tNf iT?RRPff % TTPTRT ifeRT 37^-37^ ^ t^' f^ tt ^ ’TJR Rt ^^Ndl
yfd'A^lT % RRr«b fqqcnT.elRT 37TR)]t R)T Ri7% tt R^hii'i % 37^ Rt RRRT ^TTR eft RT «ehd1 t— ■
= i.ii RR Rt ^1 ^ ^-2 ^ 30. ^2
s? 2 'Or, = , —-^— + —— RR 37fRR:?f ^ tt 37«7fR n, R n., < 30 ^ ^in,-I ri^-l . " ‘ '
RRT Sg RR n^ RRT ^2 R7R?I; RRR R llttR % f^ RW f^TR^ 1R STlRTR
RRT
Rftf S
ti
114 30^(77
: ■% "gRT y«iilqci Wjf%^TTPT^ (Pooled Standard Deviation) toff^, ^ «ncfT "ft Wi=6
^5i'40i 'R^RTFrl % 37^ g)t Hii^h ^TTcT igg ’t—
nla'^^fiT % ■% 3T^ iTP!g5 ■5[fe ■% TITg
_ ^ n-i + n-2 ^ TI2
^D-
1 1«8TgT ^D = ^t-----+ “\nj %
■5T?T a ^l^=h, hiH'^" fq-cjiri^ (Pooled Standard Deviation) ■! Pi*^I<nPa<i
^ ^ ^ t-£(X^-M,)^+£(X2-M2)^
(rii - 1) + (riz - 1)
2.6 (t-Test)
ypRVf ■R^Prnf (Sample Means) ^ 3T^'SlfcfggH (Sampling Distribution) % % w=m ^ ^ -RwmHf % 3rr ^ ifn^fen % ^(t-Test) f I ■4' ^-ST^qm ((-Ratio) ^ wn ^ ^ ti ■^-a^T^mcT ■^' H■qwriTHf % 3RR ^ ^ 3FER ^ TTR^ -gfe (Ratio) fl STcT: ^ Sl^qra,
Mi-M^t =
cpte TtORnf % 3RR ^ ^ WT «R (+) 3T2?^ "^ (-^) ^ ^ (Sign) ^ ^t, ■?^TfeTT ^ fp=l -5^ ^ -311 -^IgKn t-
Mj » M2 .1)t = t =<^D<^0
■gF #n % ^ yPdMKH 1908 ■^' ‘nf^ (Gosset) m g«n
D-0(Mi-Mgl-Ot =<^D
PT^ i 1% iTKPTPflt % 3RR ^ ^ ^ % WOT.gF
^ tl^ % 31^ ^ -HTatoT % ^ ai^qra ^ wrr ^ ^ •r
^ ^ tl ???% %R ((^/) irr ^ 373WI' ^ mnfl ^ ^ tl ^ %3Rn: gft Tn^sfEiun ^ %t ^ 1^ ^ %i (d/) ^ ith (n.j - i) + (^3 - 1)n.j + ^2 - 2 "ti HR'iP^rd (Calculated (-Ratio) ^ "RH 'NiTa'H %
RH srfiRT f gg MPt^ffuiF ^ '^n«fq>FT "qr t^ qPtd^Tqni, irwpTpil % ^ 3rr t, ^ ti t^qrtir■qf^ -qftnpjm ■el 3Tgqm -qn tth 'HT^Jqnn ■^ir^ tth ^ qm ^ t qq "qftqfqq^-sT^qiF qil ”3^ ■RT^fqRTT ^ "qr erar^fqi ^sncn ^ ■q^T "^ifq hR’^jcmii "qj! Wl«hi<. qR "ti fq^ft
wRsjfffhj /q/qqit itsd^o<
^ '3T«f t’■^T’ift’T^ yfa-ciqi3TT^ ^ ■HTqOT ^ ^ ■^) t '3^% ^ aTJ^RT % sm^ ^ ^ 3T2f
■f ■% |?f^ Zf-3T3W % '^qoi Wql'iqiji ^ yfaqq-i d'c-cliqq'i % "^TR^ 3Tr^ cbc+n -^qig^TT 'll PT^: ^ ■RTsf^ yipii'^dl ■4’ f[ ^ ^ ti .05 .01% ■Risifen ^ 3T^ ^ % %!? ^siRn ti
■5IT ^ t ^ •SrfcT^’ W RWRTPTf % 3^RR ^ ^f?rT^q-{ l^cfT^T ^»FT ^lHl-4 wifqqifli % ■RRPT %, ■?^#TR ¥f?K?Tf ^ ‘RIRT R^RTRft % 3?RR "RT^cbdl % f^i RR: WfTR yirqqicii % yrqq -in^^Mi sR^ "f I SR: '5TfcR?lf ■RtsjM (One.tailed test) t a^^qRf -qpr 1.66 ^ STfe- tr .05 ^ ’R cfSTT 2.33 ^ srf^^ 7R .01 ■RR R ^ ■STT^ 1s (Two tailed Test) ^ ft*# fti,3TT™ ^ TTH 1.96 ft arfe ^ -R .05 ^ R tT*TT 2.58 ft 3#T^ R .01 ■RR R RRl Gift'll I yPn'viil % ftR t "^T®^ ■^*nR RT gFTfftRT (Critical Ratia-CR).ftl yftW ^ f I -iTBT *;f ■rrj t ftr rf?f^ 3TT?m (C.R.). ftra^l ‘anql'i ftrftt ft) ^ 3TTTTcT % ftnj; ftRI ‘Sn R^tcTT ft)ft R arr^ ftr'ftn*? ^■ft-arjTRft ■# ■RRft ■% 3Rc#RT ft ftt RRS #11 f^/ t^qa^il' (di) % R ft-ftH RTRTRyifqqicii ^sF % Rift % ft«t)<i 3# '^3# f'’ cT*TF n. = co RT 'ft RTF RTRTR yiftqidl % Rift % «(<iq<.
%awqqF
■'R (General Jerm) f i ftRT% ftrftt RR ft ^ ^ 3Tr%R7
yiHi-q
fRR-fRR ftft tl t ar^qTcT ft RFT af^FR ft tloqivsqi
ft ■3# qEiPn «rft yld-A^fil ftr ftni RTR: shiPd^ ar^FR (CR) cTRT ftt ftTRftf % ftR ft-3TT?R (^' ratio) 7K RF Rftn RTFi t. RT^ ft' f-ai^w rtt ft RRftr 1%ri Tpin f ct*!! ftt
% ftn^ RTft^^rar d/ ■'R ft Rift Rft Rrrft ftft ftf ft Rft RTRT^ yipqqidi% RlftTRKn Rift (H Rft^ ftr 1.96 R 2.58 RRT TJRT 'JEftR Rftftft % 1.66 .
R 2.33) RTT RRftr RR% HPoiP^ld ft-RH Rft RT*ta IftRfiftT Rft R^ ftl ftft ft ft^lP-ddi ft Rft ^ yPd^VlT RRT ftft ft .RTRT R^RRlft % STRR # RTftFRTT % Rft^ ft Rftf ftH^ 31^ Rft' ftlftft ft ft' IftftT Rft TT^ ^3# Rift ‘SftfRRF fqd<«l ft ftl■RRH ft, aRR ftiRcT y^«Ki fftfR^F R^RRlft % aRR % %TT ft-#cRUT affRRT RRT*f (Exact) ftcH ft RRT df % R^ Rift RT 1rRRT RTRT^ !iiifqq,dl fqdlu! % RRTF ft# RfRT ftl Rft RRRT ft "% Sfcl^JJlT 3TTRTR ft Rft R ft. ft-Rft^ RTT RRft rIr ft ftRNR (Safe) ft# ftl RT*# ft-arT#f "ft RRft RT ^ ft ft) f^ Rft 3#Rft! Rft ft*# ft‘ RftftRT RTF (CR) RR RR# RR %ft, ^ ftl RT^ ftRT ftjR^ ^fRRT Rft fte ft ^ RIFT ftl Rftr RF ft fftcf fttRT 1R) Rft yRiq^iT ftr #1^ Op BTFRRFI 3
ftlRRft RRR ft ftft RT^ -^JR RF RRft ft IfTRR ^ ft 1rtRT ^ R^ft-ar^FTF Rft '^TfR Rift fti ft R7^ ft ffti ft R^RRlft % aRTC Rft 'RT*tRKTT rIf Rift—(i) ftft
R^RRlft fti R# aTRft^TF aiRR (Observed Difference Between Two Means), (ii) ftft yPi^^iT % RTFRi 1rRBF (Standard Deviations of the Samples), RRl (Hi) ^^tft % 3TIRiR(Sizes of the Samples) RT fFft Rnft ftl R^RRlft ftr R^R 3RRT tftcTFT ft# ft, fttr-3TTFR RR RTF RFFT ftt ■StPrrt ft# ft RRT RfhFTRF: 3TRR ^ RT*# ftft Rff RTRTR#^arPRFi ft# ftl ###' Rft #RcTF?i#FT ft) arfRRT ftft RT R^RR# ft) 3TRR ft) RT*# ftft Rit RWRRFt RiR ft# ft R## FR aiRR R)t RTFR) RR RTF a^pRRi ft# ftl RfFR# fti 3TTR)K fti R^ ftft Rt aRFT Rft RTFR) ^R ft# ft ftt R)T RTF arPyRT aTTFT ftl ft # 3TR#%F 3TRT^ Rzn -sriFR# % 37TRiH ftt-ST^RTF ^ ■RRTJR# FRT RfFR# Rft lRRFrF?f#FT ftt-3TT^ ^ iRfttRT^F# ft# ftl ft aTRvftfsFF aTRT^F*TT yPciF^fiT % aTTRTTT R^ RT ftt-ST^RTF RF RR q«dl ft RR# yffl'^^iT Rft fqqciF RTtvIFT % R^ RT ftt-aTJFTF RF RPT ftlRFFT
116 3^rf< «//<WRj?q folfk^
if ^ 37^^lriftd 37^ ^ % -q^t^ ^ Wt 37Tlff^ <i<^|^<U|)‘ •;^ ’’jRw^rH'ii spt >5/7?
'•iWrg7U|_f^^ ^STT ^WRt' STWpJRTtT ^ ^ ^ 3lfd^‘5l! 1^' 100 ^120 W -^'f^ 3Tfq?f^ ■qr iT^^nTPT ^ tth^t f5rWT ^T^m: 74.5 ^ 8.2 ^ 65.7 ^
9;8 8^1 ten fT^TT ‘^TSFR % % 37^ ^ :?n8fq7WT ^ ‘qft^ 4>ir*j^ltIM "t—
Hte
Mn sWchi<^ -100 74.5 8.2
^T5Prt , 120 65.7 "9.8T[«mHf 37^^ ^ ^ q^’ f^ -m t, ’^ynny.
(Two-tailed Test of Significance) "57^ '^iTqi 'gWll 3777: "q^^ offt wft Mf<«t>cry’ll4' Pi*-iqci ifqt—
Ho:^l-^2 = 0
Hj : Hj - ^2 0
'77T«f^Kn ■qft^ '^-q«fT Tij n.2 ^ «r5T ^ “qr "qfq ■qftqf^ t 375qTq qq ■qrq i.96 ^ aTftiqr ^ t ^ .05 ■7<R qr q87T 2.58 ^ sTfe ^ t qq -3^ .01 ^ qT w w t qife 1.96 ^ qqr ^ qr 3T77T«fqr w wr ti
qwTrt 37qR.D = Mj - Mg
= 74.5-65.7 = 8.8«.J q«7T n^ ^ t, 373; iTeqTTHf % ST^R ^ ife.HMch
4 .. 4(Jd = .V ^1 ^^2
8.2^ 9.8^\ 100 120
= /6724+\8003
. =Vl4727
= 1.21i qtqqrq)’ ^ 3r?n: -qr^fen % ^-aT^qrq
D,t =
^D
8.8L21
..=7.27. ' ._ ■q=!fc qRT t = 7.27 ^ -qR .01 ^ q^ qista % ^iq^qqr ^^sjqwq ^-qn 2.58
37fqqr t, ^qfRTi .01 'TfR qi qT«fq7 fl V^v if^ q^qqpf ^ ^ qq 37qcifer ST^R .01 T7R qr qrsfqr ti 373: qRdicrHHi, qqfeq! % qwrqr^' % ^ st^r ^ t,. q4 1^r^3 1^q«F3T "t qqr 4=h(VHq> qRqicqqr. % qqfe q^qqr^ 37qR ■^, qqtqrR qfi ^it qqRit "^i
3^37R !17
^ TTTsfen % ^ 3T^ % hRuiIhI' ^ 1¥=T ^ ^KUlf^iS«iK«Il«t* et)<.4> 3^^ 'sn h^] %\ ' , •
^KuH —1
% ^Run4)‘ w wn?r
«f«Ichni TfTT. TjfrRTf M D ts ^Dn
100 8.274.5 .01 ■¥?K ^1.21 7.278.8
9.8120 . 65.7
.01 "m ■qr
' .01
Ho : = Mi - M2 = 0
H, : = Mi-M2 = 0Ptwb'f
. 441^<'J|-?T?Tt IMWt ■% "CR fTR
mft .2025n
M 87.56 82.04.9.82
^ TT«1T ^ ■RwmFf ■^‘ ar^ t? ^'WlT^ "^i^hiiT % SRlt ^ 1«lv^T "W
7.30s
t, ^ ar^l^(Non-directional Test) (Two-tailed Significance Test) ciMiii
STrf: ^snF^Msh HfichcrH'liO. a7%ji%fe?T
- ■ . Ho:Mi-M2 = 0,Hi:Mi-M2^0
c(ffecl ^4ebdl "^RR ■RT df^ ^ -RH ^ ^1 -m ■Rt df~ n^ -t n.2-2 = 25 + 20-2 = 43 t‘l df= 43
.05 ^ .01 ■?TT8tar TRT^ % %T3:1s %% ■3ft f+^d t-
c//-= 43 % %1T I q- = 2.02
(i/= 43 % ■f^ foi = 2.69
•ara: MpoifJld t iTP % 2.02 ^ 3#?^ ^ tR .05 Tt 2.69 arfii^ ^ ^.01 'm ^ irnr ^n^i
^ ‘Hwqpqpi! i( 3RR
Hj ‘cT^TT ^2
D = Ml - M2 = 87.56-82.04
= 5.523q!f% n-l ^2 ^ "t 31^: % 33RR ^ Rmr ,^.
^ - *>1 , *-2 On ” « ____ ^ ' ■•
y III - 1 112 - 1
118
9.S2^ 7.30^“ V 25 - 1 20 - 1
= V4.0180 +2.8047
> = V6.8227 = 2.61
^ TiwTTTHf % 3RR ^ iTr«f^ % aryM.
Df =
5.522.61
= 2.11qfinfoR! t = 2.11 ^ ■RH d/'= 43 % i 05 = 2.02 ^ ^ i qi = 2.69 ^
05 ■qr ^ t -qr^ .oi ter m ■=# i\ m: .05qr Pkhi qft qq ■RqRft 'll ■qft^ % hR^ihT ^
qRT "t. arqfqF qH . qR+rMHi Ri<^ q)t TT+Eft "t ■qrg .01
^ ^ qq T?qKq ti
TTTTqft—2
% MRuHflt' ^ WR?T
^TTSfeqT wunt RTT■Srfrf^ M D ts «Dn
25 87.56 9.82 i 05 = 2.02 101 = 2.69
.052.61 2.115.52ynfN 7.3020 82.04
.05 TER. qr .01 ter qq
.05 ■RR Ri .01 ^ "qr
\
^R? aqs ■sr^raff % yRi'^^fiT % 1^ wtre ?qq ■qft^^ qi yiKn«t>fqRq^RR ^•
30, 25, 33, 19, 20, 18, 26, 27, 18. 12, 14, 22 .WT^-28, 26, 19, 15, 20,' 11, 20, 13
qqr ■^' ^ RKRTHf ■^' 31RR t?
RGRPTf 3RR qJt ^ q^‘ qff qf f, '^m (Two-tailed Test) qq "5^ qR^T '?tqTi fg^^q ^ qqi "qftqRqqi^ IqHqq
#ft-^0- ^i-M2 = o
■m m df-ny+ 712-2 = 12 + 8-2 = 183RT: % %3; rf/‘= 18 TT ^ wfr ^ m
i 05 = 2.10 ^i 01 = 2.88
■ 119dCT/CT<
^ MRilPuid ttpt-2.10 37^1^ -JT!^ ^ -q^ -3^ .05 W ^ ^«IT 2.88 ^^ ^ 3^ .01 TR RTsfe ^ ■3TT^. -STiffe 2.10 ^ W '^n^l
■=^11^ rtj 3«fT n.2 ^ RB 3>Tqft "t, WjfF^T Deviation) WR3T R«RTBf % SRR-gft RFBT ^t^TTI 373: S(Xj
Mi)2 327T. S(X2 - Mg)^ 3rt ^ 1^ ^IT^l .
(Pooled StandardRPRT
Wlft-3
^I3n3[
(X,-Mi)2 (Xg-Mg)^X, (Xi-Mi) X2 (Xg-Mg)
81'30 8 64 . 28 925 3 9 26 7 4933 11 121 19 0 019 -3 9 15 -4 .1620 -2 4 20 1 118 -4 16 11 -8. 6426 4 16 20 1 127 • 5 25. 13 i -6 3618 -4 1612 - 10 10014 ^8 64 ■22 0 0
I X, = 264 Ml = 222(Xi-Mi)2 = 444
1X2= 152 M2 = 19 I (X2 - M,)^ = 248
I:(Xi-Mi)S2(X2 -Mg)^(n^ - 1) + (ng - 1)
■RRjfW^r fq-qci-l, CT =
444 + 2485
\ 12-1 + 8-1
69218
= V38.444
= 6.20^ RWTRPTf % 37^ 3ft Hliet)
+ ^21 Th^7X<^
OD=a.
12 + 8= 6.20\12x8
•!>120
/20= 6.2096
, = 6.20 V.2083 = 6.20 X .456
' = 2.83
Mi - Mgi =
22-192.833
2.83= 1.06
qftnruM i = 1.06 ^ T^Ff d/'= 18 ^ i 05 = 2.10 W io^ = 2.88 ^ ^ ^I, 3TcT: "tl hR'JIIHci: ‘^feir'FTT Hq ^l5t>K 'll
hR'jiihI ^ «KiVi Pi^^qn 'ST^ ■^PFkTT t-
HKyil—4
% TTituil4)’ W 'HKl’vr
■^TTSf^KTrIlfrT^ WTOfl frPTM D tn o
12 22 ^ 05 = 2.10
^.01 = 2.88 .3TOT^^^1.066.20 2.833
98
Hi -. = 0
Hj;4j-42^0
3TT3 50-50 ‘Mil i(fd<9f P-iMp^d(Controlled Group) % 'W yRr^i!'! ^ 'Sr^TTr'T^ (Experimental
Group) % ^ 'm\ % 3T^ ■^‘ crI Tfzt^ hR^ii^ PiHqcl ^1 ^ ■^«TT yqlMirH^ % 'RWJHI'fl 3TRR ITTsI^ "t?
31ietiR
M.n s
yyVutMeb
RitlPstrt
50 69.27 4.83
50 62.85 7.10
yqRiirH'^ "W ■§■ §«Riy. ‘SFTtWT^ ^fW51?r ^ 3Tfe i\ ^ 3FT: (One-tailed test) ^'S#'! <iRHa "tt ^yRiy, ^
Ho: = Pj-P2 = 0 Hj: = Pj>42
«/'/teV'*7<v f¥ipjf 121<s'^a<
yrdt^viT^ 3TR7R^t 3m: MUjfT^m .01 ^2.33■5^ .05 RR 1.66 RT«ta ^ ^1
ch^ll^ 37miR "t, ■sm: Rt^jniil ■% 3RfR HTW
s? sf«• - ■“! , ^2°D -I---- + —
% ^2
4.83^' " V ■ 50 50
= ^.4666+ 1.0082
= Vl;4748 = 1.21
R^RRmf % 3RR ^ RT«f^ ^ ST^mcT,
M1-M2 ■
7.10^
t =
69.27 - 62.851.21
6.421.21
. = 5.31mm i = 5.31 tJim msta RTt^ % %T^ .01 RR m: Rt ^
Riefm RR'ift Rm 2.33 ^ t. 3Rf: RB' .Ol "RR RT^fm tl ^-RH % RT«fm ^ ^■=hK*J| '7^ Rft^mmr f=RRT ■^^5lcR^ RftmmmT T^tmR ImRT mr y=hdl "tI■% RftRTRf mr RRRT RRRft TR^ff fmRT RRT 'll
RTRjfi'—5% RftWRf ^ RITf^l
RRif^mr RRufd^vf M D t<^Dn .- s
4.8350 -69.27 - .016.42 1.21 5.31
50 62.85 7.10
.OlRRRT'feR
.01 RR RTHo:Mi-^2=^0Hi:Hi>^2
■^cTf^TRT—8 RRT 10 % % UfRR^ RTRFR tFH RTt^ m RIRfimt % R^RRRRRT Rmm ^\ rrt w lo % ^ mr r^rrh 8 % r^rrh ^ ^ ^3lfRRr t? '
• M • sn
11.7525 55.24RT^ 8
RJS^r 10 10.3828. 62.76
/r^rtT122 d^rt<
^?aT 10 % offi i T3:^ ^^!Fr (Otailed Test) ^ ^t^ni 3TrT: Hp.'turH'ii Pf^qci ■?Wt
Ho:^i = 42Hj ; Hj <^2 .
. . 'm'^df = \ + n.^-2 = 25 + 28-2 = 5l■ 3T?T: df= 51 % 1^ TT^ ^ TR
^.05 = 2-01i oj = 2.68
3ri^^!frT ^ 3TT^ t, 3T^: T?%2mHf % 31^ ^ TTH^
^7 jJTWne-
4 4=rtj -1 fh'~ ^
1L75^ 10.38=^V25-1
■ +28-1
= V5.7526 + 3.9905
= V9.7431 = 3.12
■q%ziTrpff % 31^ ^ ^ ST^qm.
Ml K M^t =
55.24 - 62.763.12
7.523.12
= 2.41TJT^ t - 2.41 (d/= 51) T3[^ ^msfen % %tT .05 1TR ^
3Tte f .01 ^ m ^ i SlrT; srr^ ( 1TR .05 ^ ^ cil t .01■qi 3??n8fqT "ti 3ici: <i>qci .'05 "qi qft^Rqqr fqr^ ^ ^eblrM^j qfqqjrqqi w1<t>d ■sn qr^Kitti
hRwIIhT MK^il Pl*-iqd i^qi IIT q^uTf "t^-
Hi^ufl—6% xrftigrmf ^ wtm
^RTT TTirnfl qnM D t- n s ^D
25 55.24 11.758 t,. = 2.01
(.,-2.687.52 3.12 2.41 .05
28 10.3810 62.76
fqtetjef .05 qr iq<H ^ .01 ^ m
.05 q^ qr^ -Oi qr Rii-tdHo : Hi = M2
Hi:Mi<H2
hiR^' Jh Mvdf 123-d^^cTT
(Steps) . '^ irapTRl' % sRiT iJ^ "^iB '^>7^ i\
"^l ^ImHI fi^qa f<ri<^I (0 hR^icth-u ^qiRrHqi ■qftoRBBT ^ Rt^Rb «t»<.'ii
HR«t>rH'ii, Hq : = (ig^«hfrMeti hR^xth-ii, Hj : |ij 9s ^2 ■qftOT '^)
nRqirM'ii, Hj : ^j,> jig 37*7^ Hj : |jj < "3^^ ''TTtfiirq tI)(ii) -TEnJif^ ^ Rrtta wfi .05 3t«bt .01 ■4‘ ^ 1^ tr^ ^ ti
7RR ^ RtsjRw ^girgr ^ ^ #tt ,(Hi) TfKp^'% 3RR ^ TTB^^ tfHT
(iv) WTT '
(u) '^^Ri (df) W(vi) ■3'q^ d/^«7T ^ "TTn: tR ■^t TTB’^7^1.
(vii) • MRnrmd •^-tTH ^ ^^^^/3raT?^w ^7§bi MRq^rd tth % ith ^ m. ^«7T ^ ^ 3TOT«f^ ^TTTTI
(viii) hR«iihT ^ 041041 4>7.ill sRmRio ■*TH % HR«ti<rHii Hg ^ Rl4.«i «h4.4>
nRch^ii Hj «q!<4in. <M.dl hR^iRio "TTH %
Hg ^ tq)«M< '4i<iil
t-^e^ai
2.7 iTKltlHf ^ 3T^ ^(SigniHcance of Difference between Related Means)
snft fsni! trwRPif #r srr ^ "q^T ^ ^ ■qf t ^ ■qtm(Independent) 3T«BT 3TOt^R«rt (Unrelated) ^^1 3T«rf<l^'?^' W<\ ^t ■5t1 TJJiftBT 3TeB-a7em ^ fl IBrR % ^ i^uMti]■?ldT "tl 'Rf 7T«n ■QfBT^Tt' % srfd^ Rt^ 37^qi<d ■% !>tRi<VF, RqR^td?t«n y-iiVn^d> #r -qr^ ^ t % irf^1^-^ ^ ^ f a7«lf^ -3^ W? WTT SUTsift ^ f! 3lfrB?ff it W<['qwTHpff ^ 4<W sttRsb "q^zTinq (Dependent Means) 3727^ 'jqwT^ (Related Means)TO TOT ti -Jrafq ■HT^fro iTKmHf % stbt ^ % "qt^ fq%3 (T% ^ tot t ^ 'T7WPTT^‘ % 37B7 ^ ife (<Ju) 'TOTOT % 37^ (df) 'TO tl 37Tf9tcT 37*7^
I^Rtb R^Pd^T Ri*^qq^ tiqial ■'t— ,
(i) T^ef ^77j^ (Single Group Method)it ^77 it fTO-f^ 3T^77^ felt "q^laTB ^ yJ^llP^Td ^)7% ^ TTwpnq
TO f tT^TT -3^ TTKI aTBT qTTSlTOT TO ^ aTT^^TOF itit tl TO: T^TO■qqtqf (Single Group Experiments) t W TOR ^ ■^chH tl fq^ hRR^;
37^TO. PsTTMtui 3787^ % "STTO ^ 37WTO ^ % ItlR aT^RTOTOf TJTO TOIF^ 37^7T7Tq TOT tl ^ 1^ Mq ■qRRaTfd (Treatment) ar^T^T ITt^TTO (Special Training) 37«rqT a7«ITR (Practice) 37«7qT PdT^I (Forgetting) .%t "a^nq t^
124 3«qd<
3T^^i'i'+idf % 1^)^ yfd<\JfI IT Pl'^fd H^fiiRid ®h<4i ■’??Nr PmPs ^ WTtTrqrsfra ^ "3^ T3^ 1TI?‘ cT^ fqsfi'^ "Slf^T^ 3^81^ 3?«tri 3TST^ fq«*<<«! 3T^TTT ^«TT "PTiT "3^ "'TT
Pi’^Ri ^ y^fiiF^a «t»<<^ 33^ ^ «IR ^ ^ y.*^^ TT^ yqVi ^PTTI
11^ Pt^ ^ Tlf^^ % ^ TTT«ta ^ ^ 3^ 38|T '’W^ %PmPf JTKTXTRf % 3RrT ^ TTT«f^ ^ ^JT^ ^1 ^ "qT ■^‘ TTeWT TTEF ^ TPJ? ^ Ur3
t, ^WpITJ, 1^ "TO^ TT^P^ ■^n^l «rii«n ©nlVt yiK^W.
"^iT^ (Pretesting) 3«IT 1|[ffP7 3W3 ''Tp^ (Post testing) 'Sn’TT 1^■f yi'qfqi)' % (Sets), ^ ttwtttr 3IRf ^ t, T=^: (Automatically),
TT^TT*^■% 3RR ^ Hn«ti "5^ "t"! <=hK«i "t IVqfd P '^P^TT^ 'I'^ni «h<^',■qHTTTT^’ % 3RR ^ -RTW "^fe % ^ ■^Tan^TIR TT^MMcT ^T tl Tj;^ T(f^ ^ ^
■q^zjTTPTf- % 3r=r .■^ irn^ "^fz "m ^ rnwq’a t-
Od “ v'^M, + “ 2 r a^, CTm^
'jT^ ^M| 7*1^ cf fgdl^ HMqi ^fcqT "t 3«1T r 91*^1^ % '^Pft
(Sets) % TT^TT’^^ 'yrf^ "tl
^ W =RT^ ^ TJ^ % ^ifdRcKi, y.«^<n llpR^ ^ ■Riq^i'il '^ 3RIT ^■qp^ % 3T:5r TT^ TTlqH ti ^ ^ 1^ ^ MRiiPid t-ai^qm % ^(n.-l) % q<Ht "tl "qr^ T^33T iPiT % y.qiei ,yfd<Vf ^ fT«lf3 IIFT: T2«F‘TtrsfenTT (One Tailed Test of Significance) ^ IPlPl '^jT^T qTwdl*! 'bPtt ^l.
33TBTqT—50 yfdq;?! "qr % qliii ’(Treatment) 38?TWlfrin "q^l (Pretesting) % 3«7I HM«t> sFCRT;
67.57 ^ 7,24 38TT 'q?^ 'qft^ (Post-Testing) % %3; ^m?T: 74.92 ^ 8.41 ^1 3«1T ’q?^% ■qft^ qikll'q^T ^ .60 ^ TTFTTT^ 81TI ^ ■qtzPTRf P' 3RR T^S!^^ f?
■^-■q?! qr ?m t Pf
Hiiqi
M rn s
50 67.57 8.24.60
MV^Id 8.4150 74.92
c[fe^ (Treatment) % ^TT 3?Tq7^ ^FT^T t, '?Tl^ (One-Tailed Test) qiy-ilq f I 3T3: 3«n ^chFVnqi ■qftqirqqiq PiHqd ^f>1t-
^0 ■ Hi “ M^2<» '. ■
Hj : Hi <|i2.
■q^ qr ■qwmrqf ^= <^i ^8.24^8.24
V^~ 7.0702_ _ 8.£1 _ 8^
- 7^ " ” 7.07-
■^' 3KqqT3 ttij^ % f. aid: ■q^mRf % ar^ iTFr^ -gfe,
^b ” V®M, +~ 2 r CJM| CTM:j
^M = 1.165
= 1.189\ d«Tr, ^M2
^ifis^<*il^ /qp/qr 1253«qnT
*y/<ehrwt/ ^ = ^1.165^ + 1.189^ - 2 X .60 X 1.165 x Li89
= 71-357225 + 1.413721 - 1.662222= Vl.108724 ,
, = 1.053■*T«!ITTHf % 3RR ^ -^8^1 .
^ ^ M^-M^
3T?T:
74.92-67.571.053
7.35 ,1.053
= 6.98
6.98, .01 ^ t •JTFT 2.33hR'^kthii Pi<«i <♦»<<* ^=r)RrH«t>i, 37?f: ^ .01 "qR Rn8f^ ti MR'^uqrd: .01 -qR
mfrt Hn =
■qft^iwn ^ fi
3T^ 1^ (Difference Method)-'^^ RT^ ^ ^ RRJ? ^ ^ tRiWff % 3RR ^ .RTT«tal % aiRTR f^«r (Difference Method) % ‘SRftn ^ g(tzRn ^ ^
-SSRIR Mil'll ^ R7M 'll "^R?^ "q?^ yiKH^hl % 3?R1^ ^ TTWjtfH HH«h Ri'^cii^ ^>R% 3TRT^’ ■% RWIRPT ^ ■RPT^ ^TRf «F)R^ "t! fRT^■gfe ’im ^ ^-sT^qm -qra w ti srrr fEif«i Rtr i{ Rq^ i\ rt%^i
■qR w % sTRTRr^ t^' ■qsp ^ qR q?TTf^qqTi qrqrfq) Ri'-iqa 5^—
ysmqr^ 51, 60, 55, 42, 62, 50, 41, 37, 46, 49
■flRft^qRt^ 53, 67, 54, 47, 56, 55, 48, 40, 52. 58
qqr qit fn^qlrl RTT«f5> MfcT ff #? ‘ '
t Rnj? % 3RRR qrt Rn«[qRn ^ -sifq %3RR f^8T qrr qqhT tl 3T?T; HIRTfq^f % aRRI^‘ qn RtJqqR cRn qpRF -m qfR^ %
fqtd^d RTHqft qR '
R7TRqft-7
3RqR ^ Rnf .TTJjqqRt^ 3iqfRD2D = X2-XXx 1
251 53 4' 67 7 4960
-1 155 542542 47 . 53662 56 -62555 5..50 '4
7 . 4941 48
^ifh^ehlii /5/yqf126 3WcR
37 40 3 946 52 6 3649 58 9 81
SXj = 493
M, = 49.32X2 = 530 M2 = 53.0
202 = 31520 = 37 Mp = 3.70 .1
!ir^' % 37^' ^ •J7«TOH. Mp = 3.70
% 3R[^' ^ TnFRT [qT^CTI, SDp
\22Z)2 ^20\ n n J
\2315 [37V 10 Uoj
= V 31.5-13.69
= 4^r3\= 4.22
•qPRT ffe,
^ - ^PpI------ 7- 1
4.22Vio^4.22
3
= 1.41
3T7fT^* % irWWR ^ ^ ^ ST^'TTtT,
Mpt =CTd
3.70L41
= 2.62
3^iff^ n = 10, 37cl; = 1 = 10-1 = 93TfI: df = 9 ^ "5^ ^ ^ m ^ m
*.05 “; *.01 = 2.82.
gAr W ( - 2.62 ^ ■HH (05 37ffeT^ -qr^ ^ 01 ^ .05 W ^
t .01 h TTOT: ^ .05 .*'•’<.
^{ffwf^ fkM J27. 3^Wn<
HR<*7rsii. ^ {ii) f^.(Matching by Pairs Equivalent Groups Method)
TR? % ^ t-(0 % f*T^ (Matching by Pairs) ^ ^ ’(ii) ^ ■'TT^ ^ ^ ^ «iHcir«<
■^'. ^ ■RJF % llr^ ^ (Variable) ^ 'RJF ^W tl ^ 'SRTR % ^ ^ feft (Pairs)4’ ^ f^ aTrar "I ^ iic^ ^ ^ TR? 0*1^ ^ ^ ^ ^% Wt ^ ^ ^ 1^ ■srmr ti "^r ^ ^ '^' ^■Slt^ ^ -RJ? -4' ^ t, ■§^Tf^ ^ 31^ ^ ^ ^ '5^' ■#r ftcTB ^ -R^ ^«TRIT f I ^ n. ^ WR!l^<4) # f I ’SR^TTR^ ST^’UPT PimP^io '^ 'JRt’TTR^ti*j5li' % «iH ■% 3RR RRR 1^ 3R!TT % ^3^ ^1*5^ «i'iiet><, t^p ipl Pi<<p5)<i
3RW (Experiment) 1^ ^ tl '^rI % ^ ^■^fs W ^ ^ ^ ^ ^ t. "sit
% ■qHRFlIi % aRR -qR^ -gfe ^ tl SRI: qt ^ UTR RWfTTRf % 3RR^ RPR) ^
R ^ HqViirH«t» RPRR ^ TITR R«Rpff ^ 5?pn tl R«Rpt % aRR RPRT
*^D “* V'^M, + “ 2 r Om, CTMj
^M| ^ ^M2 RtRRpff ^ RPR) ifeqf t.RRT RPRR^ ^ (T
Sets of Scores for Pairs) % R?RR=«T IRR) tl% IrPTR "t Rt «*jpl % RVhhiiI R)^ IPRT tl RftnfRR t ai^nm R>t d/ R)T RR Rt
R?§^ TRT RRT 3TRf?T (?1 - 1) ttR tlRRTlTRT-lfe Rt^ % a^TRR RT ^R fRRR % ^ r1 RR1?R RRit Rtl ^ Rt
PtrPRcT % RR t <<ai RRT Rt Ir^ ^RRIRR) RfR^iPR RTPfRR (SpecialCreativity Training Programme) Wl Rftl^ % "aMiW PlRP^d R RRtn?RR7 R^ RRlf^ "^RRTRR) Rt^ % RftRR pRTRR ^-
wo
Mn s
fRRpRR
RRlRTRRI R^100 52.67 15.18100 61.42 16.35
Pi*|p5ici cfRT RR^RIRIRr R^^ % % ■^RRPRRTcTT RTRTTRt t .70 R?RR^ RII RRTPlRp^d RRT RRtRTRRr RiJ^t' % R«RRHf f 3TR#f^ aiRR RT^Jrt t?
^cT—RRff^ yRfUtTR' RTpfRR R)1 RRTRRtcRTT Rt RIR RRRT t ^qPril< ^IRT I^Ir Rt^ (One*tailed Test) RfepftR tl aTcT: RJR RRT ^«t>PrH«h iRR)cRH|0. tHl-
Ho.:Hi = P2 Hi:Hi<P2
Rtf RT RGTRPTf Rft RPTR) tW-
•^M, =-£^ = i^^ Jm 1002 16.35 16.35
10
15.18= 1.518
= = 1.635RRT.
■ 128 TERRTRri'SR^ /RfW
%. 3?^ ^
^D " ■\/‘^M, +^M^ ”2r a^, C^Ma
= Vl-518^i-1.635^.- 2 X .70 X 1.518 x 1.635
' = ^2.304324 +2.673225-3.474702
= Vl.502847
= 1.226
^ 3i'^< et>l ^^chcH ^ IVlQ, ^“3h^HIC1 ,
Ma - Ml ■t -
61.42-52.671.226
8.751.226
= 7.137-STRl t = 7.137^ 1TB d/= 100 - 1 = 99 -^7 % f pj = 2.36 ^
^ t, 31cT: 1TF .01 WT; tl .01 ^ ^T^ HRetirsii ^ ^
{Hi)ilWJHFf rrar *7Hch ^Tl|^
■^tl
(Matching by Mean and Standard Deviation Equivalent Groups Method)(hqiI'I f'lnl^ 't><.*1! 3T?2TB ebf«iH ^tll "f I
RWTiTB (Mean) W BB^ r<44Vi'i (Standard Deviation) fRBB ^TT% ■SBl^ ■fl W '5BTR % ■^' 1R ^ ^ iTWniB TT^n fBB^ tl TR WR % TR^’ 1^' n f^-f^ ^ tl ^ ^ tRBB W f
^ y4lJ'irrM4> ^ eft FBT t Irbb (Matching Variable) cT^TT !RTBTc*Br (Experimental Variable) B'T tl TT^ BB^ % fRBB^ TTTB ■% 3TBT Bft ITBB t¥=T ^ BTeT ^ ^ t-
D~-J(’^M,
M, g O^Ma R^RTBf ^ ■RFB? f rf^TT r "3^ TRfe % fBT?,t, fuBB (Matching Variable) ■^«TT 'jnfBTRTB (Experimental Variable) ^
RWr f I
■RispiB cT^TT ifH=b % ftBB ^ ejBii MflMp'th t^ d/ RB (nj - 1) + (^2 - 1) - 1 STSrfel + ^2 - 3 % ^TW ttB fl
■^^T^TTR—WTBT ffe RT IITB 3T^ % R^^^RTB cT^ hH'4i ^ t^BB fO;eT8TT HkBH' % i) ^TM^ M\(Attitude Scale) ^
IT ^fBTfeet WI % tm ^6;^
(l-r^)a
? cBW7
k/W 129
vi
fq#'! wirtch ^mch
^ wmr
wir?r Tift^nr tjT
HW^ ^fts TTft^I TR arf^f^ -qr3Tf^T^^ Mm-il ITT m*ich fqnci’i
■qf^ yim^ ^ qrqn^' tmr aifq^ qmqt yikii+T ^ qwr .52 qn qt% 3Tfqff% qyqqr^' ^ qrtf^l .
■5^—=hlf^ q^qqpTt % 3RiT qi^ fq?TT "^w ■§■, "p^rQ;. qrl'^ (Two-tailedTest) (ri'inj qttS’il'H Tlqil 37q: "^pr qqj 4«hRrMq» MRqjcrMHlQ] ^iinfetid •
Ho:^^ = H2- Hj; ^ ^2
‘ qgt qr sTTq^ qrqql qrqffqrf % q^WRl' qrt gaqr qrrft ti sm: % qsqqHT qflqnq) ^fiiqT Fnil-
64 60
75.94
12.36
40.75
8.62
75.8712.41Hin«»»
35.50HwtHiq
8.58
^ 8.62 8.62 = 1.07758
8.58 8.58<^2 _______________yfn^ 7.746
■q#^ q«qqR qRqr fqq^ ■% fq^ 'snT^ q^ t, 3m: q^qqpf qqriqq^ % qrqr qwiqmf % ^
= 1.1077=\
qiwHTW
= J(1.0775^ -t-1.1077^) (1 - .52^)
= V(1.1610.+ 1.2270) (1 - .2704)
= ^2.3880 X .7296
= Vl.7422848 = 1.32
qtijHi'il 3{m< qft qr^f^qj ^ '^-si^qm,Mj=«M2t =
40.75 -35.501.32
5.251.32 ( '
= 3.98
q^fe qrqr t = 3.98 qq qiq = (64 -1) -h (60 -1) -1 = 121 m l5--5^ q^^ % %tr sqq^qqr / pj =2.62 ^ t, sm: qF .01 qr qi«fqf tl Fq%q qRqipq'Hl'^ .01 qT^fqmr qq qc Rqqr 1^ qq hq^m
130 Ts^RfT fl/Agy£#)?<!/
2i8 3^^ ■^TT^opHT
(Significance of Difference between other Statistics)
^ ■qwTTTpfr' % 3T^ ^ TO [• M(i«h ^ '5rf%^T?f'r,^ ^ ^JOTTaFTf ^ ^ ^ejpii ^ wm h
"4 1%^ "^IT^ Pifed fT^ ^4qd '§■, -S?^ %cr^ Hid«h ^wn % ■srp?T -^f ti ^?gcT; ^ ft
(Extend) ^ f— ‘
3Jqtn'lIV>n 37^31c|^lf^d 3m ^ WT^ ■gfe
qWn?T ^-■tTTq ^ df fi^n ■^ffecr ’TTr^Nun fr tr
■Jt % qrirjft ■RR ■qiT% ^ %qT ^TTcH f I^ % 3RTT ■^'^i^fcti^I(Significance of Difference between Two Medians)
'qT’^-'qT'qt qVqdl jjcidi '*T^*7h1' % aTRiR qr ■qwsn^ (Medians), % 3TTiiR qr ^ -^rmt ti ^ 'SfR: ^ ^ f f^' ^ ^ wn
q^' qncit ti ^ qffR?ff qr^ ■q«rt % st^r ■qrqqr ^ ^tptsri iM t-
^Md,-m.i = ^C^Md, +'^Mdi;
t =
■5IF qsrr CMd^- sFm: ^ qwTR^' ^ qrqqr fi
q^T ■qwTfqj q^ f Tra q^^rf^n' % sttr qft tth^t ^ qn f^qfrq^ %qT qq ti ^ f^sTfq ■ff qKTRTii % WT qr q«7qTq! % 3T=fR qft ^®qtt 3Tf6Rr qq^ ^ tl
TO 3tM qTKqqf % fcitHdiiT 1^ q^ qffT^-^' q>q?i: loo q«n 120w qft^ q^ ft% qrqqf qr^m: 70.50 q 8.78 qqr 65.95 q8.62 ^1 q«Tfqit % arm qj) qi^q^n qJT qft^ qrtf^l
^-•5Tm t- • ,
Mdn s
qrsqrr unr
qrsw isnr100 70.50 8.78120 65.95 8.62
TOfq^ qrt qHq>
- l-253si 1.253x8.78 11.00^ -fm io"
1.253 So 1.253 X 8.62 10.80 . .. _——-^ ----- —-=----=--------= 0.986^ 10.95
= 1.100
^Udj =
d^di /q^/qf 131
^Md, - Mdi = ^CTMd, +‘^Mdi,
= ^1.100^ + .986^
= ^1.2100+ .9722
= V2.1822= 1.48
TTtTif^ % 3m ^-ai^qirf
Mdi«; M(i2• t =-Mdij
70.50 « 65.951.48
4.551.48
= 3.07MPoiPini (= 3.07 .oi m -qr ^ £ = 2.58
37f«^^ t, ■3TcI: .01 m ^ tl 31fT ;01 m IT ^ ^ TmTT f % i^WTfsF^■^' 3m f I ,
fq-ciM’ll ^ 3TnTT ^ «i«I«t>rti
(SigniHcance of Difference betwen Two Standard Deviations)’jjf f't '^rt^^ (q^cinf'ildni ■^tRT ch’^-ii
3T5?TftIPm[f ^ WRT ^ TT^ t 1^ ^ ■q#T fmm ^ ^ ^
^<nii ^ ^ Pii^Ri ■qr^TT f'rmr srfyqr qp7cTT-^q?T q^spq?
H6T«(
(Parametric Statistical Tests) TRH qTTT^ qmn (Assumption of Equal Variance) q=n ^ Siiq^Meriai ITT y«bK qfl' f^fd41 ^ HM«h % 37^ qfl
■?7T?fen qrt^riT ^ f i ^ ^ftI ;ttf ^ qrr^ f^cn ^ ^TPJ^' ^ ■gRT ^ TT^ tl
T^rra’ ^n^ch fcfgr^ % STTIT ttR9T (Standard Error of Difference
between two Independent Standard Deviations)—Tf TT*^ qTTqr (Independent) t. ^ 'Sn’TT nHcb % 3T^ HUet, ^ -g^t W %qTTT^RT t- .
+cl^
qitt <Jsj c^ sh.HVl: cf«TT fgrtq TT^ % iTFRT fqxici-il sfft Hiich "jfeqf (Standard
Errors) tl
qr ■©r w yfn^^iT % fciq qi+^7.69 5^1 t giq?!: 70 W q«n 75 effi ggi ^fp^' % iTH^
^?1 9.85 q«TT fq'cjri-lT q 3TqrT TTT^t^ t?
132
f,n s
9.8570
■0niT 7.69• 75
■’^T ^51^ cT^n Pi^qa "FlnTt-'^0-<^l=^2
Hlish Hll'*
CTs, = .7151 .71x9.85 6.9935 _ g^g^Pfo 8.37
„ _.71s2 .71x7.69 5.4599 _Vv^ "
3TcI: ^ wci*?) Hi'i'sh % 3T^ '^,
= .630W,8.66
= ^.836^ + .630^
= V-6989 + .3969
= Vl.0958 - = 1.047
Hlieti fq-qci^T ^ pci'll
G j » ^2i =- ®Sl - S2
9.85 « 7.691.047
2.161.047.
= 2.06W<{ t = 2.06 W\ WT .05. ^ 1.96
t "7)75 .01 ^ -^HdH ith 2.58 f, ^ (iiB.05 t<R TT? I .01 w ■qi: ti stcT: .05 ^ ^
^ % 3?^ ^ ^Tpra? (Standard Error of Differencebetween two Standard Deviations)-^^-^^ ’’TTPR3T?ltrI mH«h rff ■’TT fq>«l ^ 'SiqtiO y^llf^ow^ 1%^ ^ f ^ ■’Ewg^ ^ ■gr^ ^ ^ t’l ^ fq^niT %^ ^ ^ f^. ^ ^ -SM t-
•t ^
“»,-»2 = ./af+ (j^^-2r=‘cr„
133
/
'4” 11'^ ^1
HUt^i f^'cjd'll % 3i'd< ^ ■?n*f=lT^ ■% ^ «ilMN ■'3^^ T6^ 't'l
—100 ■% yPcl'AJ?! ■’7T ^ "W % 3RRI^ ySKlPf^dft ¥«Ttt TTH^ f^-cl^H 5.84 ^ 6.59 fsm -yeril ^fan tlltfNT % aimi*^!"^' .60
^ TIBTFcRJ «tti ^ TTH^ 3RR t?
^-W-t, ?i - 100, Sj = 5.84, S2 = 6.59 r = .60
Hl’1=ti fW^iT HM«h
Pq-eid-lT Hllpti ^fiqT f tjcf r yi'^fchl ^ ■^’ ■^’*TR^
^ •
.71X 5.84 4.1464Vl^
■71x6.59 4.6789^^m■
3T^: HMch fqqtrl'lT 37^ 17PR>
= /41464^ + .46789- - 2 x .6^ x .41464 x .46789
■= V.251163
= .501 ■qnqi 3RR ^ %
6.59-5.84
= .4146410
= .46789= 10
S| -
t =.501
.75.501
= 1.497^ t = 1.497 ^ Tm df= 99 % .05 % 1^ .05 ^ .01 ‘'R
(1.98 ^ 2.63) ^ ^ t. airT: tl ^^PdU. ^ t ^fg#T ^ TIMf^' % Pc^’-dcuT 3RtR: tl
fqqci'il % 1^ ^ 'SmO'Wyr?i
TpW ^ipf "gM 'll
^-crfroff % 3RfIT(Significance of Difference between Two Percentages)
^877 TfR^ fq-qijiil' % wqi-i ■yf^TFcft % ^ f^faqT ^ 't—(i) ^d-jlyPd^viT ^ 17TO yf^' ^S7i (ii) ^ yPcKvif ^ w yf^'i ^ ItstW ^ ^% 37^ ^ ■77T«toT ^ 1^%JktrT 'STT ii
^ ^d»5( vifd^id) % 37^ 17T?fcB77T (Significance of Difference between
Two Independent Percentages)—'^ ■7«7?P5f ?d7 Ti^qPid yPd^rVlT 'R yfdVldTyro I^RTT ■^TTcIT "I, 37«R7 yfdJ^I'^
yR: f^TR-f^TR Rfferf <>4q5K oq[^ql‘ yRtifrali ^ "5^3iiq;fqq>ai ^^cit ^ f«R7^ ^Tld ^ ^7% 1^ ^iT^ ^c(5R ^ ^ ^
^ f^TR't^! 'Wd'^ ylfl^fidl ^ ^dii 1^, 37^ ^ Pi*-iqa ylTff "oPt '377^ %-'
PQ —^-^pi-p’ ~Til 12
134 ' 3^^777
■ ^ P yfa^ioT ^ (Pooled Estimate) TM Q = 100 - P tl ^ yPdilldT %
P ^ wn fFl ^ ^ t-
^iPi + ^^2- «■! + ;i2 .
^ Hlasfiel % '31^ TfFRFT 5TRT teT^ "tlr^y«h1 ^nTlJcr i?R ^ # ■3Tmt ii
■ ■^’ WTRT ^ % 300 ^SKfT -srifn % 200 o^Rd4f ^ 3TTfe•3TWR ■'IT 'srf^ ■'^ "n^i WTP? % 55% 3i^^Pqci ■^jnfd % 40%oqf^'^T ■STPtI ■^I ^ <hh;*^ cT^ % :cqi«w<HT "4 STlf^T^ 3Tr^ 'm'3TR^ % 7f?T «5*ifo ■4’ qi«tfq«t) 3T^ ^?
t-
P =
n
■^nf^ 300 55%ITTHT-q
3i^^rdd ■snf^ 200 40%
«i*^f5«ti yfd^fld,
p ^ »lPl + ^2^2 ?ll + 712
300 X 55 + 200 X 40• 300 + 200
16500 + 8000500
24500500
= 49%
3m: Q = 100 - P = 100 - 49 = 51% ^ yldJjial 3RR ^ iTFmr ■^,
i- + iTig
<^p,-p, = iPQ\
149 X 51V Uoo 200;
49x51x50060000
= V20.825 - ' = 4.56%
^ yfdJ^idT % simt ^ w«femT % f^P1-P2
\
t =
^rixjai /wfvqf 135
55-404.5615
4.56= 3.29
t = 3.29 ^ "RH .01 ^ ■% tTFI 2.58 ^t 3T?T: "RF .01 tl .-^ 3T^ f, .01RT ■f I 3TF: yetioi t’"% "RIRPT "^STlfF ^TlfF %■ 57T^Ri37WR RT mm % ^ wrfF ’4'ti
^ «*«iP«ifi jifri^irtl % 3?^ ^ «i«f«hrti (Significance of Difference between Two Raleted Percentage)-'^ "RT ^ R 3Tsi^ % tRcTH FKI
^ ^ ^ ^ yfdVf^ eFFF '1^1 ^ FFFf^%TF yfa^ial■% 3RR Fft HI'I't)' "^fe fFH ■5TTF "t-
'^Pi-Pa == ^tJp, +(Jp^-Sro
■31^ <Tr, F«F c^ii, fIf! Hi'i't) t’ >5j«(f^' r yid^ml % 'R^R "RF^R^P^ 'fRRF (({)) FHF 'tl ITRjjFRi yRufii (Pooled Percentage) FR "FiT^ RT F^ "FR ^ f^FTFF RTpIrt-
q-ii
P.-P.
<^P,-P, =^2cl (l-r)
' .
\ n •■RP^fFRi' yfflifid (Pooled Percent) Fit ’FPR 'f^PR ^ ■^' 'Fft RTTcft't
Pi + P2
^Pl-P2
P =2
Q = 100 - P3^7FW-400 ^ % FPT ^ SF8P#' RT f=PRFF
8R1 RRT R?PT "F fsdlq "^^PTf % RfF "RFRIF "^IRF 'FI^ 'ST^ ■% 3ltF?TFl 'R ‘RT^^Rr apFT f?
vrm ^8rTSTRF’^F
Igrtl<4 R58R 240 (60%)100 (25%)
120 (30%)
140 (35%) ,«5Hd
160 (40%)40 (10%)BfRFRF
220 (55%) 180 (45%) 400
FH-R^ ^ RPe t 400 W^' ■^‘ ^ 45% ^ R8PT RTSPT RFRF t 60% ‘FR^ RT«PT RFRF f I 3^: n. = 400, Pj = 45% F«TT P^ = 60%
RP^f^ RfF?TF (Pooled Percent),
P1+P2 45+60P = = 52.5%
22FR^ Q = 100 - P = 100 - 52.5 = 47.5%
MW136
_____ ^-BC' . ■" /(A“+ B) (C + D) (A + C) (B + D)
140 X 120-100x40 “ V240 X 160 X 220 x 180
= .328mi ^ uRjVIdT ^ 31^ ^ TTB^
?/75r?w ^
' _ 2PQ(1-r)crp,-i‘, = V n
2x52.5x47.5x(l-.328) /t ■ 400
335L6V 400
= V8.379 ■ = 2.895
if ypdifidf % 3T^ ^ % f^.
t =^P.-P2
45»602.89515
2.895 = 5.18
t = 5.18 ^ ■*TH .01 ^ ^ (2.58) ^t. 31rr: -If? .01 m tl ^^#TTT ^ MRcb^rMHI, 3lPdJ;MT "4' 3T^ f,
.01 ■??R ■'R f^RRT "^rRH "^l 3RT: R^icIT "f 1% ''R% HpiJfia ■’RRR RRf^ ^
% 3RR ^ ^ -m ^ ^ ^ WT ^ RHdd t-Hlish
100 >/(B + Qap^.p^- —n
^ B U«RT WT aRIBTRf tn;;:^ ^?RT ^ R?R?T ’Sm)' ^ t R«TT C URR ^8T4 * ^ WR! RR5 .3RTFRR ^ RW tl
^ (Proportions) 3TRR ^ -RTetaT ^ ^ ^ ^ RffTVTdT 3RRRI^f^KTT ■'Rt^ f^-T % ■§■, ^tlRiy, -dR^I aRTR^RRi "tl.
"RF^RR^ ■’jnf^jl’ % aRTT RTSf^RT
(SigniHcance of Difference between Two Correlation Coefficients)aq^RiTH RiFl! RTR: ^ R?^RR^ ^«i|ebl if fl ^ ^
% R^ if f^-f^ ^uiiV) RIR R^ il RRT Rf W RRRT f^-IRRT "R^
V
i^rfyRT 137
ST^RT t’l. ^R ^*^1 "STTR ^'Jii'+I ■% 3RTT
^ '^TT^t^RT fig'll ■?Wtl ^ <^'Jti=hl % 3?R^ yrMlqcfi % IH■f I ^uiichl (r's) Z ■f ^*TT. arHij-gici cTP^ Z
% -aTRT ^ ^8taT f I ■^rf^ ^ ^ ■Rpff % ^ 3FtR f ^ r's % ^ft ar^■?rT«f^ #ni ■EFn % ^sFm ^ ^ 15^ Rncid t- , •
11<^z,~z, - \ni-3
RFT n.j cT^ 1X2 ^'aTRTR ■fi
"S^T^TR—200 % ■'^ 'Sri^R^f BIKIR "^fe Ryicn^l Pi'^fn % .70 ^■’7RT "W "^aRf^ 250 'ST^aTlf % 1^ RH %itwt .62 ''TRT 'Wl
^ tJR 1?M3Tf % WTRT [qtiici^l Pi^hRi "% f’R f ?
1X2 — 3
n r
200 .70
250 .62
^eti*«<A6T ^'j||'«til f'^TTR % ■^’ ■sI'aCiA ^Kuil "t r = .70 .62 •%Z ^ TIH Wm: .867 ^ .725 tl ^rf '^’ % 3TR^ ^ iTPRr
11/^Zi-Z^ - V^i ” 3 n.j - 3
1 1"^200-3 250-3
= .096
STcT: •^' % 3RR ■HI*teT % f^ ^-3?^TO,
^1 ~ ^2 ',cfz,-Z,
.867-.725
t =
.096
.142
.096= 1.48'
t = 1.48 ^ .05 3TR?^ TTB 1.96 V -^ %.^'nltny. %\ ■^>?T Rl IRRT ■! R'^IR^ T?n w cT^n ^RT3Tf •% %TT f^-'f%r ti - .
^ Tprra % SRIT ^i«Ichdi (Significance of Differencebetween Two Related Correlation Coefficient^ TlR^ ^ WtT "Rr ■n^ ^ T^lf^ ^crlil ch<.il "tl 'HHT1^ Ri^l yfcK^f ’'TT 'HFHR ^Hs
(X), aiWr^ -^m (Y) ^ (Z), ^ rW f I 11^ ife (X) (Z) % ^
,VJ
138 •
(Y) (Z) % ^ “fcTr^I 3117^ "t "^ff^T^ Pi^mIa(Z) ^ ■?miPT ^ (X) ^2TT 3#TtaT (Y) ^ ’5^ '^8? 3TflT^ f ? ^■37^ ^ ^ % %TT r^ xT8^T r^,^ ^ oR^ ^ Tn«l^ ^ Pwf^iJT ^RTTT '
WR*^^ ^'^n=h ^ f , 5tilcrii< ■'RPR <i«^PeIcT "t ■^8?t '^f^ld■RTsfer ^ ^ ■r^IkTT ti ^RT8f^ % %T7 ^ wn PHHc^d ^ t-
q/7^cw ^ ?fi^
% ^«iP<TfT ^[PTf^ ■% 37RR
(^xz - fyz ) V(^~- 3) (i + ^:>y )
Vzd-'-^3TTRT (-sT^qm ^ df= n-3 ^ tl
■35fT?TW-100 -ST^' % r^ = .50, r^ = .70 - .60 ^ cTiq?3RR ■^8^^ tl
■5K- n = 100, ?' = .50,— xy
3Rf: r^ ^ Vy^ % 3RR ^ -RTster % zt ai^qRT.
«.=— r^ + 9 r r r• XZ * 3^? “ ^ ^ ^
r^ = .7.0 = .60
(.70 - .60) -y/goo - 3) (1 -K .5^I =
V2(l - .50^ - 70^ - .60^ +2x.50x 70x 60)
1.206.8
, = 1.51c;/= 100-3 = 97 .05 ^ .01 -RT^’-qfOTt ttH sfRT?T: 1.98
2.63 f ci«iT w (= 1.51 ^ t, 3m: 'q? 3Rn«fqr ii ^ ^ 1^% TTWT 3RR-q^’tl '
2.9 ^ TTT^rffn^ (Assumptions of t-Test)
■^-■qftsinqStatistical Technique) 'I'l ^^Priy, '^-'qtl'^ ^ IRlhT "qF "^R #TT '^if^ 1^,■q^ ^ Tt wp ^-■qft^ Prfl^T RRmrs^ ^ ^ ^(Assumptions) Pf*^qa "t-
' (i) yfdq^lT ^ f^Rft RRfe (3?8mT ^HpiqT) WST 'Tm 'tl
(ii) ^fRf yfriqj?iT % yiMiiWi ^mPT-^RR "RIRRI Hilq«t>ai (NPC) % fqaRa f I
(Hi) TirffRYlf % yi^iVil ^ f^ ^’RT TWR tl .
fq cfN^ Hi-Moi37lf qRR (Random Selection), WTPT tqcil'Ji (NormalU8RT
RP^rar qt^ci: 3lf^«h(rH ^ ^*qptm f 'flqPt) ^ yiyiiVl "% fqq<.wi «wiP^ ^yPiq^fiT % f^ ■RR: 3ffmR RRcTT^ ^ t "qp^ ^ yPiq^iT % 1^ 1^% ■<jqf ^ ?Rn T^rfl 't‘1 5^Riy. ST^iTRqRrf qrf ^-iqil% llf^ ■R^FT qif^q., fq^fVqq>< yRiqjfiT qil aiRiR
^ ^t -mFl ^ RFmr qfl (K.S. Test) 3T«RT '^m(Chi-Square Test) % Hqhl ^PiR-qq 'tl q^PTH ^ '?rq«Tnftqrn qft ‘qp^fTT ‘'jf^Tqit ^ ^ 1^ qft^ (Bartlett’s Test) qn qqtq ^ qn ^Tqmr ti
<jT5^(7< J1ff^s^Wt^ /q/c/^r 139
(Variance-Ratio Test-F-test)"SIR: ¥*T '^RTf««<.«! (variance) % ^ SijHial '4' SRIT
^ ^ afHc(J?tT '?T? «ehdl ■! 1^ "3^ UWTP^ ^ ^ ^crfR f^'l«t)l "SraT^
TRH t, ■^‘ RTT %\ T^? 3R7R % f^FTK. ?KI ^iRPTlf^ F-"^^ ^ WOT ^
T3^ tl "rtt^ ^ ^ t :F = ^ 'JRWT STJOT■31^ s5 > Sl ^ F =
_ Larger estimate of varianceji ^...................
Smaller estimate of variance
Si ci^O< IRWT 37^^
,g2 ^ S(X2 Xa)^^ N2 - 1
37f^ 'SRTW.'^l^ ■% %T3; WTcF^q (Uj) tfSIT '^T^cR IRTW ofT^■7?R (ug) ^ t-
Uj = degrees of freedom for sample having larger variance.Ug = degrees of freedom for sample having smaller variance.
F-^td«+)i 5% ^ 1% ■^TT^fsfFTT ^ ■'R Uj fRTF V2 F Wl W f^RT '*ii<i‘iil ailVi4)iF % MR^lPuid 1TH ^ tl F ^ TlWro WT F 5% m y
^*=71 V2 % dlHrl't)! RH ^ 37fiR7 ttcIT t (F > F.^g) ^ WWJof ttm tl F > F.^g (Uj ^87T ^2 % f^) fft W OT ^ 1^ 'Srf^T^ ^ ^ Wmf fOT ■RH 11^OTT WTH tl , , -
_ S(X^Xi)^Ni-1 ’
S?1 -
WkF^
1
SRIW
(Illustration)fd<7^fl t 10 A, 3RRJ "t t RrT
It :10 6 16 17 13 12 8 14 15 9"si^R 112 ^ B <%<.i<*i "nf, "3^ 3Rfv t "3^ t Rrt "^fe
^ ^ :
7 13 22 15 12 14 18 21 23 10 17«lN Tiinny. % Wife Pis^iT WTPT Tiwn ■^au?. (F.Q5 for = 1, Ug = 9 is 3.1112)WHSfR (Solution)
^TT^A
£(X-X)^ 120 120■ N-1 "10-1
Si ■" = #
I(X-X)^ 314 314N-1 “ 12-1 " 11S2 =S2 = = 28.55- = 13.33
' "ST^, Si = "^ wnsi=w^^^^.
% ^ Rot? f ^ W8fe 3RR ^ tl
28.55= 2.1413.33
F^^^ 2.14 < F.(J5 3.112, am: Wf Wife IRRoT
140 3WfK
(Assumptions of F-test)HMdR to ^ ‘F-^5T^’ 3TMftiT t, 3^ f-
(1) 5r^«t> ^ ytiHi'*! ^ ^ IqdRd ^'^'^TT 'i'l«hi "HTwr ^ "SraT^ ^1(2) "aftot ^ ^ ^ fan ^ TT«n ^ ^'i
(3) (T? n«TT a| ^ 3fTfqRT 1 'm 1 3Ttor .'^1 ^ ^ t ^ cTffR3RTT^ ^ nm to ^rmr ti
(4) ^ F-to^ % aifqm ^ tofer to t, am:■^FTicn^ ^ «H=hfli 't^i F-f^wT to f tof a^R ItomT Ito to "ti
L)j ^ ^ifeMi atcd) '3'i«t>1 n^fn-Tramp? fqa<Ji «Ft afR "^to "tot 'to^ ^i(5) 'SRpn % toff ^ ^ ^ ^ -toTotal Sum of Squares = Sum of Squares between the Groups + Sum of Squares
within the Groups)
fnto>riT xRtto^ to ton^
to
(Limitations of Tests of Significance)% HiHi«hl ^ "Pivto % Ito 3to) nflfinn fto "an^ ■f i
"to f.i ■mtom nrltof to tonqnrto f, to to f--
(1) ■mtomr toto' ^ toto tor ^ tor to tor ^^rmi i ^ tor^ ^ anvR to wr• I (Ya-lun-Chou) f^rto t, “ t^iPT
to' to ^ ^ f to ■^' tor to to' ft ^ifeton totof to torn 'SRtotoanf^mi 1toto ^ to^ '^ntoto
t ftor
IWlW 6
toto TT^rto totoi”^
tototo ■^, titoobcii tosnn Itoto ntopm? to "to to^ ftom torm to ftotomn mtoF ^ qiwifqch apR i^mr ws n to to> ftoto uqirH4j' to "^rto ■fi HRetitTH'ii tofertoRf !4mi«i to ■?mT to toR toto toFq, arto ^ citotom toto toto3ii to tof ^ f^rm Inciai t. ■5r^,.HR4»c^'ii toto/t, artoi^ ■armw to ai^^ "t crit ^ ntoton't ftoto
^totor ^ "Rtotor irH-'^Tff to toto toto fi totortof to’ mr arraiK '^tht to^i
nto toto f t ^ tototo?
RrRrqdi to ^to to aRR mtotoRr '^?’7 to ^n^to to toto nto tom ^ ■JsmmTi ^ toto^ -Rf
arto RF to to amtoftocT 3RR am^R to rnitol to ntof toi "aiR srr nto ■mtotoRi ^ to Ri«to mm ■Rmr to to iRmr arto r?" to to rf rir fm toRRR to?RiR rr (95% rt 90%) r^ to Rto toi
Romr 3RmT
(4) ■mtomm ntoann toRm rf to rrt^ to to apR totototR ma to m*to to, rt^ apR toRTRoif tot nto RRTRi toi tor to ^ aqk atot RRR RR -^ttr tr^ct tom to to^ afk^^ ^tr ftoto^1
aHiJq^i totRTR "R? to to totot fRtoR RTRto tot totototR wj to BT«tom tot tofR mto ■RRR, RRtoto mr Rton rrt Rtonto mr totoRR totoR nto toRR to itr to "rtr toRi rurf "RifFR, toR srmR totot to apr tototok toto mr toRi Rnm ii
mi^jtok towto I4i
wf-cpf (Chi-Square Test or test) ^
■^-■qrf (x^) ■qfr^ ■sh-hI'D ■'KtsjTTi "t cpi^ 'Prq#? (Karl Pearsqn) 1900*?II Iq^fc^ojuj ■^' ■^-ycqci ^®n oZITW
oqqeifl^ "tl <£in'-sd ‘JuiR*iq> HRqurM'ii ^ fHieo
% 3T^ ^ f 1 W\i-W\ '^m ^ aTpTefcT WfeR^ 'NtFI '?[^ t^cK^T ,ar^tTT ■?TT8?qKn "m ^ % 1^ fen t i
■q? fern ^ t ^ 13^ ^ fef^ ■^‘ d'4>4yl 3mml t TT«ii si^eilfer^ 'J3?? 'SIT ^chdl ■f 1^ ^ "fer TPU 3?T^f%
fqcROI ytHiHi-q fqoRa '^RTI ^ "t^? fH ‘51?^ ^ "3^ "SIM fe( 3TT^f% fqo<.''iyyiHI'^ 3^rafeT fef^ 3TT^f%qi ?fRT "^n ■^T%, ^ ST^^ftfer 3TT^fe^
"^n opT^-'^ 'T^ts^iJT 3iqcriir=hd (yql'iicHq)(H^qiirMpieb "qr fqa^wi % 37^ % iTFT"% ''Tft’^Tfe 'feTT «iq>di 'tl'
qr^ 1^ ^ fqa<'jil % -sRR ^ ^ ■qrw 'fer ^ ^ ■^’Tife ^<di
fqa<«l ^ ^^Tqpl 3^
t-(0-Ef-
x = 'LE
fert—O = yr^qi ^ 3Tqcill^o sn^f^Ri (individual observed freque'nCies of each class)
E = yr^ w\ ^ 3n^j%?^ (individual theoretical frequencies of each class)
^ ^ ^ q^i?q><.wi 'A "Sr^qi ^ ^ 3TT^1%2Tt (O) ^*1T3TF^1%^ (E) 3^^ %qT ■^HcTT ’fl O - E ^t^TT f cD Pi^qq '€\ (x^)
^ #TTi O tT«TT E ^ am fenr arfsT^ qrrf-^ ^ ^ sfl mn t arfirsfr ami ■qr fq^^l ^ ml fe ■^nt f i amf ^ ^fer ^ ^ -q? t a^q^r wRi^^qr -qqjwiirHq) aRR 'q<i«R alk Ml\«iHtq<!^, arqvilfeT qqi '^^Tpqqr aq^f^ qq amq'i«q ^Iqii 3RT^ qq q^ qR^ % q^qr^f, yo^q) q^ aRR q?l "aq q'f qft an^fqqi (E) %
"Rar qrar t, fer^ aq^qrfei qR fer qr q^i qfe fqqci'iT qrl qRRql^qq ^*(I^«t> 3TT^f^‘ qm -^Tjyf fqqTTjT ^ qft^ -^sm tiqF^^rc^qqqH^fiqqrf^q^fqrt<^[ ^ -qj^ qR% a^qO ^dil jc^-lqTRtq '4, arq^rfeff qft ^f<sHi, 1q^ iqRFqq aT?T(degree of freedom) % qq 4 qftqrfe fen qM t. % fe feqq qi«fen m (level ofsignificance) "'K 1^ ^4 qn 4 q4 qi^t "ti qf4 fq^Tq :x^ qq qftqfqq qn 4^ fnfeq q?T 4 qjq 44ti "f 41 q? Pi«t>4 14qq^ ‘qrar atqcrilI4>a qq? 4^n14Tq> an^fqql 4 qpTqq qqHcn "t,
t4 qfq arr^fqql % 44 fqqT’4 (aRvrlfqiq q 43114^^)arqfcf 44 4 am qr^qr qi 4 aRTqrqqi ■!■ 4 a:^ qq qfqrf^qcnf^qq 4 ^4 4 arfqqr 4qii qrqrr’q qq 4,qnf-q4 qq arfq^ sR^hryil (null hypothesis) q4 aRqlqq^ qR4 q^T qRT ’^mlqqfe^ qR4 % fej; '^^iq 4tt tl
qq^-q4 4 '^ifq qftiqqqr qil ^ qqqr ^iqq fen qnqr t: “ arq^fe arqqi 4^ qqi r^4i-ti ?KT ?nq 371444 4? q«T q4| am q4' ti" qq^-q^l qr^fen q4^ ?q q?4 % m qRqi t-qqi amtlfei 371444 (0) qqr 4^qt4iq> aiqqi ycMtl^ra 311444 (E) 4r qlq 37^ fqqi arfq^ 114 ^ i3.chHiq aiqqr 4 qqt^ qm qr qqnTT t? 44qqq Ri4w 4 q^ 4qr t f4 ^ ■^^qiqqq (44qqq % ai^R) 44—^dq>l qrqi^n 4^ W4ilqq> 4 ■§—m<'^ q^ "^qqiqqq 1^4 3714^7
M&Tq
142 wf^Fi^ f^rfm
t
f % ^ fTctiddl % 37^ 37%^ f[ 37^ ^ f, 37^ ^ 3T^f I ^ 3T^ -ziF-sfcnFr t Cqt=^ t) mm a^^Mxn 37^ t,
^ aTftRT ■^.'PT^ ^v(cTT "t STqcrllf^cl 7T^ yoHiRfia 37T^t%^ ^ 3P7TT ^ chK^i 3icR^< % aimiqi 37^ 'f’l
(Characteristics of Chi-Square)• 37^ iqVl'^ai't 17^n7 ■^-
(1) ^ ^ MRJiruid ^ ^ t-
. ^ ■ ' x^ = sf<Oz^l ■
^fi<=ttrH'ii 3???
E(2) ^ dirt4 "377 '.iif4etiai-'7»cid (probability function) f 37?T
(degrees of freedom) yi-^ci (parameter) FtdT "tl qi-wiq ^hRcIU F^ 't'l WcF97 37^ % F^ 'll
(3) '71^ fqcK.wi ^ 1%F oiiNi 7^ 3^ehi iTT^ 3f?T ^ Fbni
. -TT^ ^Ici-^ 3T?I 7^ 2 ^ ■^, tTCT^ ijpJMdi ■^' ^1 ^
^ F^ f 1
(4) 73T3f ^id'^ 3T5T (d.f.) 30 "TTr "3^ oFTR fIfT ■!■ ^ X^-tFF73F.tW7 F^ tl ^+>■^1 f^cR^ R 3037^ X^ % 7^ IRTRFF 'F'7 14nf<t1 ’^l 37f^7FifR "SP^ 3^
^ WTT 30 ^ FR7 t, 37F: ^ ^ 1^' FF ^TFFR X^ IfFT^ ^ ^ t ft!fFFF tl 30 FJR ^PTfFR 37?!!"% X^-t^FT’F 37f^ IFFR F^ ■!, 37F: 1^ 30 FF^ % Wld-^ 3?7T ^ X^ % 37Fm-3TFR ^ fl ^
-3T9ff Ffl Vm 30 ^ 37fi7F7 F^, FFT X^ % "FTF FTI 3RaFf^ 1^
FTFTrW
3Tf7JFr
««<lnoq
FIT "RFiFT ■§-
Z = ^-^2(d.f.)^l
X2 = FF^ 'M F^ #r aT^FR X2 FF HroiPuid
d.f. = FFTF^ 37TI■JFRT FfTFTTFf FF 'SRTM^ 'SnfFFRn FiFR (normal probability function) '^FRT Fff Fn# t, 3787^ .01 FT .05% ■RT8JFTFT 'm m FF (two-tailed test) FTTclt fl
(5) FF^-oPf fFFTFT FcfT 77F^ (continuous) fFFFF FtFT "I, FT^ FfF "FF^ Pll^d 37T^fFFt Fft <70^1 FTF, F^ ^ W<sqi37t 7§f7^FFI FFTW FTTt FFFT % FF '^TFT^ F^ FTIFI ■!■T^FT M ■% fgFF-tFFTTTT Fi^ ■RFF^FTFIF FTF^ FtFT't^l W Fif3FT| FR^■% %T1, ft^ cft)oo 37T^[1% 10 ^ FFT FtFt "t, "43 Fft "^ife (Yate’s Correction) FF TTF^T fFTFI FflFT ti
■^IfT "3^77 FFTFT FH "^FF X^-FiFTF FF 4>cici FTF^—TFTF^ 3^ Fit +7^1—sldlfl FfF Wld-^ aTTff Ff ^Vll FF7 Ftcf f Ft X^ fFFTFT FPft 37^7 ^FTcFFT IfFFFI f^ FtFT f (positively skewed to the right) ‘v (IFTF^ 37^ Fft Wo41) FFFt fttfI f, fFcR’F FitfFFFFT FF7 FtFt FnFt f 3fR fqdOF dlsJdl "f TTfFFtFFT % fFFTZ 37TFT Ffl^ ^ri 'aTf^FT FTFT % ‘u’ %
37^FRF: FFTFT^ FtFT f-F7 fqa<u| ^-77^ ^ 1337 FT^ FIFT tl
fFTlf;
^r^=hin 143<iT.^Aai
fTR to ^ t-
100
V = 1
y = 5
ij/ = 10y = i5
25 x25 10 15 20
Chi-Square distribution with 1,5,10 and 15 degree.s of freedom
0
3Tyr (Degrees of Freedom)
tl 3T=q 7T^' '?TTFin ^ -^^ujrrf 3TT^f^ ^ ifWFT ITH^, wfi 3???^
^ *rt an^-wn ^ "sn Itot t? ^ 31?=? % ^ ^ ■sttctt^1 <s<^ie<wi|8j. ^7^ 5 Wo^iit, ■RTtocT "f-
22, 17. 36, 9, 7 ^ = 91
<<iid-94(
■qf t % ^ ^ WT qJT ^ ■3n t -q^ 9i qn -qtn
^ to ^ ^tor qgq; i\ toto i\ tonti ar^ 'to '^, ^ 5 ir^aff qn ^ tor ^qrar t■qqfq^tn*^ af?^ "ait WoHi 4 ?Wt, alPdH ■torqn
% 1^, ■tofftq ^aoidi '^Wtl qft ‘■qnJi' (constraint) % ^ ■qft’TTto 'toTT '^nqi^KTr tl ^ tor qqi t, aicT: ^mr qq to t, artoto aFff -^ftaqr (5) t' ^tvtqrt -tor (1) ^ ^ qq 4 to t, am: 5 toraf qn to to qqtl to tofq ■f
61<.=b<.l<
ato ^ tor 4 toam ‘N~r toffm to to ti-+qia-?q
diRnetJi t, Itot arto qto (rows) ^to (columns) tf. q^icpaq amr qft mmi fto^ qft tot t-
r = qto (rows) c - ^to (columns)
d.f. = (r - 1) (c - 1)
q¥ qjq ^qmr 11^ qto qft torr ^ i qrq w tot qft ton ^ i ^ am toiq^TcF^
■^to sfito qi7 to mRf ^ to «IT%int (constraints), qto tl 2x2 anto mftoq12 qto fr«Tr 2 to to t am id.f.) = (r = 1) (c -1) m (2 - 1) (2 -1) = 1 toi tom
qmcT^ am ^ dfrt41 fqtoqR «t)ai<l ct^t qtot % toftto tthh qto qq§^ "qR qnto t 4iqci q>t ato% totffm ■qto to q^q^cfT tstH ti "to qq qmrqt ort tor. mqt qn tor
cf87T ^ to 3 ^atf qq qmito tor qqi t fri: w qqm^ amf t ^ to qmtr^ am m«nan % qiim qrq tt to tl
, qm^r^q
144 Tsqffq mffeitoq
3TI^I% fqa<«i ytiiHi-H ^ 3TRT^ ^3n?IT't 3 ■^TVf3^
troTsFr 3TRTf^ "^n;^ ■% 1^ ^«&il^'t> 'STT^f^I^ W (X) cf8?T TT^rFf
(ct) r'1«t)H1l '*ll^'ll, a1«0 ^TVT ^.&lpi1't> ^ qit^fq«h % ^<\<^<.
^ ■^l rft^ ^tn^-iTTWT, 3ITTN cT«TT 3TT^M ^^ X2 % ^ajtJT i% -sm w ^ f ([; = N - 3)1
^qcilf^a STT^f^ fqci<.'Ji ■% 1^ ■’^fPTOR % ITTK^ ^ WTT ^ ■sncft tl "iTTKiT ^8TT "tef^ 3TT^1%^' %yiMMd: % cbKui X^ Sl^dW-lT -H©^! ^ =h<.<^
3f?l W ^ f (u = N - 2) I
^ 37^cfrt«Fj^ STT^f^nTT qiiql «ITtft't, ^RT-'^-cfiTl
i ■?^TcF9T 3RT ■^STTcn ■§■, <W)T^ ^ RRR «SI’1I "^Wi t^TcFft '^NT^ff
■^TT^^, '3?T^ ^ WcF^ 3T?TfilV'qd RTC ■'R a::^ ■% difciehi RTT '^. •sh’^Vi «IT^ 'tl
3^m0^ui % (Uses of x^ Test)y? ^ 'SPTt^ 37^ WR ■RRPTIst t f^RTT R^RTT tl IRW TRiR t :
(1) wdi^Hi w x^ mliyui-wd-^idi ^ t’ wwt t % ^^ ^d*^ tl IRt HStq'^'jf ■RRF7T RRH t iq^nn RiT fd^k^ «t»<.'iiti t % ttt RF 378j .Rtf fHclil^l Rn RRkR RRR f Mt 37TR?^ RFRiSRI FtRTI FRRR RF t % RRR "t Rf^ tf RT ’^R^TR Rt tf, RR RRR 4 ^ R^tR-yid<J^f ■f aSRRT RRTRt’ % RiRR '^Rt "RiR^R y<^f5ld Ffl 3RT: RfRR^ "t RRfflF R^R^ fFRI ■RT^ R ■RF^’jyt FIRT •ctiley. fRR^ 'RF 'RTRT "RT "R^ qiKiq t '3'^ RRTR RR R^R^ RRR "t Rt
Irurh tl
■Rfe
^iRf
, FR TR ^ ^ Rft^ RRRT RTFt t % oZjf^ ^ ^ajfnTRT (RRr■RT:) ■RiT'R^R^ R>pf t fR^TKR-^TRRT (3RR RT) tl 'Rft "FR 'tf 'RR! RT 'JRf-** neilq^nnqf RRl “RFTfR?IMR f rW^ RFf” t fRR^ RR RR^t fl RRTR %t4 fR^RTRR 87RR1Rt ■RR! 't—* ‘'^’'t IrRRR RRit tl F^ tf "Rt "t '^RR^RT "Fft 'RR Rff^ 'RF RRfflR 'RitRT
RRT Hglfqtildq t 'SffVl^TR tft RT R Ftt RR Iqitll M=hi< ^ RRR ^ R ^ t RiTt ^ "R^^ t RT RFfi RIR ■RRt RTR^ t, "Rf FR RTtt % '# '^RRRT 'RFf t 37^RT ^ ^RR^ ti RkR^^RRT RR -^m ^ % 1^ % # ^RR^ t, FR RRf RRfRTfRt % RRR.f ^ ^R-RI^ ^RTT ^ ^TTRR
3RTFvR«R^
dlRddi! (Contingency table) t 'RftR Ri^l STRctf^ 'RRT yrmHfid STT^fR^ ‘^'RRC y^ Rft 'R^RRT 'Rft Rlttfl 'RfR y^ RR '^TftRpRR 'RTR fdfVqd 'RT^fRiRT 'RR RT RRT ^P^d TRTR^ 3TVft RT 'f^ "Rt :i^ % RT%RR RTR't 37f^ t TTf FRRH RR tfRT yrniPjia RRT qiKlfq=h 3R^fRRf % R%R 3RRT SrfRRi t str Rt %RTT 37RTTT '% 'RFf "RTRI RT "RRiRT, RRT 'tHf’ "RT TRR^ tl '?R% 'fRR^R, 'RR y^ RR ypRlf^rd RH RTkRRR RPT RRT FfRT t, 'Rt FTT^ 3R?TR RF t 'RRfRRUT % % R^R R^f
RRTR
fRFIRTR RFf tl
(2) ^rjiKidi 'RR x® Rff^IRr (a::^ Test of Goodness of Fit)-R^-R^ RTfSIR'RR'RF 'fRRf% 'Rirt 37R#f%R 37T^^T^ 'fRTTlf (qp^l'^ 'tST^RRT 3T[^iR "fRRTR 'f^RR.
WqTTR, yTIIHM 3TTf^ % 37^?^, t RT RFfi RF Rffy^ tstr R?n "t 3TfRRT4 FTRl t RRf^ %RRI 3lf^ RffRRRt Ff "SITRT Rff RR TTRiRf tf RT^ IrR ^ RTR: tfRT t. % RffRTTR TftRRTRf Rf
iRdq^l ■% 'Ftt tl RF RR RffWRf TTRR '% 'Rlt't 'fRW^ 'fRRRRrt't 'RfR T^RTT tl 'STR: TTIRT^RR:RTh
145T^RfTT
^ ^ ^ ^ t WI t ^ ■3^7^ ^ri^ ^^Ipdd^ -d^n
■% ■’T^fiPT STRiVd «hW) ^«siPd^ fqfl'i.vi
iiin<^J{[ f^ci<»i ^ ^ ■'7^^ % %Ti ^iTcTT f I 'SRHI
d^otai "tl ^JTrTI "f ^ yfas^f'
37Wf^d ^ 'ST^TWf "t "^l
3iJti'*n ^ ■Jcfn^rlT ■'Td^inJT ^ MR=hcHdll ^ IVi<h1 {qa^^i i?'7^, ’^T^TOd, 'SJTTFTRT 3nf^ ^ WT tl 71^ yPdi^vl ^5R ^vTdT 11¥ ^ yPd^J^f H^ciroT MPidiC'MHI
■^■.■^cn^ 7T^ f^?Rtrr % 3^^ f -^n MR+<rMHl ^ ^ t 'STt ■^'
"f f^ff^ r^^ehi fqd<.'J|
37?T % aiRn«M "^TR 't, STra^ ■RRT t ^8IT^ ^ "31^MR+crMHI ef^ ^1<^i< ■RfTzn ■^TTcTT "fl 177% fqwOa :)fi ^ ‘^fT^lRfR '*7R :i^ % aiRn=hi "R^ 37Rr?■?hn "f cff mR'^itH’II 37^%^ fd»<J| ■^fRI't ?[*7T PiW4 Pichicni ■^5TRT "f 1% "^Nt f^fRT "O;^
3rRRR ^ 1R#T ^ "^n;^ f i
'ie?)'>i
(3) 1^ jeldTUT % Ufii^l % ITToTRsrt X^ "Cffi^nT (X^ Test Concerning theof a Normal Distribution)-177 "'TTI^ %f 37T%nT^ nMdi "t f% TTiTIf.variance
f^ITT^ ^ yfd'q^^'i n»i*ti ■f, WRP? fqdRd 'f I 177%^ ^Id1<l IRf^R TT^RT
■f 1% ■5R "TTHTJ ITTTWPI ?R RiaRd hihi'T)—
(n-l)S^X2 =
(n ~ 1) iMM^\ 3f?T X^ % 3737TR f^Tfer ^ tl 37cT; X^ % TTR %t ^R=TT 'qfeRTIT % % 1^ X2 % dlPddil RTt %t W TRRI tl ^ "3^7^ TJ^ f n W 3^^ f ^
fig'll efiRsi "^tt 11% S^ (^rfcRlf ^ 3I77RT) irfd^-iR 1%TRT X^ R7 % fqa<'J| % SIc^RT tt Ri=b<i , tl 37rl: S^ % 37T^ ''77 X^ fqa<.«i ^ "Jplhl '9177% 17^ TT^-TORT't <HM(RtR hRs^ct'^^ ^ ’’Tt^
Wl tl ■7n«NRn-W (a) %T W t afR Wd-^ 3T?T (u = n.- l) W t, cT^TT
■STRtPT^^ ■'TR^vRRR 177 y«t»K t :Ho;ct2=:ct^ ^itn Hj:a2 = a^ ; •
■Sftf ■'77 cyo»'9o ^ 37Rt3 IfR 37«raT nRctiurnRieh WT’^T t, 3771: ‘ftoRf ^ 37T^ "IRtT—
^0 3T7%d. -%^ X2>X2 ^«t>frMeti HR=h<rH’ii % Ri‘jf'4 37TVR 'FRtT—
V
Hj 377%^,,%^^ X^<X^
TgirPSar 3m (d.f.) 30 ^ STfei tt% TTTt^(Chi-square Test when d. f. Exceed 30)
diRichi 30 T^R^ 3T?T cRT '^nTft tl 37T7R TTlf^RiT (Contingency table) t imt ^IRl^ ^ 7§T^ t t% (r - 1) (c - 1) > 30, 'Tit -^-w\ Wl't' TWt^R RRT t ‘MTOHM ^ % 3RPfcr ^ -dlfddil ■^,317%tt %7TT 77% 30 ^ StRrt T^TcI^ 3m
tt^ ■'77 ^j2K^ f%R'^ Si^hma: "aTTlR^ fqci<u| FtcIT tl v2X^ fqa<.«i ^ ij2v - 1 cTR RRR
Ri’dyn 1 % ^7R7 ttrTT tl 3R; ''7^8^ "RT "SPiRl 7T7R "FtR t '^TPti ■yj2v - 1 "t v2? %
%t 1?Fi^ 3IR'7 fq-q^ri’i "STTriRR-l^gTR (normal deviate) % t ■fMf%T 1%R "R TT^RT tl
(iJ, a)
{v.a)
3787f<(
146 d«qrt<
~ ■ 'Z> = ~^2v-l
. ■STf^ X^ ^ vr<Tiru'M 1TPT 106 t cT«n 3?^ cTTf^ ^ 3m (6-1)(11 - 1) = 50 f ^ dH^cKI TFfN^m % 3i^^-
Z = ^2 X 106 - ^2 X 50 -"l = 4.61oe^fctrOra S^f^IrlilT ■^jJT ‘'IT c^^Hch^yi(Grouping when Individual Frequencies are Small)
% -spcid ■% f^, yrMin'W 5 (10 ■?NtfVlT) I eFR 'HH'WI ^ Chi'll ^ rH?11«r)<, '^n
"t slh imK ^ 'Sr^'OrHsh (discriminating) '^^TPTT /^l ^^ -qr am (d.f.) ^ fq^tm ■% ■qr^'^ ^ wn % m
. mm ti '“'tTnddT % ^ TmhFT (Yate’s Correction for Continuity)
x^ 'A' ^ wi mm f3m ^ 3T^^t^I^ ^2x2 arram diRiii ^t ^ mr- "Hmim % fmi ■?jpf mnm ■qmt. ■sit X2 % ■qfmfmr -qn ^ mq mi:..'^ t-
[|0-E[-lf
rni qm?Fm
X^ (corrected) = SE
(|03-E3,|-|)^ (IO„-EJ-yE E2 ■ E„
•rfe— I O — E I aiqtrilf^d, 3fT^fq qqj aq^fq q>T mirdcb m ■^mrfqm ■f^mR ■q^ (absolute) 3RR qq WN^ "tr
Ea
X2 % -qftqfqcT im mt mq mr ^ qqt=f^ mf -q?^ 3rr ^ .i. qr0.5 qq? qq mm ^i qRffqq fmm mm "f qqffm ■mif-'mf qmi^ q "qm (continuous and smooth)- Ftm ^ ‘Sfqfm aq^f%^ % ■qmq mt ei<s4i armq FtcTt sfRqq^-gif ■% ■qftqf^ "mt <aru5d •'mm m ■qmm 'll ■qlq ■qf?K9f mr aqmR ■^tm "t, 'eft ■qmtqqmr"RFm ^i fqq ■qfq ■qfqq?! ■mr 3nmR qm ■qftqf^ ■anfmm (criticalvalue) "cit mr qmjm ■% ■fmq 'qqtqq ■mr "qqtq ■qlW^ ■rh mt mq mq ■qm "t fqq^■?jm qftmmm mt ■^qtmq "Imm m qmm, 3FqqT armt^Kf 'mr ^ m^i ■
2.10 W-TOTW (Analysis of Covariance)
qiqPTd: ^ "q^ (Experiment) q^qqdf (Intervening Variables)fqqf^ ■qrr^ "mt arrq^qqrm ■?tcft "qqtq ■% mRuhhI "mt arqffed "sq qqi^ q ■mrq% qm- nU«mhT mt ■qfcmqq (Sampling Error) mt qtqr % aiq^fq ^(Treatment Variable) % qrim "qM mtmR fmqr m q%i ^.siPdch ^ qqtq ■qiqt "mq^q 'q^ (Treatment Groups) "4 yql'^T 'mt Wq ^4 (Randomly) 4 aqqfect ‘fmm ■mm ■qif^ mqqf qq^^ q^t (Equivalent Groups) qtr 'oqqttf ‘imm "mm ‘mflTii rniq^iftm
a^s^cJT Wf<s^<^l<^ tq/yqf I47
^7? ■4Rr^rd4)'^ (Experimentor) ^ rft ©qq^d '4 "Snii^ ^ ^ ^
WT: ■'MTI 3T^ MlW^lq qiK'Ji) ''©^IqeiRet) qifdii^qTch<H
dHtri®*T
(Available Intact Groups), ©q^ (Disturb) 1%^ smi '^rc^ %it^ •q^ ti
fq^t^q^i ■^li^ "^Tf
Wn % «Ml'iirHqi q-h (Experimental Design) ^ ^nq^T ^ SlTRTq Pq^c^qiJi f^rqi ^l^ii ^ ^*^eT ’4’ yqVil4<i'd 3Tor?rtf^ 3T=^
<^qvi f^RT==T ^d^ni % 37^ % ^TR’q Iqiqi '^IT ^chdi "t qqllq) [qr^’-i "^PJF yqld %■qR^ TFRH aT«T^ «H^crq q^"II %ft "Rsifq 3717^ ■^' %, SRRiq % «b'K'J|
% Fcriy, ^ 3iqvilf^d 37^77 (Observed Difference between two Means) q®7T 'SRnqTPq^c^q-Ji '4' % qtz^ 37^ % ^ TfpT (Sum of Squares for Difference betweenGroups) qiKirq-q) 37^ 37R7qr ^c^di f f«ra% qqTqr. 37«7^ .qn ©qq^a q^Tiq % . qr^fqr -cr q^r 3RTT8fq7 ^ qr qrsfq? ^ 377 ti qRqnqq: q^ % mRuiih ^ ^'■qqiq f 1 qtq Minrf qfl ^ 'spr 1qr^ qn#
\
qr^ ■fqRft qqrq fqfqH'’qr87T qqf (Class Sections) 3T^-©qw (Disturb) ^77% t^q qiq ^3787^1 tq^H (Matching) % 'SRI 'STSfq.TTijf qqr^ 3i^Hfd q?TR7f^ qqpq w^: qqrqraRf. ■q^ ^ 'f’ q®7T qqViqiqf qq qr^-'qqf (Class-Sections) qi^ q^TTqq "^qR qqr-q;^ qi’^-'q^ qq Tjqi-qiqi it %5pq ^ q^ q^) fqq?T #TT qs qqRn ti q qr^-q^f 37^ q7 it wr^t'qj) 37TfSiq q7 a7qfq ^ qfl 1^Rqf% qTl qqif^ qR^ tl ^ fq^ ^ TPqRqq TT^ -■
Wcqgyf qr ^ t qqr Mq^ qr^-qjt qlto ^ i fq^-fq^ i\ qq^q ti qR % f=Rqfq qrqRjq ^ % qq^ qJfeqi.qR ■^’ 37qqR qiq ^^-qqf qq qlq fq=i-fq=i Iqfqql' ^ Mcst^ % dHd'd qqqft qv^HM "^^iqr Id^MPd '^qcn 'fq^ iqfq^ qft q^iq^Fleidi STRR ’^yffTcI q^ SWt Ttf^iqr 1q^7f% % (Covariate) ®^feq> TTR % 37qR ^ qqif^ ■gWtl 3Tff:
qfeqi w i f¥q^ qq wtf TTi fqqr Rpqfqqra mR^ihT ^ .©^itsmiqp7q» "SR^ qR qqqft i\ qs ■qrq'qTTqff 1 i q^ qftqTqqq qrqqil Tq^ ii
. ^TRqft 1q ^ qfTqrfqqq mRuUh
pqlTt % -57^ Tlfqijf qst qsqqiq •^RsdlRaj ( 37*1^7
^ fq^)
. yqtq % qqrRT ■TTTJ^ MSilMH
-Rpqfq
IqRT^wjUj % ^^hrq 1?r^
^nf -f^ w
©qr^TH ■fqfq
qi^ ■jTqqi Mn
fqqiR-'fqqTf f¥q
160USJTT 120(gt^q
gcfl'M
110 15090 140
mptt 1 t % 1q^ qfrf^rqR-lwf qR7 qq^^ri# qqrq ^ r# ti qr^ qq qft 4\[4^ qqqqi/^"^^f¥qfq q7 qqr f qq qm ^ t ^wnq fqfq q q^ qi^ ^qq qqffqqr qRq ^ qiqf^
1^-f^ q^ qr^ W ■rfqqr ^\ TrnTq t qft qRfiqq^ ql^ % qqTqr
fMq qqffqq? qqrqTiT# qqrwTR^nq
/q/vqf148 q^qqr
-Srf^ 9f^ ^1 cfNt % ‘STjjt ^ ■RTCf^^Ri '^FRF ■^’ % qpK^ <^qci Pl'^Pd hR'^IWI’
^cRT ■^)T=n fH’4)'^ 3?^ ^‘^)dl f I "SRiR^ ^ ^ TO W 31^ ^ TO TOlRf % S^m m TOF^ f^T^ ^ "TO% 'a^ RRlf^ 'R ^ffl’«hT R^RI <Hll<sq=bl4j ^ R f^fi|>^ RiTPFcTr■^itMi R?-3RR^ (AnaWsis of Covariance Method) ^ inftn f^RflW tl R?-W^ PcjV^^ui TOT^ '^' 37R. TT. fqr^ % ^ amt 3RR^ ^^b<TT] ^ ^ TJ^T1%RR (Extension) t ^ ANOCO R ^
qiWc^
% sn'
f I ANOCO VK Analysis of Covariance ^ (Condensed version) t AN 3t^ ^ Analysis % f^, 0
of % "^^IT Co 3T^ Covariance ^ 'f’l WytR'^i % i^ ANCOVA icT^sF^ 'll *,
^«hfll
W3ITOT R’ TOf^’ ^ ^rfTO "4' ‘SRhl ^ TOR(superiority) (Inferiority) w: d<d^l< 3)^ % strt ‘4’ RT
% 3|T^ ^ RRtfeRT fro TOl ft 31%^ % ^fRcq -^i ^ -R ■^■■HHi^ilPdci '^<’11 "t "3^ ^ (Criterieon Variable) t 3IR: X R%rTT^ Fcks^tl 3Rm % TO R Rt -qr ^ TOT ^ t 3R Rlf^ ^ (Dependent Variable)
' =h'g^ f cTRl 3ITq: Y y'^di^ f^<=J^ 'f I 3{^T: 3lKhRqi l^fTOiTTsff (Initial Differences) "'K f^Ky-nl'T^l (Subjects) RT 3PThT (Experiment) % 3nfJR "4 RT
(Criterion Variable) % "TOT y^ilftid qfl^ (Oiterion Test 3TR^Pretest) ^ ajcR) ^ ^ ’rora^r (Initial Scores Pretest Scores) TOT■fl yqle % 3T^ ^ (Treatments) % 3ITO) 'qfl’ ^^'di % %TT,3T7f^ % Rm ^ 3T?1Tf?lcf
TO TO ^ 37f^ -qftTO (Final Test Post Test) F«n Rt TO ST^' ^ RfTO yi'^ii'b (Final Scores Post-test Scores) "TO “qT '^F 'TOTI 3P=Jd FFTTi^ ^ cT8TT 3TTf9m Rt FT t cT^ 3PTFT % THTO F«TT TO FT) Ft)^(Pre-Test) FFT TOtf Ffl^ (Post-Test), % ^ FRTftm fro ^ RTO tl 3rFlF % 3TTRR yjfflHnd F%R Ft TO Sf^' Ft ^ FfF MTO % TOTOf t TOt F^' t cTF cfl 3R7tF[ ■fF^^FR (Anova) FF 3RtF ten Fn RTO F, Ft^ Ff^ tepl (%. FVqFPfl) t 3FFT telTO Flm t FF teTO tr IRI TOtF Ft^, (Post-Test) Ft 3nFT ate % F^FFPTf Ft RFteTO
FRFT fTtTI FtTO Ft teTOFT % tet 3TtFF FFSFF % F^FTO) Ft FRt> 3rFFF fF?^TOFTRT Ft RFFmR te^TO (ANOCO) FF tl ^ ^T^FF Ft^ Ft TO ate' (Y)% ite Ft^ % ate (X) FF FfIffFF (Regression) TO, FTtFT fIft t FFT FFJFTt 3ntteF> tetroi (Initial Differences) % fte arfte 3nFTte Ft tteftTF (Adjust) ten TOT tl artet Ftter Ft TO ate' (Y) Ft ffIffff % ^rtr tnj^' Ft IMtoft ttef^ Fte %TO ttefFF yiwT (YO % tet FtRR tetem (ANOVA) ter ff 3rte ten tot ^ti fF^^TO Ft FF 3iten ter tnte 2 t te Ft trot' % %t tro Ft Fn ttt ti
FFtIFTtTFiTtTvF
tete i49a^FFt
■«Kyil 8
Y' = Y + 2.85 X y. Y' = Y-0.95X Y Y' = Y+3.80 X Y
18 9.053- 9 12.80 . 7.80
9 10.1511.15 .10.1512.15 12.15
7 105 8 7.051 4 IX) 14
8 9 9 8.0512.05
5 6 9.80 134 6 9,so lo 15 10 13
9 10 9.052 5 8.80 13 15
50 40 50 45.2515 30 49.00 70 55.76S
8.0 9.05M 3.0 9.80 10.0 14.0 11.15 10.06.0
1.79 1.67 1.671.67 0.89 0.891.41 1.67 1.67 .■ s
wn TIM #1-M^,.= 7,0 . a. = 8.58
= 10.0.:y =
^8?^ ■% Tjcflwn
N = 15 0, = a.37
3?<T:. X ^ Y ^ ^y- .
3.58= .897
= .95X -3; 10 ^ 8.0 t ^
ti82[Tn^ 7 ti 3Tci: aT^ftqcF 'qO^ yimidiT ^ ^ ^ 4 ^ #i?n>ji<q(«»i fgalq of ^^cfhq ^ sb*li^l: 3 ^ 1 'ti Y yi‘<n<^'l % X ^ HfftWTT
'3’JTf«F 5- % Y yi^!«hT ^ yKl^ch % %q; ^(^xi ~ ^.t) ^ 'HHiqlPjiaTOT m 5 = .95 t. 31^: % Y yikll+T 0.95 x 4 = 3.80 "qr ^«(1^ ^rft^T % Y yiKii'qiT .95 x 3 = 2.85 ^ .95 x i = .95Y yi'fli'qi siqfd (YO 'te wtqFTRFT % ^ ^ la^ ^HiqlPjio (Adjusted) yi'<ii<=h'l (YO ■q?l ■33^' % X ^ (Equivalent.) ^ WrT '^1^ Y MixiiVil ■% XFfqr^
^ ?f8n y^<'J| Iqj^c^qui (Analysis of Variance) '^57■fctfifR (Treatment Effects) ^ 'll RW f Y' ■qMfsFt %
^1^ ^ yK9^<+i 5l*idi ■qi ■5^.(Free)•^1 "qijfq ^ w-y^<wi rq^'c)q'j| (Analysis of Covariance) "qn 71^7 (Theoritical Explanation) f "qt^ ’4? (Exact) ^ tl RWl^Ror■4 ^ "qf Hicjeji) fi^?c:)qq ^H-qPiled «n%j ^ -syqVjl
^<^qil qi?-y'H'<.'Ji- fqsfQiqu! '4’ wwnf % yl^qi ^^aipqq ■% "if 1^ 93 '4Tpn tl '
3.37
150 <s^d{
«15-M^14-
13-
12- /My.2 = il.15-Y^!i
I^y3l0=9.80—^
Myig = 9.05 g
8--
k*
7-My, 6- -
. 5-ffyS*
4-
3--
2--
1-
. . , . . I-----1-----1-------^0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 X
M,3 M,2
I I----- 1 »\
M„
(Graphical Presentation of the Adjustment Process in ANOCO)
(ANOCO) ■% #n:i 3if^ mikii'^iT (Final Scores)■SratWPT (Regression) % ?RI wtftRT ^ 3iicj^<4=t5fli 't, 3#?TR SiiMi'ehT
(Final Scores) % ^ ^ WTI ^ ^PTlTTtf^ ^f I 1^?^^ (ANOCO) ■4'WTT % fsff^ TTWH 3TT^ .
■3n T# f I ' ' .
2.11 '^-TRTWT fq^^9tui cl^ "nuHT
(Computational Process of Analysis of Covariance)
^ sJtMu'qi) (Initial Scores) X TT^IT 3Tf^ ¥i'^l’^T (Final Scores) ^ Y ^ cR 'H? TRTT’^ IPTPTtf^ Y TTP^Tf^’ (Adjusted Y Scores)
% ^ ^ (Adjusted Sum of Squares for Total) tPTRtf^ 3iRTf7^ ^^ (Adjusted sum of Squares Within Groups) ^ WIT ^ ^ f W fqrT
ITRT^ftf^ ^IH(Adjusted Sum of Squares between Groups) UlcT ferr '=311 F^kTT tl ^ (Adjusted Sum of Squares) ^wn ^ f •
(SS^■qT=SS.. =SS..-Adjustod y SS,^’-A- y
151
Wjt ^57?^ X ^[■<11^1'. Y ^IklfchT X Y ^ ^ ^ ^'. tt^ ^ ^ ^^TT 3{l^f<-cti^ ^ ^ W ^5ncTT tl
(0 X w^i<^ % ^ ^(Various Sums of Squares X Scores)
c =£1^ '■ ” ■"■ N
^ ^ -qtTf, SSt^.==LX^-C,
SSo. = I-'• N, ,srrm ^ -c.^*\ <<Vl, — SS'p^ — SSg^
(ii) Y ITRTf^ ^ ^ ^(Various Sums of Squares for Y Scores)
P _ (2Y)^S-^^- ■
= ZY^-C^
gY,)^
^ ss T,
=z^ "gif SSn -c.Th, N,
^i-nRch cfif "qt^, SSw^ = SSrp^ - SSg^
(iri) XY %-f^ ferf^FT #ri .(Various Sums for X Y Products)
gx)(zy)N
SSt„ =SXY-C^^^
,y(SX,■)(£¥,-) ^•v Zj • ]^.'^T^r SSb
3TRTftg> TpjpTT^H titiT,
(iv) ’HHi^iriirt Y "^Tt^(Various Sums of Squares for Adjusted Y Scores)3TcT: ^T^TFTtf^ Y yWI^ % 1^ 'MHiqlP'Jia ^ ^ f^TR Wrf '317 "t
^ (SS^■ SSt,
(SSw„)^
= ss
3<iTiRcb ss w<.:.o=^^w,,-^ ss^AiljiisiiKl Y SSw.
tHHitilPAd «n?r ^ ss - ssr(\ SSb =SS w,BaiI)u»i inlY .v-v)(y-x)
152
(i;) ATTEST ^ ^ ^
(Various Mean Sums of Squares and F-Ratios)cTT^ ^=kii;^iT (dfs) ^ ■*TT^ W
t'l ^ <jr^d f^ ydln^lHi ^ ^''I'li % W-'
■4 3iRrit^ ■^«IT ^ "^Tf^ (SS^ TT^n SS-p) % •i=KiRl (df) a^ ■^rTcfi
tl HHNir^d 1TKT th^T ^ '-qk Wq^Rf STRfe q^qqrf -q^ ■;^ qm -HqTqtf^ Y iT.EqJTFTf
, qit <i<?iii qft F arjqRT q^ q^qqr qft qfRrt "t f^Rrqft ■^TTsfqjcn ^cwRil (dfs)' % qfferl ^ m gmr q7T% tqqfftq qft -^rquit tl Ff "s™: lIFcT
^ qlwqqr (H) qq '^m qn^ t % W^q qW^' (Treatments) % 'st’qq qOTC q^' ii UTRT qi^-a^^qrq % ^sfqr qftq^^ ^ qnqt t^ qr qp qftqjeqqr qft ■qqft ti
nfi.'^trH'ii Vff W^
fT q;q ■^rqiqtqR % qqrq qrt wq^TT’q % qft^qrqf qfi ^kiVi qT%qq qrq: x, Y qqr TFnqtf^ Y qrqffqrf
% 1^ qM #=Tf ■qqi-ar^qRff qr) q^ f^qr ^qrqr ti qra^B! qft ■qBqwr %qirqTqif qft ^r<i^i -qr^qq qq qrqq ■^rqqit '4’ q^ IqqTf qqr 't’l
Wrqft -9Iq^^qqT % mHuiihI «kiVi
WiW:
#rr ss Fdf MSqrqTfq?
qirfwTq) % qtq (Between)
■q^’ % aqq?: (Within)
^iqi (Total).
SSb. MSK-1 B.« MSB.SSw, MSN-Kyl^rticb w, MSw,SS MS(X) N- 1 . T, T.
srf^qq % q^ (Between)
% 3^^ (Within)
(Total)
SSb'-’y
MSK-1 By MSByMSw,,SSw''yN-KWIKIIct) MSw^
SS MSt^(Y) N-1 'Ty
% q%q (Between)
qrj^' % spqr (Within)
(Total)
SSb,.. MSK-1 By-. MSB.v-yarfniq SSw MSwN - K - 1 MSwy~x y-xy-x
SSt... MSN-2vimjch Ty...
(Y-X)
(vi) ■Rqrqtl^ qsqnq
(Adjusted Means)■q^qqqq ■4’ qftqjeqqr (Ho) % Pk'W qr qqrqtl^
(Adjusted) Y ■% qft '^Rn qqqt '^tqt 'll ■^qqqtt^ (Adjusted) Y qftMil'll % q?^ f^qq "^jq Y qpqjqft % X qn qqtqqqq (Regression). ?nq qi^ 't’-
SS“'.xyb =
SS «'.r
wfwr^ 1¥iFlf 153d«^n<
^^adjustedY; = M^. -6(M^, -Mj.)
= i % 1^ Y TiKm
M^. = i X R%2TTin
M_^ = W X■^T^ ■2IF i\ #TT ^ 3T«RI ^ X ^ Y ■^'
■% M«H} ^<e}di ^ 't^<^)'d! f'l X Y %
mw^ ^ it^-SSt
xy^TtiLnl
^SSt_^ SSt^
% 3^:^ X ft^n Y % ^ itm- .
' SSw,
■JSS^v, ■ SSw^
(vii) Wf4l'f^ h«aihH1 ^ ^difi (Comparision oYAdjusted Means)
HI'll ■’FRT % "^WTr v5H=^1 (t-Test) ^ ^«t5dl ‘ft
^*Tt 31RTTT '^RH ^ 'm i^^ft ^ "^RRif^ R«2RH ^ hH^
adjusted'^^ ^ RRl ^ f 5^1^4. '^RRtf^ RTWlt ■% 3RR ^ HIHcb
a^ = ■ Om
EwiUiui Group
2 MSW adjusted^ . ■^D “ -y'' n
% 3TRiK % 37^RFT ^ ’’E 0j^ EH Ihmi^HK W '^IE!—
1 1 — 4* —rii nj
H^fn -HHiHlf^d Rt^HHIil % SRiT ^ EFR)
^ EERtf^ EVjhhI Eit Hi'i«ti
E^ % 3TTE)R EEH ^ El
MS^D = W adjusted1F EE ETH^ E^ ST^EH EiT^ f, EEg
^ ■'jyf EET«I (Exact Formula) 1eREE t-
[2
rRT E^ % 37TEE^ % 3TEEH ET •
MSWadjusted
1 1 (M., -M )^---+ •----+ ^ adjusted<^D = SSw,^J
3RR ETt EPTEv ^ ?IE ETH % ^FEEEf ^-E^^ ^ E^ fERT-ER EEFn fl •37<T.*
E«RHf Eft ^HET % Zt-S^^ETE,
^SjffWf^hJ /EftR7154 ■s'^ai
t =37^
•D %•<^D
^-3T^ ^ df= (2n - 2) 3787^ (n^ + rij - 2), ^ tf. m WTcH■H'titfl fi TTf-’srar^ (ANOCO) '^Ft 3nT^ f^ ir^f
^ ^ 'w ti4c0^yi—■
^ W^. ■^' 37^-3773 ■^' ‘^m •SRTTftra
f^Ml "^l 7^ '*71? 3^ “TTHf ■eTJt 'f^T^ ■f’T^ ^ ''7^7^ % '3’7TF77 33 "37 f^wnfrl '3^^ HJfijf^a
1^ wi ^ ■q^t^ "qr ■sttrt stsf! qfj^ X 38n "q? ttm a^' ^ ,Y ^■qq7i ^ ^ X 387T Y wqr lo % 373^ ^1 i7i#*7qr Mw=^ (X wqff m)
■MHi<iir>ifl ■qrr^ TfNt '773^ "pT^Tf^ (Y yiHH=h) 'qjt uhIm ■qftf^r •
q?7
wnft 107iF-T7777^ Pnf^rm % wrr ^
WTO*T
X2 Y2 X2 Y2 X2 Y2X Y XY X Y XY X Y XY
5 25 9 81 45 8 64 11 121 88 7 49 648 5661 1 36 6 6 36 9 -81 54 4 16 6 36 24
9 7 213 49 . 7 49 13 169 91 36 1006 10 6064 68 36 48 . -7 49 64 56 648 8 7 49 56
6 36 8 4864 . 8 64 256 128 2516 5 8 64 4016 5 204 25 6 36 100 6010 3 9 7 49 21
9 81 10 100 90 819 17 289 153 9 81 819 81604 16 5 25 20 255 12 144 6 36 819 54
£ 40 248 66 298416 56 404 96 1224 690 48 316 62464 392
Mx = 5.0 s,= 2.45
My = 7.0 = 1.73
Mx = 7.0 s,= 1.22
My = 12.0 s^, = 3.0
My = 8.083; = 1.22
Mx = 6.0 s^.= 1.87
^-*77? 11777^ Pcliic^qui 377 UKN* -q^HT 377^ 2 ^ W tl 'TTTT^ ^ ^7*7^ t
%
^ ^7*J^‘ %
N = 8 + 8 + 8 = 24 £X = 40 + 56 + 48 = 144
1X2 =248 + 404 + 316 = 968 lY =56 + 96 + 64 = 216
IY2 =416 + 1224 + 524 = 2164 SXY = 298 + 690 + 392 = 1380
. = 6.0
My =6.0
mT<3»'^l<4 fqfVq? 155
(0 X ^ irt^T(Various Sums of Squares for X Scores)
gX)^C. N
144 X14424
= 864
SSt^, =SX2-C^
= 968-864 = 104
SSb/=X -c.Ni
40 X 40 56 X 56 48 x 48 -864888
= 880 - 864= 16
SS - - SS
= 104 - 16
= 88
(it) Y midfeHT'%(Various Sums of Squares for Y Scores)
(LYf
B, Bx
cJ N
216x21624
= 1944
SSt^ =SY2-Cy
' =‘2164- 1944 = 220
SSt„, = -c, ,Ni,
56x 56 96x96 64x64 -19448 88
= 2056 - 1944 = 112
SSw, =SSt,, - SS
= 220- 112 = 108
B.
156
(fir) XH8IT ^
(Various Sums for Products of X and Y Scores)
^ (sx)(£y)
yRttirHii 9^
cxy N
• 144x21624
= 1296
sSt^ =i:xy
= 1380- 1296= 84
(sx,)(i:y,) -cN, . •
40x56 56x96 48x64-1296
8 8 8= 1336-1296= 40
SSW — SSt — SSn^ Otv
= 84-40 = 44
(iv) Y TIIMIcb^' % (cifii-i grf i^tTl
(Various Sums of Squares for Adjusted Y Scores)
(SStJ"SSt^ = SST, ~ SST,
84x84= 220-
104= 220 - 67.85 = 152.15
(SSw,)^= ss. SS - SSw,
44 X 44= 108-88
= 108 - 22 = 86.00
■SS = SS -SSB W,.,,
= 152.15-86.0 = 66.15
157
(u) ^ mi
(Mean Sums of Squares and F-Ratios)"^iFFr ■^vf^ 'll 31?I:
^ -m^ 3 ■4’ 11^ ,f^ w ti■=TteH
"H^-wnir fq^»Attui ■% qftuu4y «kiVi
(Source) MS Fdf SS
% ■*Tvq (Between)
■% 3T^ (Within)
(Total)
16 8.02
QC) 21 88 4.19 1.91
23 104 4.52
•% T^WT (Between)
(Within)
(Total)
3d^ fllklleh'l (Y)
%
112 56.00 .012
10.89108 5.14 ^ IX21
9.5723 220
% 1?^ (Between)
(Within)
(Total)
.0933.082 66.15
86.00
152.15%
4.30 7.6920 'xm ^
6.9222
yiKi^T 1^ cfNt ^^5! ar^ 't' Y yi'<ii=t»l %^ vx^ ff^X.01 ■?<R 3T^ tl Y yikrf*)' % ^ 37^ .01 fcR TTT '?fT«f^
t ^ Tnfr-37^ % ^ ’4' ^ aTRl X um^' ffr=Tf v^' ^ mf^tl , ■ • •
(vi) wxnitf^ Y xfojTjpi
(Adjusted Means)#Tf '^TqpfffeT Y (Adjusted Y Means) ^ Y
Hiyii'ti) %tT X ■Srr<Tf^ ynln'iH-t,
SS^6 =
SSu,^'
44 ■88
= .50i % Y iT«W?T t—
^adjusted Yj = M^. —
m: 3I«m ^ wtef Y iTtWT,^adjusted Y[ = 7.0 — .5 (5 — 6)
= 7.b + .5-= 7.5
If\
158 dWS? wf^l^hl /cf/wf
Y, = 12.0 - .5 (7 - 6)
. = 12.0-.5 = 11.5
% %Tt Y ■tTKRPT,
MadjustedY, =8-.5 (6-6)
= 8.0-0‘= 8.0
rim TTwmH 4 ^ %\"FRiift 12
Mi»<tichl % fcTt^ 3T^mrat^l?f rWT TIQIITH
(Unadjusted and Adjusted Means for Scores)
N M. •M, M.-.
8 7.0 7.5TRW ^ 5.0
8 7.0 12.0 11.5
8 6.0 8.0 8.0
24 6.0 9.0
{vii)(Comparison of Adjusted Means)
Y ’T»WT^' % f^m=T ^ W-TIWT (PostANOCO) mr tptPt f^wT ewlTV mf ^ %3F^ TT^nm (Variance) mi ^ ^imdi •STrl: ^HiMlPjid ttMhPII ■'% 3WR HM=h
1 1 <+
fii hjMS<^D- W adjusted1
■^■qr . adjusted ” 4.3 71^ — fl2 ~ Tl^ — 8
=, 4.3 - + - - 8 8 .
■ 3lcl:.
=-Vl.075
= 1.037^ "qr ■#! TT^mqH ■§, 1%^% llH (Comparison Pairs) cim
lfW3: #l a ^PIT^ ^ #Tf m) Wft TR^ "W tl
3^^ wfm^ fkfm 159
13hw4hi*»1 '3?^
sF*T N M D aD t 'm
1 , . TTsm 8 7.5 .013.864.0 1.037
8 11.5
2 iraw 8 7.50.5 1.037 0.48
8 8.0
■••3 11.58 .013.5 1.037 3.38
8 8.0
■5I«TO ^ ^ ^ f«R:?[it •a«m ^ ■^’ 31^ ti#qFT (Steps)
'?I?-3RTT^ fqf^'^ win I'll ^ I^Hcld fd^l %
(t) X 3tr<ff5Rl' % ^ ^ TTGI ^ tT«TT aiRlfe ^ -qln w ^1
(ii) Y inWJiT % ^ ^ •qh, ^ ^ ^qr 37Fifi<ff ^ ■^Ttn W ^1(iii) XY iiUHi+><rlT % ^ ^ liW, tT«T ^«TT 3TRrf^ ^ '^l(to) Y UNI+l % 1TKT ^ ciqi aTRlft^ ^ W ^1(u) ^«wiviT ^ 1¥^ ^ ^ qm q«T ^ w "^i
(ui) qn ?m ^ ^8TT frqffer qr^i
t-q^iai
2.12 3lTlRf^(Assumptions underlying Analysis of Covariance)'
3iHTq % qqrr qF-TJqrq qft ^ qpqfTn( t'l iioi’^rilea qPTcin( fdHqn ^—
(i) 1qfq^ qjT "qqq qqfe arq^r qqfeqf (Random) "sqqqr ti .
(ii) fqfq^ q^el ^ MKf^«t) Mi'<ii<t)l (X), ni'fliWl (Y) trqi tiniqll^d Y jlikiiqil cFT
qrqpq a(l^^di ^sf (N.P.C.) % .315^ ^i
{Hi) ■N'fi % f^ yuNdi (X) qrr mm wft qqR ti
{iv) Mq^ %. 1^ yudi+l' (Y) qn -mM w\m qqpT ti
(u) fsrfq^ ^^jel % srf^ wi’fliWl (Y) qn "siRf^q^ yi^i«t)l (X) % qcftwrq (Regression) qqH fl
(vi) qqt yPds^iT % stfqrq inqifeFt (Y) ^ YRfwRi yi^i’qiT (X) % 1^ TralwrT(Linear)
160 3cqa^
(Random Selection), rqcK'Ji (Normal Distribution) (Homogeneity of Variance) ^ ^
Hi’^aiy] ^ Hi^dii< "f ^ cT«n 3Tf^ «ivni'«tjl %yal'H'i‘H’1 (Regression) tl ^ (AdditiveNature of Treatment Effects) ^ 3^fdf<=W nT^dl
W-y«<«i I&Hi'il (5(Hi*iI [qci<,wi % ^ f^tga ^ 'tl^ cfTT^ % ^ ■^■^jnftwTT (Multiple Regression)
f¥N! W\ ^(Statistical Technique) "t ^ ^TFf yj;t«i
^ ^ ^ tl
tl W-yW*^i fq^ci^q'Ji yilqP^
^“<ci1 tl tHIHin»I=h
I Researches) W-'Srer^ 31?^ H6Tq‘iyf ■jrrfsrRr % 't ytj.'W tt# t f(Treatment Groups) ^ % Itr? ^ tt 'mrn tl
■f^vEncB^fm (Student Activity)1. ■'ilWrW (Null Hypothesis) t?
2. wm (Standard Error) ^ t?
3. 1908 f '^MZ ^ yrdMlP^d tt-aT^qTcT m ^ M^l
HlT<s^’=t>l'^ 161
4. , (t-Test) '*TF??n^ '
2.13 «ikiVj (Summary)
• ^iT^s^etO^ HR=h«rH'ii ■*TRRTT '^TOI •% ^ih1=ii1 % '9)8?^■f"! «kT^«h checuai t fq-qK^a hihi^iI % PiF^qa ‘TFf ^ 'fl
’^ifsK. ^-¥iRsq=h1<4 "^l «pfff^ l^«t)4 PiebKrl'^ ISFfV f, ^>ri^, ^iTisHqOq M^ffl ^ ^1
• ^ yfaq^nl ^ «t><^ qiloi ■§■, ''TT^ <HiHi'qa: '5f^n?Tl^ ^ '^■'*TRT
. ^sm t % ^ Trf^ ^ a^RTR 30 ^ ^ ^ ^ "Slf^ liHT ■5IHT ^EfT^I• ■*TT^ ‘SHTN ^jldq^i-l^^nTf (sampling errors) '’TN 'll yioq<<'i ^
■f%^R ■! fRTI snsf^W 'sjiicftid -dwqlM ■JrfrR?! ■% ilPflV) SijHi'i^=[%rl tl '
• UT^ ■% 31^ ^ uRiqqi tqcK.ui TTPH^ Hirq<^iaif. ^Rnk. ^ ‘Sriroff ^ ^WRpjf % ^ «j4«b^ % W(: ■WIMM yiPj^hdl■% «rqq |'l
• ' FTq ^
• Hi-qcii "5^ MiqRi«t) 3i^HnicHq» -HiTisqchlq 'RTf^fv (Parametric InferentialStatistical Technique) 11 ^
-^rr ■1:1 tI' -ffe 3if ^ ^ ^ Ifi ■• (x2) ■^^m t f ■f^' ^ f^PRhl (Karl Pearson) 1900
■^‘ «?TI '^tRsqchl^ rdVclNUI ■^' ■^' WT ^ziTWoqqeiRd "'Kf^ 'll
Hi-qni
W9-H<u| PcIJ^crlM'JI ■^ y|idl4>T ^ TT^ ^ f^^HH^'>c3fii (superiority) .sr^rar 'el'jdi (Inferiority) iqqK 3Pi^ % 3RI
sjfm ^ -yr^ r?1 -^rntr^ ^sm ti
3WRT-W^ (Exercise Questions)
1. MUq^rMii ■RdsjR % •4\iq^qq^ 'apl '^\
2. ■RT^IRkH ■^ ■^ftRT'STf RTT ehIPnql
162
3. ■'Tra ^ ‘3?’ <^<|cb ^ ■^’ 1^JT=T 10 6, 16, 17, 131
6 <^«tvnfl'l ‘■^' %<\'^ Tf^ “^[fe ^ 7, 13,22, 15,12 "sn^ ^ ■^'TWH ^301 ,
4. W-^l'H<.wi 3T8| ^llNiy.l W-y«<^ °Fi' "TTOHr yfsh'iii ^ 3TnT ^'t? • - •
TP^ (Reference Books)
1. Tfffer^ afk f^ w^, ^ vPd«?>ww/
2. ^iToHetTl’H f^ftof-^.TJ ^cTcV^, ST?
3. afjrp /q/c7<v*/ Tt^^, ST? ^7^/
4. yif^«t)di fqcR'j'i—/«*/)% ''T/^T^TT?/
5. ^hIcim tt?7. ild'RI yf^<^!^HI >
nff<SH'=t>l<{ /wfsT^f 163<3w»(T<
3TSJJFI-3
^4) 1^—1
(Theory of Sampling)
TR^HT (Structure)
l.T (Objectives) '
1.2 ’ (Introduction)' ' ‘
1.3 ■^?1T yi-^<ri (Statistics and Parameters)
' 1.4 (Population and Sample)
1.5 Wdiji (Summary)
• (Exercise Questions)
• (Reference Books)
1.1 (Objectives)
• "STf^ ^8TT ^ amTTJTi ^ wr^ ■^'i
1.2 (Introduction)
^ 31^%TH ^ ■gi^ . f^cST’d) -STgigT eTTHT "t
ITH "5^ eh.<4i RPia TlRFllsff <H6Nc1I "SI^ ^ ■^1%!^ ejchd'i t. eiH^l ^
% "gKl ^ ^ yfnTiPi.n <hl4-. ijtH^ TO ti oin^^fe ^ -fgf^ tr^ ^ ^ ^ ^aram -HHIRH -ato I w -^4^ (Universally) ^ ^ mw\
fH, ¥Hivrf?Tff^ 3p1t eiH'Wiy. Wl: ohRki^’ f rTsiT lira ^ r: ^ ^ ■^Rpqra^' ^ w^iu ^tri '^i t
3ig?TiTRrf ^ 3R^ % '-^tr f^ ^ ifT Rg? ^ Mpn ^ dsn ^ ^ "zn
164 TsgR f¥im
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(Method of Samples) ^ ^ "t’l % 'SRThTar^R^^HRRif % SRI RRT "sifh^ TPTfe ^ MPdPifv?^ "RiRn "t ^ 3dd<?f rr^%ct■^^3^ ■RFRRIT if ai^Hid cRIRH ‘^TI R^kTI "I cT«n (ii) '^^Flf^
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ft 3lf?T^RR fqdi«il (Sampling Distributions) ^ yi’dd, RRfe ^ y|d'\<f, xld-q^d fqfu^T cRR Tlfif^RR
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RiJF fRRRR ar^RRH^ RTRIR f 3T*RRR RRm f R RTRI-'^^aif % 3TTRR m RRfe % R^R^ if a^^RH ciMiHi RRIT f I IRf^ RRfe RRI yfds^ ftft % ft fRfRR [qjftRdiaf aiRfcT R^RRIR, R^RIRT, RIR^ Iq-dcid, R?R*R^ '^RT^ ^Tltf ^ ‘RRf RT RRuft fl %R^ RRK^ % %R ft fRf^ R’jfRTcRRT Rift Rft RRRT. “Ft '^Rlft ■§■ RRffR> RRf^ ft '^Rft Rft ftft "t % "aR^ RRt RRRlt’ ^ RTRI RIT9
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Rft R'fRTcRR? Rift R^ oqqgK f RRRT R % RT^RR Rt RRfe % RRt RRRtt R
■RT RRiftRRTR ^ RRf^ ^ R^dRTrRRi Rift ft RRRI R^t RT ««hd1 f! Rft RTTRR f RRfe
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Bra ^ 500 ^ yPflqjff BRT W ^ ^ 500 (^<iMdr«ffcT. "m 3^Tf^) m st^ % (ST^T^ TraB 3TTf^) ^ tlR .
^ Bra Bf?rf 3T^' % tFWTTTB (3T«RI■STlf^) yiqcri B^BPRII 'srf^R^pjfi’’ IIPT: 3T^ (Roman Letters) oq«w ^57!^ ‘f
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'^iqfq’ Hiqcil =Ft WT: TltB 3?^ (Greek Letters) 3Tq^ 3T^’ % Pop 17 f I 1 ^ 'SrgB ^
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wlqq^I Mi^w.qBTKHq? HIHIcb
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qqT't) ai^qpRKif ■^' yPas^UT ^ wn ^ ^ wr t ■^TfFRn yiqoil BBB "tI 'RWPRT flixiciT BT 3F^HH dMHI ^
BffeFBk 373*ra (Statistical Inference) BfT ^ t cT«n TTffeqBt^ BT TI#! B7% TJT^^ % Rfft^TBt^ 3j^HM ciMi^ ■§■ 3i;jHiiicH«h 1^ff%IB inferential Statistical
(Inferential Statistics) B?^ tl
(Sample)□ A qO □ X
A X A X^OX O n □ ‘^ O
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4 MMl wVaxf%nr 1. ttb bt btb bb^i
, 166 f^M
1.4 tinfec 1^ (Population and Sample)
^ ^ ^ ^ tl T^‘ TRfe ct«TT ■Jlf^ 7T^' ^3lf^ ^ ■^’TTf^ ^ i^ff^ ^ Trf^P^ '^#1 ^ THTTO 1%^
W(\ "tl W '^' ^iHil-rt W ■^’ 3T^=2T5l W]^
% ■^’ ^let>K 1^; JH % 1%^ a#3ft ■q^ Population cI«1T ‘^T^TJ 7T^ % %tf 3#^■q^ Universe IPTW f^rqr '^i^‘\\\
375?T%TTq qPTfe 3T8?crT 'Jii^'<sq( (Population) ciicm4 §=tii§<Hl (Setof Units) f^-i^ WTPT fqsfl'^diy] Ftcflf5R% Pi'*h4Wfi 'ft<,ii ■qieoi ■§■! 3T^^^ii«t>dF "Stp "q^ '4' s^wprqr^ ^ % '^T’ft %Pi<*h4 ?n^ ■^r5?n •^ledi t ^ 3ti4i srwprq ^ "^rqf^ '3. q. 1t«7ct ■qr^^rfq^ f4aivi4!’ ■%qpTH ^ "^5 'f4tti ■’JFpqqt’ ■^, "^q fq^ln ^ nllsd- oqfwqf qfl irqfeqT ^ '^li^dl 'll '■rftfqd (Finite)^ ««ha1 t qsq arotfqq (Infinite) 4t <4=hd1 f l tHlfnci (Finite Population) "4 ^^■=■'^•4
^ FuhA' <4'<siii •^Pin^qa 'll qiTcT '4 qr^qfq^ ^4 '^n^,Iqjfqfqsji^M 4 3T^iHch) qft W^sqi, f4i41 ’q^TFR 4 hR^iA ^4 4^1, l^iAl PqRhrWlviq 4 APiqlqft W<s4i 3qf^ qftfqq qnfe % "^^FFTW 'i’i tqq^q 3T4t14H ^hRc (Infinite Population) 4'i|'i4(s4I t f4q4 W<sm ^hRiR^o sprt ?tc4 "ti 44 stt^tt^t 4 ^14 q4 <h’ohi, i4>4tqr^T 4 414 '^i4' Tf^’ q4 4om 37Ti4 3T4tf4d 4qf^ ^ f i q^ilq 4 ^ 41 TRfeSHtPlRid 414 f qpc^ ^741-^741 "^rqfe 4 i^nfql q4 4^1 344^7 ^IHl 1147 754 fqm ©qqsK4 ?fq4q 3i7qwi^ "flrTT "t ^tp^t 441 'pqfeqt' ^4 oqiqgiR^^ 4 ^IcTT tl 44 3TTq7T?i 4 ^4 ^4 Jtwn 37^ 4141 qpg ^ w q^qqi q 414 4^ q7P7q^ st^IThci Tqqf^ inqT qir 7r«ft'Hi ti pqfeqf q4 HRdiRrMd q^TT q^TT^f ■pqfe 4 41 Iqqqq fqvqr
qTwq
73R ^qifli tl 41 ^Hpi ?7q 4 fqttHn 41411 ‘34 qsiTSif '^PTfe (Existent Population) q7^ tl ^ 4 qr^ ^ qqTqr t % qqrsl 'pqfe q4 -pro iqTifqf qpqq 4 fq^RH 4141 ti ^ ■®i41qfl ■qqfe, ■j^qqA q4 Trqfe 3727^1 HdlARiqT-qt "TTqfe 37rf4 qqr^ 4qfeql % qqr^Tq 41 pqfe 134 qq 4 [qtiHi'i q41 4l411 vRchR^ri (Hypothetical Population) qTWt
■PT^ 11q7 wRqiRrqci q4 fqiTfql qfl chvH'ii 41 q4 Tin ■pqitl tl 4 '7741 qiwq 4 fqsjHN q41' 4141 tl 44 f4q% q4 m (Head) q qq (Tail). 37t4 qfl ■p41 fn^qrfqq 377^144! qfl
pqfe qftq44qq TTPfe ti R+q^ qt 3%si(A fqqr 41 441 pqfe qfl mR.^ivs’Ii q4 qq ■Hqicfi t _
tl w47fqq4q
t
ql4q4 qH ■=q7q4 41 q7i4 ti qf4q4 .(Sample) >wRd qTi qf qrq t sr^qprqTqf % ^ qi^gpcjoR- 37W7qq % 14fq; qqPdd IqTqr WT tl q14q4 'pqfe qq q1q1qfq?q (Representative Miniature) tlcff t q^T ■?774 qj^T ■7J^34’ qq qqqFql^TTqT q7T47 pqfe 47 qil
' 4 sT^qrq emrqr ti t % ypd<4 ■qqfe qfl fqylqqrst .qq qfqf^ 41qi ti qfqq^ % 374^ ■qqfq Sample qq ^f4q 4) Exemplum 4 'g37T fqqqq 374 t Exmple 37qfq qqiFTqi ^ ■?Tif^q7 374 4 41 ‘TT^TcT Rddi t"% 3714^4 ^nRd qt 441 fqTifql qq 4q7^ t f^nt pqfe qfl Iq^fl^diS^ qt iw qrc4 % Itrq qqiqrq 'TqqTq -^jnqr ti sTd: qn "pqTqi t q1qq4 qr^qq4 TFTfe qq 37^qqq t^ 'qqfqq fq^qi qqr qf sr^rt fqpqq sT^qrq^qf qpqq 4 37sqqq qTM t qqr 14774 qRT qiqq4 qfl fq4qqT34 (qfqq441) qq TarqpflqiTq q7747 Trqfe qfl fqffqqist (qrqvil) qq37^qTq ePirqi qnqr ti TTPfe qfl §«hi"^l qfl 'Sffqqqq iqq^ (Sampling Units) qTft t qqr 1441
/44qf 167dpqnf
% '?n|F ^ Mra'q<4'i ■31% (Sampling Frame) f I yfcKifl ^ ^ 3rf?R?f (Sample Units) tl yfd<!/[ ^ o^PcW^T atar^ %
, ^ 'Hsb^ ^ ^*71 $'i«hi "^rf^TTzif "an ^^arf % yiKii'cn) ■% ^ F^qi
■% anvR ■'Tt Twfe '4 '^rffer^i^T ar^pp ,aiPiqi4 itf^rtt %■?4tt "ti 3iPiqi4 Hi-qoi % ar^^TR y(a<iJf[ *ah\'^ "34 4 yfdfnfVf^ 3Rrn 4 a7a7fcT ■yl^'^'vf 4
ipr TJl 'i^ 41 4 "tl Mpmifl "RRfe 3JT 34 4 ylflpif4^ ^ WT
4 arai^T Il4p4 4 ^ qi^pq^n iq4'<ai hRciI^ an ylalqHqd HFf 4t3t 4 33 ai^HNicH^
■^4R9£p43 14f44t 4 ai^nn 4t ^T3i4 4i 4 33 ai^niPra ur^p 3R3f33T'STr^erl
4 .f33 4t ■H3J4 4i 44 ^tr 37^ 4 ^rgr 3^7 4 3^ 3t4 34 afRp ■^3Tf ^ 3r4 % fm4>33I ?1f4 4 4*73 RTaql^qi (qamql 34 37^ 3R 4 ara^q-Kd 33 31334 4 %3T '373 33 3F 31334 337; 34?T 4 3R 4 3^ 3l4 3JT4 34 "RRfe 33 3f3f3f3?3 3^ 3R ■r43TT 33ff^ 17t4W
44 4 '4*73 'RT®443j f3irRR4 ■% t34 34 ^7574 p44p3 3Ft 13731 33T 4l "33 337 ^4t y«t)k 47 34 31334 4 3ff 'SJT^RT 33 337 31334 aT33f "RRI^ 34 14433734 34 4137 'OT 4 3l4e7%3 3^ 37T ‘34)37 337 ^ '33RR 4 ^14 "34 aiyfdPiIVc^ ylds4 (Unrepresentative Sample) 4 37RT 3lt3741 34 tiiHi'q1«h<«i ‘4) "1^ 37737 |[fe’j4 '4371 a73:
4 ‘13)41 yfdfilqcq 31334 "377 ‘333 3)t ^734 yfds4 4 .'577R7 ‘443731 377 34* 433 ‘33 4 ‘773fe 4713T3 ‘3137^313773 13737 ‘377 7T47I ‘31334 ‘47 y1dPll3?3 ‘?14 31337 ‘3^ 377 ‘3?3 377 ‘3^ 3)k371737f43F 1357p4 (Social Sciences) 4 ‘3^33 773fe3l’ 34 14133 ‘f377|3l 4 413337 ‘?137 ‘ti 4H337
f3?n3t (Physical Sciences) '44 771733 '^TTT^, pqf^c^i, «F)l3 317I3 4 3l ‘31334! ‘33313 13737 '3737•
4 ‘37^ "3^7 "37 ‘31334 ‘34 37f3f3l3R337 ‘37 ‘4377 ‘3^ 34 4 33ff37. ^3^ 3^[33 TTRfeafi 47T41 |377?3T 373: ■^: R37 -474 ^ f I 373. ‘3lf337 -37 fnm 47 -7741 3rtr7 3T«T37 ^3l^
47 ^ 34 ‘7141 aT?137 ■fe41 -43 4 3317% 4^47 47 4) 7141 374 37333 % 71373 f I 1^7147
3)R’3 ‘?334 ■47337 7737 13)7^ 7I3fe 377-^qa yfdpllilc^ '3773 4 7134 ‘glcll tl 441 TRlfepI 34
7133Tr47I TRlfe (Homogenious Popiilations) ^trti ^ 37^ 7113ll3l3) 13?7737 ‘44 33113373.
?1R3, TRIRWT^ 3nf3 3^ 7T3fean 313: 4l34ll73f (‘3733 37 3^ ai7f3) 47 414 fPddq)*! 14133 ‘?377?37 ‘37737 133-133 ‘?1fl1 't’l '44 13774 TlpF 4) ‘j573l ‘34 '^fe ‘37 cqfqdcq ‘37 31437
tl^ 713737
37^4 ‘f I ^ '5137R '34 773^31’ 4 ‘31334 371 ‘333 37731 314^17^ 3)16*1 ‘3737 4 '47 31331 7R%
3)1 ‘3143 33 4 ‘5i4rf3f4c3 377 71471 3)1^ y(d<A^1 31331 TRlfe 371 yldPl14c3 377 TFl 4 31331 '3^ 177377 l3VllTJ7 ‘^731 373^ 317R«T3 73 3774 fl 3774 47 % ^l1%f 3^3 TlRfe %‘34 14433734 377
■'IJJ! ^ 3113?3^ 33ff37 ‘33 €l 3t3l 4 '57137 3l7'J||*4 34 '53137 34
744-744 '5773 '34 ‘37 7137cf1 33 yld'^Jfl 4) "^ap 3 ‘3774 'SITRl mRuhhT 377 7T73R3l^77VF 3774 377 "STH
‘3^ '3337 ^1 37733 4 71Tfel343 4 71^ ‘^, ‘5713391 ‘47 3134 TRlfe ^ yfdPlf4r3 3774 ‘34 yi1^«t)31 ^3731 f 3 I37 '371371 Pl^lU^l l37aT7 '3137 ‘ti ai^ 3777^ ‘t 137 31531373137 7lffl51343 I3f4a4 4 ‘517^
71373 ‘314373 ‘517133731 ‘4) ^ 4 ^133 1374 ‘3l4 33f% 7lffe7343 4 4B74 ■5fl339l
(Random Sample) 47 'gTTl 3133l TRlfe 371 ‘57l3l3f4?3 3774 34 '577133)37 71311437 ?1cf7 t. 5kfViQ. A"^3 57l339l 34 ‘5747141473 ‘5ll339l 73l3)R 37747 717R7^ll377’3 14)^1 ‘3737 ^1 37733 4 3153T373137
7lffel343 34 3l143)f97 3|513f373l3l4ai7%^3rf339l34 ‘5P’^‘J17^ap7 (Fundamental Assumption) tR ail37l73 ^ tl 31i4 47 -^7 llt339l WT 34 1¥4P I3l4a4' 371 7l4^ 4 3^3 13737 37 7?7 tl
168
(Simple Random Sampling)
fV'wl ^ 3rf?T^ ^ cT^ ci^Pi'>t5 'MvMv (Simple Random Method) ’it ^snflT f I W’T
(Random Sample) ?T^ (Random) dlc^4 (Haphazard),(Unplanned) ciin^qiel (Hit and Miss) "t cRTT ^Tffer^ft^'iTPK’Jlt ^ "^fe fjHM yPd’d^H Wt ^ tfhcT ^ % %T3; Wt ■?M ^ 3RThT tl Wt 'SlftRr^f ^t^^tWT ■! ■% TFlffe ^ ylh«\^[ ■^', ^ fRTT yif^«t)Ol Btdt "fl 31cT:
^ ^ w 1'H’it % WT yiRjdidi ■RTrpT «it ?ra ■^'y[a<vf ^ ■yfcRf^f (Random Sample) ■^*IT?TT 't’l
cT^TT■% 3T^ c( •spqn ^ ■?Wl "^TT^Mfl ^Wr TTRi^ (Simple Random Sample) ^iPqchai
(Constant) ’’C?^ ■§■ 372?fcT ^ '^nRT "RTPIT ■gtcIT 'll rT^TT '^ri? f^TTf^'"% WRyifq'^)dl'¥’lTf^ "^tcft f I 'iVR<u| ^ 10 ^etil^-nT "^ft tlHpC
2 ^ yPi^sif ^sTcdi "tl '^fK 10 2 $=hl§<<l ^ "^fel 'JII<)*!I^ ■STKRT wi ?wrl’ ^ WT ^ yiP^4)di 1/10 ■gt^ ^3r^ ^ 9 ^«if^"+"dl ^r<qidd 1/9 ^ '^TT^^I WR cit ^ yfd^^t ■^TS^ Wl "ni ^ -3^: ^ ■^' fh^ rfKTT fTR fEFl^ ^ ^rft ytci^^l ITTcT Wr 'Jil'i^'ill "yfcT^ ^ ylnwiHn T%fT Trfrf^RiR(Sampling Without Replacement) ^ IjIcRSirRT TTf^ uftl«rad (Sampling with Replacement) ’it oRTf «n?TT^ ?5fe ’T^ ^ ^ ^ arotflRI "RRfe ^f^HFit ^ "^t^ ^ TUrfhefuTT ^ 'R^ilqcl ’TRTT t’ Rtfhd(Finite Populations) ^ '^' 'SrfcfSRrT TrfcR^TTRT Tf^ ^«TT fF^rf '^Nt 'JeRK opn ^ «=hai "tl
tritR^TFRT.Tt^ yld'd’H'l 1WtT ■^' ^TFTT “<if^d 'ShTT f^ ^<iH<,RI Thi vpR qiHiq ypist'^tc^i'T) (Conditional) TtdT ■!■ 3T«7hT 37^^ .itl^^/Ph-el ■^qFTff ^ cR ■§ 'd^Plci ^ ^l "% "^TBR
"5^ V(fdsi'^irH=h yilVehdi (Conditional Probability) RH R’it ^cbi§dl % fyif^'^tidi % ^'Ji'iyici ■pRRT (Multiplication Law) W f^RIT ¥chdi f I ’iM<)«Ki
^ |cf51^ % "^ipT ^ irf^RTp^RT yiP^chdi ^ RH 11^ ■^' ^3^7%■FTTfh^Rn ^ "4 ^ MiP<;i=t)H ■% '^'Ji'i'+iQi ■% oRRT 3Ta7fd (9/10) x (1/9) = 1/10 #
W\ '^R 1I8RT
■f I HIdchl cRt ^ ’RTT f (Infinite Populations)
%3T8|fcT (9/10) X (8/9) X (1/8) = 1/10 ^1 ■ .
ytd^iff ^ ^ Rt’TT^ 'HH«F yPas^fiT’t Wr ^ ^ R^ yPaq^l % ^■4' ’it f^^RT "sn ■RRkTT t"I f^RTt RRfe n 37TRiR ^ RRRT yfd<^ifff %Rt
■?R RRTR 'Sfsi RRT f % RitK^Tf % RRR epfl yipiic^dT RRPT ^ fR RRiK R^ yPci<^t ^ yld'A^fl RT?T RRkTI ^I Rf^-’RR IRTT^ (37, R, R, X, R) RT# RRf^ 2 "fRiTf
RT^ Trfd^ RTT RRR RRRT RR ^ ^ Vif^- (37, R), (37, ^'), (37, R), (37. R).(R, ■77), (R. R), (R, R), (R. R). (R, R) RRT (R, R) #tl rIr |R RR yPd^J^lT if 7^ RRT R7T
RR«KI'l'fxn
3«R{R RTifejRjtR 169 -
^ cfl "ciMria ^ ^WR ‘SncIT "t ITr^ % ■5ITf^Riun RRHep^ >311^^111 ^ ^ t «t<<T«ti ^ TT^ Mpci'A^fl qii«h<■3^' Tj;^ ^ ^R=rT '^’m i\ fT:'H% ^ yPd^im yPd^d■f I ^raP? ■^^ipTl't) Sl^HMirH^t) «lT<SH=hlM ■% Uf?!^ ^ Wr T33>3{|cj!j^ct)dl ■f t"^M fsrf^RTf ^ 'SrfcR^ff ^ "RRfe % ^ ^ ST^HP! ci'ii^^1 «R^ ^ ^PiP^fqa ■^RTTT <jP^a "t % "yfcR^f "WP^ ar^fefT «hK^!^ ^ TFT f I turrit TTRfe "si^t^ 'MMt "ti ^ i=TftRn—Tnsrt■fsrfv (Lottery Method) Wr 3f^ TTTT’^ (Table of Randoin Numbers Method) TTofff^PF TRPT, TTl^FTfiRT 'F^TT <JH<i«Ki "tl "^N! TTRf^ .'^ TTRTF l§4l ?TH fHt
3TT=R^T^ ^ f^TTT% «tJK«i <^qQi tillHci TTRfe^ "OlFF^f fPIR cIT^ TT^iFT
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PTTZft ■fsff^ '4' TRfe TT^ 311%^^ §=614^1 (Sampling Units) % aT^M ^ WisMi arpTR-arpR ■% RT nn<aq>^ Hl-sq>< cfl ^ncft "tl fR y'f It ^hIW
^ ^ ■^TTcfl f 1^ ^ TT3^ ^ 37^ cTTF ^ IrPTT^et>l‘
TR^ % q<.iq<.-4041 4f^<^T ■pT9RT?T ^ "t [qail % 1eTT^ isle'll fI?!! "f 1
fR 31^ ■^fsl Rf mPcimT ■'TT 37«I^ shHtKS^I 3lfcl^ T^ '^RTfl 't'l500 TJTjl! f^TTft TRTfe 20 ’STilf Wr 3TfF^ ?9i<i'ii F^ fTi TTRfe % TT^ll 500
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■315^ % ^ITRf 37PFT-3T^ WTST ^ Rt lci<3«h<. Hlqql RT fcrua^jt 1^ Hpqql31^ Hl-seh< f^FTlt ^ •sniq)< 37^ cITF i%Pn ^1 cPSTTO ^ "^iT^ 20
^1 20 ttW -qr #1 ^ 20 3lf^ ^ TTT^ ^ cT«TT F^20 % yfas^f RT 37«Riq ^jt% h^\ TTHfe (ST^rfF 500 '37^ ^3T^qF1 FTTF 1^■^TT^I ^TTZfl im yPd<vf ^Ipt ^Hich< (Rotating Drum) 4in^«S l^f^(Mechanical Method) a^ ^sF q<riicb<. (Moving Number Wheel) F^^TF^Tl %^ 3rqhT 37t 1^ ^ tl
3T^ WH 37f^ ^fqc3 g ^ Wl 3Tf^ 3JTRT ^ Tn^Pt tlqi<.*qK yt^Ia1."SH 3T^ «iK«n qiwq 0 9 cT^ % ^ 3T^ ^ slq^ql T7R*t1 F^ yqiK
t f^.1¥R?r Rtoif W T^‘ TTRt afqjf ^ 3TI^ cRT*m tRTH ■^#333 %Tfl ^ %■^■. w 3721^ ^ ^ TH arqr an^ f^ ar^ pm arrmTET ^ ti ^ Wr apF TTTt^ ■4’ toft afiF ^ tor ^ ^ 'qr ■3Tt% o, i, 2,3,4,5,6,7,8, ■qr 9 ^ ^w^jm TRTiq tl Wt af^ Trot ■qftoff ^ t<t3^' ^ ttw toi^ ^ Ft Ttot ti •^'Wr aRj TTiT'qt qq ■qq^nFi fqwK ton ■qq TRicn ti "^ItoT.% Wr 37^ tttt^ % TF3a^ (w qr^ qr’ft Ftoff ^ st) ^'■qr ■qra % tt3J^’ "f 31T5F q^rt f 1 qq afqr TTir^t % Wtf
qiT qr qra ^ Wq Ttoraftto fPtct q^rt ti Tto qF^ 33^. 3?q. Tft. ton (L. H. C.■sfsqrr "aqqft TTFiqqr qR-qR ato qrrft 10400
Wr «<sHii< tqR qft qtl qjq "t tospr (M.C. Kendall) q toq (B.B. Smith) .FRI fqiFR (R.A Fisher) q (F. Yates) aiTfq t ^ 1."sh ato/TR^aff qft Tnrfqqr tqR q>ti % arTfq^nr
3Tq> Ttor TTTTqt q'lMi 3^ qipid qTT^f «TT 3R^ "t F^ tol tlaqq q»**<i<:< qft TTFIRFISquaring Method), RTF arq^ iqfV (Power Residue Method) anfq aqtn**! t RT^ ar^RiiTqqrfifqft Wt arqr/TRsqi Tnpqt qqr^ q>t aRq^qqRn RFf t qqff^ artqr Wr arq^/TRsq? Tnrqt wraj ti ar^'qTR totqf qqr Rffeqqft qft ppm RRt ■^' Rt Wi afqr/rtoT Tnrqt qf Ftcft t qqi aT^'qrqqnrf F^qq Rqtq qR% arqt 3rto^ ^ qqq qrr pm ti ^ TTftoqftq 4>wj^d< % ^ tqR
qr
170 awc/t niJ&i<t>l<i fpf^pu
ufa^x^f ^M ^ ST^/^n^ (Pseudo Random Digits Numbers)f e^q'l'Rt) ^ ?RT "Wn (Completely Determined Calculation) (Generate)
f W7T 1¥v ^ WR % fRtta ^ ^ ^ ^ ■^mft t'lWt ^ ^TFPmr ^ ^ ^^ "RRIRTt TIc^ 5«t>l^ ^ T^-T^ sFR "t^l pita^
at^ (Digits) ^ t ^ RcT^ sfEFf ^ a^ 'SK fer 3T^ y^s^l/^KuH
^ f RRT ■3R TRsin % ^ RT RI RT RT^f aik
RfT^ "f I fR RRTR RM yoHiarf ^ RRt «oqn<, Il'ich! RTR RRF^ sFRJtm % RRR^ a^RRT RJR #n t. RfcR^i '4' "RM^ RIT^ ITR ^ tl Rf^
Rjil arfRRT RR aTTfft t RR %R^ R?^ RR^ ^ f RRT RK ■^' RT^ fl ^RR>R ■% 3RRiR % qiTwa RTSII RR RRR RR 'f Rf RfRR^ RR Ptnl^i■^Tcit tl ^ R7R ^ t‘ fRR^ 1RRRRR R^^f^RS ^ Wr RT^ % R^ % fRTi 37T^?RR7 ^-RRR, RRRt RRR R^S^TT RTT RRR ^ RT^ RRT 1S?itR. RRRt •^‘ %R 1^ RRI RT^I
Rte-
^ 3TRff RftWRR
Wt aTRT/R?^ Wft ■4' RRR Wn RR RRR RR^ ■% f^ Rf^ RtM % RR RRR RRRT
■?tRTI RT r\ <:rli<iO "ST^ RT RRicft't aTRRT RtRR^f RRR R^t t«\RlRr, RrSF, ftR2 "J^ 1%^
RR RRbr, RT RRRn tl R^ IRRIRT (3TRRI R^ 3TRRT 1W fRRRR ^ R^ RJTRT ^) ■^' R?t
^ arfR^ ^ Rt f^RTRJ % 3^’ RTt Rt^ RTRT "^nsRI RT^ ^ RK^R 1^ RT RRRR h ^aiRRT ^ aify^ ■^' ri^ rtM % f^ ir r^ rtrrt rrW ■4‘ rt rr^ ti fer rtt# ^ RTR^ ■>f3 % RRR ^ ■3RRR R^ 1^ (Point of Entry) ^ RRR RTTRI ^l IR% aM
T% RTt RTRjft R5^ RT "feRR RRT^ "f RRl aTRT RT '^P^d (d«tJo1 "t RR^ 3TRd-RR^R^ Ri■% 3T^' fRRRTT RR^ RT^ ■hVshi r^ RT^ (Entiy Number) ^ RRuft f I RT^ ark % ai^ Rj\- ■^R>T R^ RT5II RRBI "t 3TRRT Rif 3Tk % ai^ fR^ %R R^ % ^«t>"S RT^ (Second
Hand of clock) ^ RR R^ 1^ RT RRkR fl RIR>fl % RT "^pRd ftRR^ % ^ ■^RJ^ Rft^^^‘lRfRRfR^%RT^’RTR'k'l2'^6^^^RlRTf3Tk% aTR^ R^f'^RiT
RRR RTsR RRTf RT RRkTI tl RfR '^RR^ Rt'^RTt'Rmfl2^6%'#R'^RtRT|atT% afRTf RtI RRR RT5TT RRT^ RT RRRft tl R^ TR^TT % fRRfTR % ■3RTRT R?' PiqlRd R^ tkTT t% %TT ferr t' R^ Rrtl "^TT^ Rt R^ Rt % ll, 12, RT l RT ttt RT RRIT Rt affT, ■^4>"S
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Rt TT^TRRT ylas,<l RR RRR '^Tt Rt fRfR IRR '3RT?TR TR^ tl TT%^I
RIR
a^l^'OJT ^ %Ti RTRT %Tt iRafTeTR aTWJRRTR 500 wtf Rt TTRfe (N = 500) ^ 20 ,^IRt RR R^ Wr r1rR?1 (n = 20) ©k-ii tl TTR^ Rf^ TTRfe % 500 '^IRt R^ RTT^ 001
500 R^ ^ RRT Tk^ RRTR Rt -RT^I RRff^ TTRt R^ RRT TT^ 500 % 3TRt' Rt t. '^TTf^ #T afcFl' RT^ Wt Tk^aff RR RRR R7TRI ttRII akp TTTTRt % ^ ^ RT ^ 54^ RT^ ^ TRWI TTTcT R aro RT akFlf ^ RTT»T ^T#» ^ akr
RiRR*
Wr TkSTTk Tjfet RR fRTRRRdR>T
1r>rt RTk RR fr^ Tk^nk fR^pt—940 746 169* 767 897 299* 561 455* 332*' 532' 054*. 585 959 571 630 279* 345* 862 400*. 007* 477 442* 911 929 805 224* 764 938 461* 729
171TSRoT
477* 946 041* 179* 520 219*804 399* 202* 909 .895566 967* 854 416* 626 ■
3T^ 219 tl 31^ ^ t
^ "Pm "t flfd'ASfl ■% 3iiq!{q<^ 'll (Starred Numbers)
cFf ^ ^79 tH<2qn< yf*-Kf^ci chi'll t'l ^ 001 500
«<sqi3ff dKir^id ^ ^817 Wom3Tf 13;^ ^5^^ Wn tl 3T7T:
t''SH qqpid '37^ POHqd fci<a1 >i=hd1 "t-
400, 477, . 455, 007, 224, 399, 367, 169, 332, 041,202, 279, 442, 179, 416, 054, 345, 461, 299; 219,
'^TK'^ ^ 3lPd4 ^ 31T^?^T37 W^s4H< Hlcl H(d1 ^ cR 37
^ ^ch<. ^ 31^ ^ ^ W=hdl ^I ^'
1^ *Tt ^ ^ ^ tl IWfe M 100,1000 '^TT 10,000 .
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Stratified Random •Sampling
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Simple . Random Sampling
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SystematicRandomSampling
ClusterRandomSampling
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Multi-Stage Random Sampling
f%T^ 2. yl^<4'i f^rftRn
3tT ^ -sm'-Ocki. qP'id % 3ifdR«fd ^7^ 3PT i^-Hl ^ *
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172 J%/V4f
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(Systematic Random Sampling)yRi'^^'i fsrfv (Systematic Random Sampling Method) ^
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% ^ (N/n. = k) ilTcT ^ t ^ ^ ifyPd'^^f % S'^Pid 'Tx.'il "t^l "4 k qdidl x[d<^Jfl Tfc^h^F faplf ^
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^ t" 5«bi^ ■% "sIB yr^«ti ki(
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shHq.S shH^ogi
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stiH^OHl ^ t cT^ "^7^ "ST*? «ioqii), suH^iSi ^Hcft t', 177
462
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tl Wil ^ 77TB cT«7T 37B7 TT^T^T^ tl
wfl<j>d yldT^qn (Stratified Random Sampling)
TTRfe ^ "^>75 ^<rs°c ^77^1 37*M TTift (Strata) t «fTHI ‘311 TT^hTT t cf«7T . a7^’%7H=Kclf n T7^ 37*7^ T<7^ ^ "^irTT qi6di tl rR 3IR: wflfd
yPd^iH ^ -51^ 1^ ^ tl ^ inwrfiT^ T7TT ^ ^ ^ ^57T 6, 7 ^8 % % #T 378RT tlFftB R TTFTl ^ ^ T77^‘ ti 37«7^ ^ ^BT^' ti 37«M 3te R fTB # #7 RTjf ^ -giT -^TcFr^ tl ITft 'ERTR ^ ^
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'■kiTl^d Wr TTfcTePER 37^ ■RBT ■^fTcTT tl KiOfid yfdqq*i foffv t TTRfe ^TTRI qt^i'tx, yr^qi T7R I^TTf^ oFl ^RTT '^TIB t pHdqit yPd^sfl 'SFt TRB ^Rcfl
tl f^TTf 'BTT Rtiddl 1«hl1<HT ^ 1T7% PiqU^i % BIT 1^^iB7—
(i) T7RP7 3iiq(i'i (Equal Allocation), (ii) 37^Rict> 37R^ (jproportionate ^location), (Hi) 37-31 anq'd-i (Disproportionatc Allocation) TT*7T (iv) 375^BcW 37T«r2^ (Optimum Allocation) tl TTBt tl
3«W<' 173
■?TR^ 2
HrfcTEra^ '?nT M«t»M % % 'S^ig'<wl
Mfnq^t % 3Tra>IT
ai^Mjfn«t>
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Allocation)
3T-3i^HjIn«»i3TT^
(Dispropo-rationate
Allocation)
aiTsfe^(OptimumAllocation)
^T*iH
(EqualAllocation)
imroT(Variance)
3TTW(Size)Stratum
2820 30200 25 33
■ 40 55500 20 34 50
30 . 30 17300 3310
100100 1001000 100
^ ^ ^ WT WsPT wf f cTW WT ^
"tl ^ ^ 'JHI'+iR % "Sirat^ ^iMifdch 3TT^ i'\ m ^ ^ ^
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aTTcRR ^ (kRRfhT) "Rkt 3lf^ 3TI^oRR fq'qci'iJfilciai 'RRf f^TT^ ‘ST^ Tit 'tl
sT^^cTtR? 311^ ■^' % -RRl % 3n^ (N) ^ umnT (ct2) % iiun4>ci1' % ■^‘ -qf^
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■q^ siife ^% 37STm 3i^/qks!Tr ^ 3RfW
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t«s*l (Cluster Random Sampling)3T^ ^ RpRT 'SfkR^ ^ cfH’ fkf^' 3Ttlf?r 'fkfV, shH«»i5 tkf^ «j<ler>a
Wl fkfv ■RRfe ^ fchl^zfl ^ ^-cOaj^ ^7^ ^ tl ■RRfe ^ ^wrt’'siTRf "^TTn etiPsn ■qr
qk ura qRqr■RTRi ■
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tl
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% ‘kri 'RRRT 6i^<<|>^l t eflv^d) ^sfsqR 'sqqft 101 q^ ■qr^ ^ qfiiq?f
Wn
174 cTsgjR
wn ti ^ ■fsif ^ ^ ■% it# it W? ^ wfn■f ■jST Wh "SritR^f 3T«?^ 9hMs|.a t^H Hft1<^5jl 3?*?^
°ir?T ■^n yet)fli #1
«I^-'?ftTTR nfrl^ra^(Multi Stage Sampling)
31^iVH«P^ 1^ # 3lfM^ «rR ‘Srffra^ ^^.-I'l M«dltl "So Ho i STWIHTCT wit #> HfcT^ H57 HiTi ■% H?# 'JI'IH'^, tHR
W^ H# HRH# i ■# yr^«h ei^<<|>«n .^*IT IhR W^ H# #' # #W5f wt #1 1^ ^ i W^ H# WH it ql«ilq=i> yfa<^^[ H57 "PthH
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3lfcl^ HJI^-'-hImH "tl % 1?ffiH ^ilnnl HT <sh<1«ki qf^m
f^vif HTT Hitn ■sn wm ti<JM0qd #> ^ifnRqd itt?H (Purposive Sample), WfcT HftKJrf (Judgement
Sample), SHHrfWHT HtcR^i (Incidental Sample), <sH<nc*I yPcis^i (Available Sample) ^«1T ■fwif?! (Quota Sample) it ^nit fl ‘^rit HHTR % yfas^iT cIRHi
# tl 3^g^mHHKTf %ztr # # Hftr^ ^ fiukvi
HiT3T t ^ Hftl^ 3T«7^ tl HHTR % HfcT^ HTH: HKp^q) 3TWTHit
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Hftr^ % ^ i <rq1q>K HiT f^THT HncTT t it SPn^J^f Hit -SHtnoti Hi^t tl hih % fqRr^ # ^f^Rna Ht 'HTt HT^ ^q>i§q]i ^ HiT tHHft’H H>T% lH>it it # WZ#t t HH yfd«\jfl Hit fTHHRT H^TH^ Hi^ tl HT^ 1H ’'tit HHiR % # HFH %
3TWR HT ■?mfe #r f fHTH^RtH H tv ^T^HH W HR^ HTH: Hif^ tit tl Htt HiT^T t %Si^Hiiiwqi 'HiRsqqil i fH 'Hit Hfd<\^if Ht HHltHHT (ti'I'HM Hit ■% HHHT tl
nfa<\i(! #■ /773f7HfHHiHT
Hfe
H>RH;
HTH
fjh^iichcim (Student Activity)1. Hfftg7H)tH i yfns^U # HHT HTHTi t?HTW
2. Htticf HHfe sfk 'HHfe H>T Hi HT^ Htf^l
/HtW 175
3. anr? ^?
1.5 ’HRTTT (Summary)
• ^ ^ ^ ^82Tf, 3T8?m <m^*\i ^ m wtht im tnqli "SRH «»)<^ Hi'iq «nf^ "^f <i5Nal 3r^ TT%I
• Tjf^R^f % ?TTH ^ ^Rfl" q4'iirH«t) irrft ^ yfcl^^fpl (Statistics "Q^ ^Statistic) ^ '*!«<(«♦) % 1^ qyf'iicMch "itM ^ yi-q^^i (Parameters)^ fl
• STJH^ ■^' hhIW ar*?^ '*inW<sHi (Population) ^\n4 TT % ^«iq (Setof Units) ^ % I^TT^ 1^<{lqan( "flcft ^ ?f8^i f3B% '4' ar^H^IFT^clf
w ^T=n t)• W’T 8fa<\?I (Simple Random Sample) qi«iq ^ BHfe «f)t ^
IHPt^kTT (Constant) 't 3T«?ffI W«i^ 5«hi§ql ^ "t% xRR ^ ^iPijcff^ ^T^rrf^ ^ ti
(Exercise Questions)
2. (Multi-stage)' ^ ^l3. yPdfiyNH Tf^ 3fR yPd^im ii ^ cITcq^ t?
TRT (Reference Books)WW1.
2. ■^rffei^ Trim, w^imM st? ot/3. siftjr ST? ^7*T/4. yAq>qv fqfl<ui-*^(/^ Pw'K
5. 1??7. W^, 'f/<)«/ ?/^^777/
176
2
3THiChef’ll ITOT’CH’ (q!i'<riqui
(Design and Analysis of Variance)
^TT^FTT (Struetiue)i
2.1 (Objectives).2.2 MWIdHl (Introduction)2.3 (F'Distribution)2.4 y^i^l ^ W (Theoretical Aspect of Analysis of Variance)2.5 (Process of Analysis of Variance)2.6 y^i^l (Post ANOVA Test of Significance)2.7 Tj^ yiRifcHl' ^ ^ n:«i<|ch TOTH
(ANOVA by Subtracting a Constant from the Row Scores)2.8 yPd^Jjf ^T^hnT "3^ % ««<wi
(ANOVA for Comparison of Two Sample Means)2.9 'SRR'n ^ ^TRTcTR (Assumptions of ANOVA)
2.10 ^TRRT (Summary)• (Exercise Questions)• (Reference Books)
2.1 "3^^ (Objectives)
^ 3TWPCH %
• ■^nsfer ^ ww^• ^ 3ir<i "^i
177
3?f*7WK=W TT^' JRm2.2 \rt<iicni (Introduction)
3T5?TVPT^?^ 37f^ % ■*TWRPTI ’SR^ '^TFcTT 'll■^' ^ ^ srqtn t ^ ^ ^ #ti ^ #t -^t^' st, ^
cT«TT B % tTKPTR^ % 3RR ^ TTT®fen ^ ^ 3^ cT8^I ^ % f^, 3^ ^ST? ■?? %^«?T ff #r %I3; rFfH I f=T ffNf 37, cf«7T 17 % "it^hT % 37^
^ 1TT8%^ ■'Ttt^ ■geFTIr'T^ ^McT % "4’ 3T3*7H eFTRT ^ 17%’TTI "ST^^ ^ 1FJ^‘ % 6 ^ W 10 ^ 45 W W\^ ^1 K 17^3^ ^
K(K - l)/2 ^ ?RT^ #tl ^ ^ 37r%I^ IT^' ^-NV ^ ^prm tor ■^n
T^ cT^ 9R STfil^ WTBT37^ cfT^,'^ yfd«^J?f i7%zn7Hf (Extreme Means) % '31^ ITT^
^ ^ ■371^1 ^ ^ 37r7f^ ^ ^ ^ 37f^ i7WTTTpff ^ #7^ 37^WTT "^jlTcTI "t^l 37f?jf7cRT ■^' nHeti ^TH yc^ f^FT-f^TH TTfcRyif %
HH«ti ^ 37f?7^ 37^
■^FT % it777*J|
^ 17% ^^STT
t, w ^ ■^' ^RTToik f-(!:)t cT«7T (ii) ■?17 W ^ IF’TT^ 1^ t ,%
IT^kTT
■JTPT^ fq^cri'i f 'si«<nt> 17^ yfa'r^iT % (Pool)
37^iTB fq^ 17^ %\ ^TROT t % ^ 37f^ .'qWTqpff'(Analysis of Variance) yiP^fV 77% 1%n '^JTTcTT "t I. vl77<^7(ANOVA) ^ f I ANOVA ^37l7eT Analysis of Variance ^ I7^7f% ^ (Abridged form) ’t Analysis AN 37%; of % O 37^ cT^TT Variance % VA 37% fqcT^ oRI tl TR^ 37«7R '^‘ W % ^^iPdct) ^ STT^ R^ZTiTRlf ^ % f%; 7f^ W7T
W7 ^ fen qq? ti '
2.3 T^-fqn'OJi (F-Distribution)
■SRTVn iJrani'll fIffeTT 17ff7^T%^ 171 %Tc^ R. fq>TR (R.A. Fisher)f%T «7TI ^^^ Wt ■Q37 17T27 37% ^ 'qczTqHf gft ^7% ^cTIcfl f % ^ 17^ yfd^vfRq? i\ ITRfe (37«7^ 7337 ^ %1 17qf%f) % f 37S7^ ^\ ^ ?T% ^ 3{%
qfl gcidi eh"<% qain! "t % 1778% fR=7 "f 3787^ ■%! q|(5<*)T ^ ■q?’■^TRH ^IT % 4R^ fq'qd'i (s) % % (Square) % ITITI^ (s^) 't l ■^FT W ?7q ITR^ ^ "f % y^<'J| 37% ^*^5) % R^hm) % 37RR 17T8%7n ?TTr7 ^17TfeT%7 fsff^ "t cT87T ^17^77 IT’J^ % 37R71 17i«r=%7 % 't’l ‘ yti<^i
17^--^ % wim w Wi % 1% iiif% f%7 TSTTcn t %% 17^ 1715^'% qiTl^ % RFft ‘4’ 3787f?T %%1%7 ^1% ■3%7 37787R qi 37^71 qfl
in8%in ^ ti . ' . •■yiTl^ (q^c^qui -f^ffli cii^c7 •4' (G.W. Snedecor) % ?[717 qi^^ f% % tn5Fr-f^rt<«i
(F-Distribution) qi 377^7% tl 171 37R q. %?71 17^; 1925 2= (1/2) log^ (sf / si) ^ ^TRU
171^ %q7 877! ^17 1%!^ % 377% Rl 17^ 1934 ■^' 1%%! 7717% % 373^777 (sf / s|) % t%fqa^wi %! 1%77 77877 f%71 % ■^‘ ^ 7717% % 373q7c7 % w-3T5qT?7 (F-Ratio) % ?% f%iq % qq^'l^iKui (F-Distribution) % ^iq ll%fyR7 f%77l ■eskL % % Tl^-ST^yid cf87i Tiyfi %5RTJT ■4' 11^-37371 f%R % Riq % ^8717 37371 ^ %7q7 tl % %17t ITRfe i( n^ n^ 377q7R %
17**77q
178
37/»7^cW mm1^ ■^ilT^ ^ichi ««<«] shHifi: Sj^'^ Sg^ ^ Sj^ ^ 82^ ^ Sj / s|
(n.^ - 1) °r (n,2 - 1) ti'WiJfi (df’s) 3Tc!: T^'Tj-ST^qTcf F = / s|"t l ^ ■R^iR % T^'F-37^’TRTf % fqa<wi - 1) (n^ - 1)
(df’s) R'?>-Iqa<.yi 'sti^Mil "RT^: 3i:jHia 3T8?^ ^-fqcu'Jl % RTs? ^cwij^i (d.f)itcft ■!■, "OcFr’ (df) % 3^ (Numerator) 3TT^ trr^ % ■!■ rT«T[
-5^^ (dp F’: (Denominator) air^ mm % f 1 % -rtkt
^ ^ (dfs) ^ 3TT^ ■4' f. sm % ^;^Ff?T (dD ^ ^ ■% ^^iVi (d/) t^RFf ■^ITcTT "tl F-(n.^ —l; ftg — 1) ^ FTc'Pf FR R^-ai^nm^ ■§■ f^RT% 3^ ^ FT % i^'^sRTRT (d/) (n.j- 1) R (Mg - 1) t l R'F-ftFT^ ^'UcH^h ^ (PositivelySkewed) 'f^Trf?F "t F^F ^■(H=hi (dfs) FT 'pT’fT FiTrfT tl Tr^ f^RnFT FTTFFTfe ■% FTTRF FtFT t f^RF^ FtdF^ tl aiF: (d/s) % 'M’PF (Pairs)
■ % ^ fF^-fFR FF’ tl
■ftnr—T(CF-fqn<yI (F-Distribution)
TF^ t f% "FT^: FTrCFl FF 31^FTF t F^fl ■^^iPd'dj. "tmm RITTTR t FTT TTFfe % FTRFT % ai^HiPid FFT tl f^FTT^ '^FfFF Fftf
FTTT’F FM %t Ft tl RFr-Sf^FTF % 3f?T FFT FT t y^ctd tHf FTTT^’ Ft t^FT FI TTFf^ % ar^FTfFF FTTTF F^ FiTTF "^FF TRTF FfFT ^l[r*t(r aRT: F^-3i^HlO FF FH TTF" % FTIFT ttFT
FFTF xnF-ar^FTff FF FR FFT "t ^ FFT atfF^i Ft tt TTFFF tl yift=bdl fF^PF FFT TTffelFtF TF^FFTT % F?FFt Ft ^jlV^d TTIFTT 'fFiTTt FTFT /Obtained) aifFFT .aiFette (Observed) Tnp-a^^qTF ^ TF^FTcn 'FT fFFfTF tFFT FHcIT tl TTF-ar^FTF % TH^F Ftt FF art t fF FTT RF-ai^FlF t> %FR TtFtFF?T FFF ttt Ft TFFTFFT aTcFRT FFT t lFfTT% FFPF TPFtFF TF^tFFTf TFT FT FF FTPFt Ft. f^TTTt FF FF aiyrfF FM ^3TT t F!^ Ft TTFfe % FTiTFT aTFFf TTFi TIFTF FTTTR FTTt TTFfeFt % FTTTRlr FF aTJFIF FFt FTFT FT TTFFTT tl TTT^tFFT % iFFtTF ■f -^tn % %IT TJFi-aT^qTF Ft WTT FTt WT TT^F Ft am (Numerator) f F^ mm (Large Variance) FFT Ft (Denominator) t FTTTF (Small Variance) TT3t t fFTit F?F ar^FTF FF FIF F^ t arfFF^ FTFT fIi FTF-fFFTR % TTFIfF F F^ % FFTR t^F FRFT ■^FTFFFT FTFT tl
arwiTF^' f TW FT ^ t TTTFTTFF: .05 FFT .Ol % ^ Tn^fFRU TTit'FF FFFt FTFT tl irrf^ FIF Ft^ aicfTtl^F FF^-ar^FTF (Observed F-Ratio) FF RTF FTT^ TFFttTF Fht ■gFFm (dfs) FT^ FTP iFFm % FTt (Right) FTF -StIfFTF ^ 11m F^FT t FF FTT F7F-FTF Ft .05 TFT FT TF^lFi FiFT FTIFT t FFITF) '%FFT TTFtTSm fFFT F^ T^'F-FTF 3Ft Ft FTfFFTFT 100 t t %FR 5 ar^ F^rt FTF tl FBI FFFT t FfF lF>Tt arFrtf^ ipF-ai^Hid (Observed F-Ratio) FF. RTF FTft TTTFfm ^ ^JFFm (dfs) FT^ RFT-fFFTR ^ Fit (Right) TTFT fIfW ^ t fm FtFT t, FF Ft .01 TFT FT TntFT FTFT WFI t FFffF) FF V^-'m % t^FeT t^hlFVI arrt Ft FifFFTFT 100 f t tlFR 1 aTFFT FTlt FTF fI# tl iFTTt aTF#fFTF THF-FTF % Fit (Left) ^ FT Im 95% ^F
t fifI
179F^FFT
tjg 1R77W -if "3^ .05, ^ 1THT ^ t ^ (Left) 1^ 99% ^ fern^ ■'R ^ .01 m 3TOT«f^ RHI ■STTcn tl cZj-ef^^ 37^#^ T^^-STjqm ^ ■RT^fsFRn ^^*7^ T377J-3i^Hnil ^ ■iHK'Jil (Table of Significant F-Ratios) ^ f I ??T■RRoft T^'F-3T'fira % 3^ cT«in^ ^ (dfs) % f^«FT (Pairs) % .05Tr«7T’.'^iciH "IIH t‘1 ^ RH ^Rcl^ 3<ciQij(cha ^ ^
"RR (Critical Values) f! 3Jr?r tr^f-STT^ ^ RR ^RTPJTt ^ ^ ^ ^ -qt #■77T«f^ TRR TR 3Rn«l^ ^ -iRH t ^«rT RtM ffR STfq^ tR ^ -RT^ta -RR ^ .
^ ■sum tl ^ RR^ ^ 3151R? (Row) tl«n 3J«RT RT^T (Column) f (dfs) "M tl ■RKR ■^' ^ 3RR^. 3f^ tf R3T W t. % %? cTSTT -SI^rR RPR t IRR^. "Wt R§T iRn t, % (f//s) ftt ^ f I 3T?T ^ ^ % g^^M (dfs) Rfer ^ rr*! ^
^ T^F-RR t f^' ^ RR .05 RR % RRT ^ .01 RR % tl‘ Rn# (Light Face) R 1tRT g^ RR .05 RR % flRT RFTt (Bold Face) ^g^-RR .01 RR % t M g^^T % .05 rr % lRg gq^-RR r^r it.01 RR % RR^ gqr RR ^ RR -tRn t, -IRf^g ^ ^ ^ RT TRRT t fe RRoft t ftRT ^ gq^-RR .05 RR % RRi R^ gqr-RR .01 'RR % %g gqr-3TgqR ^ shiPd^ RR tl?n tl 3<ie<'Jtl^ Rf^ 3TR R gcwi^fl RTRR 3 R 20 t, RR gRT-RlRfft Rt RRR Rf^ ti df= 3 RRIRRR RRR t df = 20 t^itl ^ % RRIR RT gf^ R!^ tl gRT % RTRIjI rR tl>ll RRRI ^Rl^ t % df= (3, 20) Rt gq^ R^l .05 RR Rt RRRI RR 3.10 C?^ R^TRI -^f) RRI .01 RR RT RRoft RR 4,94 (R^ RR^ ^ t) tl 3R: df^ (3, 20) RI^ 37R#%R gqr-STjqR RiT RR 3.10 ^ Slf^ tit RT R?‘ .05 RR RT: RT^IRi tblT _'5RffRi' 4.94 "t srlRRS" tit R^ R? .01 RR R7 RT«fRi‘ ttqii sTRRfer gqr-37gqR #r 3.10 ^ rrt tit rc r? gN! t ^ r1 rr R7 rrIr Rtf R^wr ■RltRII
/cr^c)«fu/
(Theoretical Aspect of Analysis of Variance)
%?R t gq>-3Tgqm, gqr-igcRq rrt r«trr ^ rrrt gfe % rcrrI'. rr ^rrIr rr#) .srtRj' Rftg^f R^RRpt ^ RTR gRRT RRt % %g 3RRR fR^^Rnr RtIrIr ^ iRRt^ fRjRTI Rfg RRfe tl 3TRiR % SrtRi RIrR^ tSlt Rlt tl 3it> R^RRR Rr1rR?T ^ RfRRRR gfe % ROT gRJ-g^ ^ fsRR tt RRTt tl fR# R RRffeS % ROT (a^) RR arjRR Rl fR=T-f«R ^Itl (Sources) ^RTIRI 'RT RRRfT tl RRfe % RRRTT RR gRT ■STgRTR tl IrIrH % RRPtf
■RT RRkII tl rIR 1R^ RRig?! RR RIReF IcfR^ s t RR "SR RRfe, '^RT^ Rf RfRR?! %RT RRT t, ■% gRR> fc^T^ci'i (a) RR STgaiPid s^jn In-1 tlRT tl 37R: Rgfe "% RRTR (a^) RR <ngHiPia
RH 5^ [rifin - 1)] tlRT, "Rtf n rIrR^ RR SURTR tl RIR f^dl ■HHp^ R^t RfRR?! f^t Rt tl, RR
ITRfe % RWI RR gRT aresi 37giiR (Better Estimate) '^it yld^VlT % WR siggitl RR ate tl
it cPTlRT
tr 37^ ROT (Within Group Variance) Rt 3TRT: ROT tl RJ^ t R7^' R^ gOT t 1^ aiTRTflR? ROT ^ RR RrIr IOT RRT tl % OT -jrOT (Between GroupsVariance) ^ R^T ROT 37RRT aprlt ROT tl R>Ft t Rpg TR^R g^ t ^77% ^ RTgl ROT RR tl RRIr feRT RRT ti
180 3t^rf<
'H=bdi fl TTRf^ %(q-cici'i (Pooled Standard Deviation) ^ ■^RkTT %\ swlTVi %
^ yId<Jf!T ■% 3?^ "SfI fq^^ii^ilcicil ^ vRRT W "t ^ ■% ap^yH^ui (Within Groups Variance) arRrft^(Within Variance) 37^^ "t cT«IT
<^‘w W'^ai^ t'l % y«<«i (a^),"^ ■^3t[ 3T^iTPT yIa<AJff■R^^HmI cFTPTT ■3TT fichfil 'll Hld=til ^ ^<u| -fhif % IT^T^
(a^) % 37fM^ 3p^ T^' JRm
awM
q*^d: ^<^*ii’il
»Ti^ Tq^qvli "t '^CTT [q-c<eii a *11-1=^(aj^.j) cr / yfn % ®Rr®R "tl 3TcT: iTH ^ sj<,iq< 'f^TT ■ciiie'^lafT^TTt M % K ytd<Jfff ■5TFrT K Iqxid'i ^TTcf "3^ K/K — 1 ^ '^’
ajyi aj^TTR 31?^ ^ IT^riTTI a^ = nc^^, 3R: ^ irtr
% ^ ^ ^ ^ 3RTT^ ^ ^ ar^crPT ^ ^
ar^TTR yRi'^i'iT % jt^hmI % r^trt ^ ^*^5) % t7%27 yy<ui(Between Groups Variance) 37^7^ y^<«i (Between Variance) 373*TR 'i'a^B ^ ^ "f et^Hl'f^ ^^TJTfe % y«<,^J|-% ^ aijHH 'WnPoe % y^^^t %
FT ^ W -sn t ^87T TfTTF % WT'»FT37T^ ^ T^q^-aT^ ^«Nf ^ ^ 1.00 f cT^ 3T8j f % W
qfl <^qcri 371^ yif^=tidl 37c^ "t 37«7f?I^ fq ^ravnl’ 'SR 'Qiqi ^% 1^ 3R7T^ 37^qH ^ yipT^dl 3F?RT 'tl 37^: «wiP<FT ^«l=hdl 'qT.qTFF ■«chdl■f 1^ W7?3I 3i^Hi'i TJiqT TRfe % f I f % yPd^J^fIfr^ 37^171^ 3IP^ ^ f ■q^ ■^Rfe 37«7^ ^T^Tfe^’ f^ M f,ft^^dT2^ (Implication) "t yfd<\^iT % ^T^hhT 37^ f '^Fi37^ t| 1wt?I ■qqr-'RH #r 37^77«f^ ^ ^ 3T8j t 1^ w TJ^-'m 'TTT^f^ m
W'n'ldqjfi ^ 377^7 ^ ^chdi "t 37«7f71^^t^ yHi^il qFI T^ ^ 37«M 'TTFfe^ % %q;37?7Fq 37517H ^TTFT ^ehdi't P>i^«t)i Pifs.di^f ■!■ 1^ yPd'A^^l '*T^hi'i'1 % 37q^lFqrd 37RTt
'TTqt’J^ ^ y«»>dl "fl '
37^ T7fc4*7idl "5^ ^ 3H<l=td c(P5fd 37flFF7T "Qiqr 3Al6<^l ^eNdl'W^l fT^Tin ^ ^Chdi f I T7H7 % 37, ^ rT«7T ^ #7 37f?r^ f f^Hcb yikll’ch, 7T«7THMqi fq-qcid Pl^-iqd t’-
37f%7^
'4iit«n—1rftH % ivTl^ Mf<ehn^r1 'HHCh
^3T
5 6 910 9 1311 3 156 2 ■ 7
71 = 4 n = 4
M = 11
$2 = 10.0
. = 'i
M =8
s2 = 6.5
T =K
f =7.5
d-^ai wfw^^fhf /^r/yqf I8I
37fi?^7W mw ■J^RT yrd«\!jiT ^ ^ 3T?t^ 'SRT 'W "tl cR rlHt yfa<\^fiT
% TT^THHf ■% 3T^eftf^ 3RR TTlcra^ ^ %^^rtj| 'ftRTI «wlTVi ffNt "tt
^Rfe (3T87^ TrrfeiT) 1^ "t ^^Rny. % URTM ^ 3T^*TPT "SR ^*\\m\tlti'^)ai
wfrt«AV>T % MtKuil 'HhI** % ST^TR
(Estimation of Population Variance from the Variances of Samples)‘^’ ^87T ^ mv^ ?fR9T: 6.5, 7.5 ^«TT 10.0 tl 3TW7R 13 f V^VH'>dd: yffl'^J^I *3?
t •srf^ % %cr s^ w ^ 3?™: /i ^'; cr
■% ^ ^^HiPici 'JTFT% TRRR ^ ST^iTH [4/(4 - 1)] x 6.5 = 8.67 tiRTI ^
%
[nl(n-l)] s^ ttcn tl 3R:3rfcnf^ '3?' STT^TR’’K'^Rfey«hK ‘■^' % y«^''i % 3n^ "'TT
^■3i^iTR [4/(4- 1)] X 7.5= 10.00 .^«TT ‘^’ ^^ 3TmR ^■% y«<«i 3ijhi»i -[4/(4 - 1)] X 10 = 13.33 tt^l HRsbtrH'ii t % t cfHf uffl^
^ Rt t, 8.67, 10.00 ^ 13.33 i\ TRfe ^ ^ 3igiTRtft ^«fT % won ^ 3r50T 3?^ (Better Estimate) in cfHf STjilRf ^ ste tHn
^ 10.67 t, 3R: ^ #i\‘ yrd<tJ^iT ^ wW % 3n^ ^ wfe % wr^n %
uwn
WdlriH ST^sfB 10.67 tf^TI ‘^F liwn arjiTH ^fd^C^lT % 37^ yiklfcbT ^ WOn ^ TIRT13TT t, §«Riy. it 37RTft^ ^jfd^f won 3i^hi'1 (Within Samples Variance Estimate)
fonlti , ' ■
wlhc^^iT % ^TtWnf ^ Wrfe % WTOT W 37^*?R(Estimation of Population Variance from the Means of Samples)
3lf^ '.3T’, ‘g’ W '■??’ % TTEWH^^PTO: 8, 5 11 fl ^ -qKm 8t TfsiT won 6.00 tl o*;wT=rf wi w won otn okwh!.'^ w onr t OKTinnf■% f^wn ■%.won ymfcrieh 3{^md (Parametric Estimate) [3/(3 - 1)] x 6 = 9 % ttni 37«iZTR 13 t 07^ f^RT t HI’ll % f^cTOn % OFR) fq^^cid. 3T87fcl^ 3IOOn % ^O^^OTPT (Oj^) «F^ 10^ uPd^i^f ■^. 3TR>R n ^ wfe (q-qci'i a t^t "nr"^TR (3 / ^fn 6<tNt tl §hRir Wfe % won ^ OR "% WOT •qifsk.l «rqiT«ft O^OOR ^ OROT ^ Ot STOfO ^ O^OORt' % Iwon % WOn % WOT ttcH t fOTRJT sT^orfno OR o^-OT 91 oor oRk?! t' oh-oh oronor t wRno oofe % won on or^oif^ -OR 4 X 9 = 36 thni ooff^ OF WOn ST^OR UPd'tVlT % OtOOT^' ■% won "t OTR ^3TT t.Tt OTFI ofoo^f won 3T50R (Between Samples Variance Estimate) ort’^ TfOT a^g't i^nitl
T^-3T^TTO OTOT WOT (Obtaining the F-Ratio)
TRifit oof^ won %'Fntoo ^ arjOR, aw TfOT 0^ tt oofe % ^ fOR-fOR wOTR t. 005 t ■#Tt' OOWOTI fOR-fOR tl OO^t tl OF OOT't oNf WOn.ST^OR
t 37001 OTTOO t fOR t, W^-3T3nR OTR 0>T% WOt OTsfORT On OTIOR %0T OT
tl OFT or F = ay / aw = 36/10.67 = 3.37 tl OOlfR ctw ^ ^ sOOlfto
t wfRO %o d/=,3 -1 = 2 ttnli ^ ooff^ ay oto ofoovif, lont ^ Ot^ %■^nom (d/) on OR 4 -1 = 3 t, or SOOrfR t WfRO fTT^ 1m df= 3 + 3 + 3 = 9 ttfll T^R-si^nR (Obtained F-Ratio> % 3m 0 2 001 FT % .9 ^oom (df) tMl goom 2
OORT
^ /O/OOf182 own?
^ ■qr'TT-wrt ^ t .05 ^ ^ shlP^iTH (Critical Value)4.26 .01 W ^ 8.02 tl TIM F = 3.37 % FI ^ ^ ^ ^^ mf m: ^ ^siT^r 3M; ^ ^ t % t^-st^ tttm1^ WJ] t, •■fl’^VlcIV'l Tt^-^ f^ f I ^ ^ ^ t % T^mST^^TR qiKiq "t
"w f. ai«M 13;^ ■?iTTfe^ yfd'^^iT ■%■f <Hq'l*i<4>ii ^ TrtrT^^ ^ ^IIT*1 'tl 3TcI; Pi«h4 =^51 'H'^icll "f 1^
3jfV4><^' V^' 7H7^
TIM
■f % y'W'^ "^nTfe % TRR^^ f^-f^ TIM FT^ T3;^-3T5qT?I TIM ’^HRI tl W .’9^ FM T^-ai^yicT
■ ^ '?TTtfen-ai^8f^ it Wm TJ1%^ % 3F?R ■^T]5fen-3TOT«f^ itit tl
•i4<|cK1 fq^-q’l ■% 3TT^ M;
M^Hi % ofTR'JT TTHF^ [q^c^qyi "% ^RI ‘•T^T’TFTt % 3M^ ^-■«T^t^.(F-test) it W ^ tl FM t % Timni TTcp-'q^^ i( ^ ^sn^irt % ^ Trfti^?i! ^ TIM .-c;^ tt 3t«m ^frIW i Wr^ TlfiT^?if TIM ttt hR'^ic^'II T^NtR "HHI '^RkTI tl 3M: 'Qi^-'yilSir’^■^’ "^Flt 4n.«t>Q^'il3T(- ^ I^TM Hnoi ■H«M1
rq5»(j^4W| % W^
t-^0 • ^1 t^2 1^3 = l^n
3M: yt^'+trM’TI qipol 1^ TlfcRTf ^ t,Rtl t 'Jiqf^ ^ecll t 11 •
% ^RI ■Q^-BT^yR TIM "^iTt't Pll^ci MIO'^) etifcJiTlcftcl ^qidl tl m<5ct) '5^ TT^Ft t'l ^ Slf^iR) ^*^5!% iIwmRf ^ tg WIT WT 'FTto ^ 315HTF ^ ^3TR9^Mcn tl
TraT'H
i TI^nM fyVc^MUl TprpiT "^T^ i 'O;^ fsrfV ^ TRUt ■^TTcTT t % yikii^iT ^ mmj ^ T^qr-ar^w ^ wn ^ tr^ ^FMl'ti ft ^
3TTt %M W TFT tl
oqqFK
2.5 THTFIT f^9^Nur TJfeiil (Process of Analysis of Variance)
fqi't^q'Ji (Analysis of Variance ANOVA) '5)1 TR*^ Tlf^M ^ TltFRt ‘STRI ^ TT^icII t ^ FT^ itWI t—
(0 wt (Sums of Squares) WcT ■^RRI,
■ (ii) ■gsRITT (dfs) W ^RHT,
(Hi) TRRTq 3T5*M (Variance Estimate) oimmi,
(iv) TT^-ST^ (F-Ratio)’^ WTT ^^TT
{u) TIM (Obtained F-Ratio) •^iomi cFmi
. FI ^ ^ i^wT ^ 3iTt % -qr ^ ^ "q^' ^ wtt Wi ^ fm
1^ Ml tl
^ Tltri (Sums of Squares)
fqf^M ^ 5lTcI TRTT^ Traff^ RFW'J^ '^>1^ tl TUfenqiN tiJHI^' '^f^ (Sum of Squares M SS) TTTM^ dirH^i TITMT^’ TR% RWRTH [qqeiili "%
TRuoT
183
% ^ f I W ff^T ^ ■^ffn (Total Sum of Squares), ^ ^
(Between Sum of Squares) aTRTft^ ^ ^ (Within Sum of Squares) ^ ^ ^^ ^ ^ % ■gro Tj^ W<lfq^’ (Directly) ^ fen
;«ef>ai "tl 3TP^ "^Nt '5TW % Si^i’^T % fe^ ^ 'fe ^1^ ^ ■^tW ^ SSt ^ fei^ t ^8?T -q?- % fe[^
(Total Group) % W^lfe qTT "3^ ■qezRH ^ fe M feR#’ % wfi qq -fe ^ tl 37?r:SSt =23^SSt = S (X - M)2
(Iq-qciiT ■% ?KT) •
■qr -
■(2X)2 , ,w q?:. sSt, = 2x2 - ‘ N
qffT • X = ■% yiyiiVi
(■^ qRffqtt % gKi)
M = ^ qq q^qqH
I,:»^ = "^nj? % N HiKii'chT qrt qsjpqq % f^ Iq-ciciiT qq
2X = ■^. N qpqfe qq fe . '2X2 =: ^ ^ tT^ N qrqrfe ^ q^ qq qW
N = ^ wqfe qi^ WisHiqm qW qff ■^' SSg ■qfeT^ feq^ t q^T q? fepT qfe?l % qwfel
qq q^qqrq q^ iq-qciiT % q^ qq ■fi fq q^ qi^ 3t^ q^i qt^(At Par) qqr^ % fe; qfef qwfet’ qq ^ q^miq fe % qqf qff ^q% qfeqf %
Tnr ,qTT% qrt^ f 1 arq; . 'qiU grf -zitq, (fefef % gro)SSB = 2[ri. (M.-M)2]
^ qqr^ qt.
(ZXj)" (?X)^SS3 = I
■q^ n- = i qfqq^f (qr ir^) '4' qfeqq qft «h'<s4i
M- = t qfel (qr q^JF) qq qmpT
, M = "^ qiJF qq q^qqR
'IX. = i lafef (qr ■q^) % ^ql 71. yikll+T qq ^
IX = N ytKiiViT qq qfq
N = T^' yi^i’«tj) qfl ■^Rsq.
sTRiRqi qrf ,fe qft ■^' ss^ qffer^ feq^ "t qq? qF qfe?f % qrqnqft ^ arq^-afe qiZTrrnf^ f^ tj^ rqq'Hql % qqf qq 4q ti arq:
- 3TRTfeq^#rT, SS^ = I [I (X; - M.)^] wi qrr^ m,
(^ ytkiiq^T % FTO)N
(fqqqFff % g[Rl)
g Xjfss„=I^'-IX.. = t ■^' ijfqq^ (qi "^qj^) % qrqqqr
M. = i q qfe?f (qi fTijF) qq q^qqpT
¥Mfq^' % gKT)
184 a^(T< /qfyqf
. SX; = C^n n- yikH'=+^T ^ ■%
ri' = i ^ C^ yi'<n'«t)T ^
(SS^) 3T^ ^ ^ (SSg ^ SS^) ^ ti ^ VM‘% ■^T^’Ttl 37cT:
SSrp = SSg + SS^.
r^<V'c^t|'J| (ANOVA) ^ ■?tW ^ ^ W (Additive Parts) ’4' ^ ^rbft % f^TH TT>T?n ^chai ^1 '^fK
we t %■^fen ti ^ %
^ 1TWTH ui\,i'^ ^ TTWTPT Mi i\iTPJ^ ^ ^ yiklf^ X X■gMT^ % 1PJ¥ % 1^WTF^ M fq^cii 1^ ^ f^TdT ^-^dl t—' •
wT'n
X-M = (x-M.) + (M.-M)■5fNf ■R#' ^ ^ ^ -qr'
(X - M)2 = (X - M.)2 + (Mi - M)2 + 2(X - M^) (Mi- M) i %■%! TT'ift n.. W^ ^ %TI m ■ .
S (X - M)2 = Z (X -M.)2 + Z (M- - M)2 + 2 Z (X - M.) (M^ - M)
.- i WJf % (M; -M) ^ 1TH fw (Constant) itm. SJP:
Z (X - M)2 = Z (X - M-)2 + n- (M- - M)^ + 2(M^ - M) Z (X - M-)
i ^ % %i3: Z (X - M.) = 0 3{?T:
, . ■ Z(X-M)2 = Z(X-M.)2 + 7t.(Mi~M)2 K -q^' ^ -zffTT ^ TT^
Z [Z(X - M)2]- = Z [Z(X - M.)2] + Z [7i.(M. - M)2]
^sqto wteq ^ -q^ (L.H.S.) ^ -qk (SS,,)^ ■q^ 3TRTft^ -STfry (SS^) ^ TT«q -JitTr (SSg) %¥W%T % ^ .'^ ■#])■ ^ WTT ^ cft^ ^ -q^Ti ^^ ^ ti oHqpK ■qPT: SS^ ^ SSg ^ WTl
SSw
(L.H.S.) ^-q^TiT tl ?fM -SRf ^
WRn ^ 3T«T^ -SftUT % ^ f! fT«TT SS^ SSg ^
■?TT^
d«TT
qTw
qrr ■qn^n ti 3t?t:SFd 1 Wn
sSt = i;x2-.^5^ ■ N
(ZX)2SSb=Egru ^ N
3TRTf^ M qjtn, SS^ = SS,p - ssZX = fvl ^ yikl(d>T qq -qW
ZX2 = wj?' % TT^ ■qrqrfqff % qq
Bqj?T
N = yiKii'chT qft
ZX. = i ■^' 'w^ % HiKiiVi'l qq
n.. = i ^ ■qRTfqjf qft
Z[(ZX.)2MJ = K (Z X.)2/n. qq
ec^<7T wfwiwf^ /q/qqf 185
Vcf jTFTfW (sx)2/n (SS^) ^5^t7«i ^^ (SSg)■^RFTT ■^HcIT "t, IwIciM, (Correction Term) % ^ ^ ^SIT^ t cf^^TC %TSl WT tl f^rSR "q?^ -q^ (C) W ^ f #ilf -^M'^ .Wn ^ f I %T W ^ ^ .PHHcld
C - g X)^N
SSt = 1X2-0
g X,)^SSb = E
3TRTfi^ SS^ = SS^- SSg
xm ^ -c^i
tl'wm'i(Degrees of Freedom)
^i«Kn’^l ■^TRT^ ■^’ ■^ ■qftfwfd 37^^3Tf "qft «OHi fl '^IT^ .yW^
"4 ^ ■#] 114 %, fq cfNf 4 ■g^cTRit (dfs)cFTHT ■^^TTI w\ i^iT (SSj) N yiyii4>T ‘^TFRcn 4 ^TTcl 14»in ■^»lTdT t, 1^1% 'gqdM ^ -^nsm (N - 1) ■41ft d/p 44^41^ 4 14^ f I -qyq; ^ 441 (SSg) ^ K iT^WTf ^ Jy^'RclI 4 WcT tqrqr "^aiRn ■!■, ^«kiki1 ^4 4<^i (K -1) 44ft f44 df^ 4%^^ 4 14r§t
ti 3imR+ M 44t (SS^) ^ K 14^4 4 31^4^ % %ti (n- - i) t, 4
3!fM 4. ??rf^ ^ 4i54i fq 4r ■% 44t 4> ^0^1 srarf^2(n. - 1) = N - K 4t4t f44 d/"^ 44Knw sifwrqtT 14jin ^ft 3m;
d/T=N~l '^ 44t (SSj) % f4n^ ,ofi^ gif 44t (SSg) % fen^ ijysfnvi.31rt1^ 41it (SS^) % 14ni ■^ttttt, df^^ = N - K
N ^ yikil4)l' ■qft 455IT cjgi K ^ ^ 4l^ tl
d/g = K - 1
’ sfTU 441 rT^’3TRlft^ ^ 4 '^ctdlVfl ^ 44T ^ 44t % % e^<isK
44n 41 4> 'iMO«w ■?j4’ 4 4t f 14>d/j — d/g — df
#4 ^cmi^iT 4) 4> «bKWi 4qci 4 tJ'WRil % 44 qr 44nr 4 ^IVmI t1?haj 4^1 -:
WT^T ai^TR
(Variance Estimates)44t (SS) "5^^ (df) ■qft w 4i ^3WtT % "SRpiq ■% 1^ 4t 3t^hh qr^r
f4i4 ■5n4 "tl TPER'iT 4> TT- ■qit in*^ ^ 44t (Mean Sum of Squares ^1 MS) 4t fl
y^<.u| 44t (SSg) "4) ■3?t4 (d/g) 4 ^FT qM f^rq?■sfM f 144 TPj^ # qwT 3RR % fero; iF«q ^ 44t (MSg) ^=^^4 f i 4^ 3rr 4r 14n3:
TTTwi g4 44t (MSg) 4i qtqqpit % 3TmR qr ■qqfe ^ t.
^'Hpdq ^4 O'! 3T8^gT Vg 4 4t 14^34 "f i q^ qr^qr f4?4q^ 4 14iq qrq; MSg 44hTT5f: qrr 4t q44T 4FqT qrmr ti 3m: ,
SSbgr?! tiT?g gif 4tiT, MSg =d/B ,-
186 • t4/4qf
^ ^ (SSyy) ^ "3^ »jeKiKr (df^y) ^ ^ W?!r=t’Mi ■^TTcTT ^ (MSy^r) ^1 q^d: ^*j6l "% 3T^
f^^FTH 3FTFIT % 37mR ^F?fe 3IW^ ^RrH t, ' 3T£f^ .«Ttf^ps^ f I ■q^g -m: MS^ Tr%m^ ^ i\ ■srW ^sm ti stcT:
_ SSw. .
3?/*7q?rqW ^ JWF7 /^7#qor
.3lWRch mS!I ^ '^, MSw~ d/w
■q^ cfkTT fn^^T -qW (MS^) ^ ^ (SS^) ^ ^ wf^^r. -5^ (df^) »tft
q)<.<^ ■qRT 'Wqidi 'tl ■'F^ ■qWT ^IRT "^TT^
ti a^ 3ts!t^ V^ t sra:
gif, MS^ = SStd/r
(F-Ratio)fgfvFT tnwi gif gfij (MS) qM '^TT^ % i^-3i^Htd gft ipiRI 'f I
% gWT 3RR % 1^ grWT grf gfiT (MSg) ^ % ^TRlIW ITT^T gif gfij (MS^) ^gmi ti 3m:
. MS . MSw
B1^-3T^QTH, F =
oqiiwi (Interpretation)T3:q7-3gqe gff ipw grr^ % "^qTim mm T27F-gm gff gf[ gmf'll ftriT ■fpgfHm
ti'WRi'l (dfs) % %q; qi’fe'd gr^fsfmT ■?<r gx i^qi-aT^qm gir «kwD grg gnm 'll 1^ ■^w 1^gi ■gi T^TP-^i^Hid ■% "f, 'q;gi' 3T?T (Numerator) ^ MS %f^ ggr ^ (Denominator) 31^ MS ^ 1^ ^ tl gigRiTO: F % 3TT^ gfte^ ^ d/g ggj d/yy f^nggiT "3^% ^qd^i ^ ti -am: mm F (d/g, d/g) grf ■grcgf
' T^ gffeg m'lfgmT gnl^ T^m-gm (Critical Values of F) grt^ gr. mm^gm-3r;jg(d gff gr^gidi-argr^qiai gffeg m^fgmr "gR ■% ‘fggfftg gff "gRTf grm ggi-si^Hia %
^ gg%
«igd
3RTT8fgr fngiridi ■§■ ■% fqfg-H
gm-sTjgm % fgmi^ Iggrcmr ■!■ 1m fgfgm % gtggHI gfg smR gr^fg!wg fs^'5*-^gur % mRuiihT grf mg: wm mg amff^ ^ ^ mE^ mr^ f i
ggmr
gTTgff~2
fq^'^qwi % mT gRim
(Sunimary of the Results of Analysis of Variance) .aRTm
grig(Source)
wr ^ ■qrg»-3T^girrdf SS MS F
MSb
MSwgT8fgr/5mT«fm
^4 SSg MSB
(gij^ % gsg)
3TRTf^ d/w SSw MSw(g*j^ % 3imT)
d/)j. . SSt ■ MSt
/gf^ 187T^gTR
jmVJJ (Analysis of Variance) ^ wn STFi 1^ ^W ^ -^T^Tfti -
'3«^I6<«I-<HK'J|1 1 ^ 'WhViI % fq?'(A'^w| oFTT ‘51^ cRT^ 'yf^'i^vi?^ hoi'll Chin*!*) I
1 T^-l^ '5fRff^)' %'m
(ANOVA) ^ 30#^ WTT3kTw
t<K«n 3
W 3T
X2 ■X X2 X2X X
5 25 6 36 9 8110 100 9 81 13 • 16911 121 3 9 •15 2256 36 2 4 7 49
ZX = 32 ZX2 = 282
n = 4 M = 8.00ZX = 20 ZX2=130
M = 5.00ZX = 44 ZX2 = 524
M = 11.00n = 4 n = 4
N = 4 + 4 + 4 = 12 ZX = 32 + 20 + 44 = 96
ZX2 = 282 + 130 + 524 = 936
(SX)^"HyftSR T7^, C =N
96x9612
= 768SSt = Z X2 _ c
= 936-768 .= 168
(SXt)^SSb = Z -c
32x32 20x20 44x 44 ^ 7684 4 • 41
= 256 + 100 + 484 - 768= 72
STPrlft^ SSyv = SS,j, ~ SS= 168 - 72 = 96 .
12 t ^ (SS^) ^ df.^ = 12 - 1 = 11 #t1i
f. ^^#TT3; ■jfrr (SSg) % d/g = 3-1 = 2 ^*TT 3TRrf?oF ^ ■qW % (SS^) % (SS^) % d4 = 12 - 3 = 9 #ftl 31^: ^ ^ (MS)
B
W?^<ib?4 /gfW188 dWoT
37/»7^'FV'// Jfmw '' _ SSipMSt =
16811
= 15.273
SSbMSb =rf/B722
= 36
SS\vMS^ =d/'w969’
= 10.667
MS B■cnTr-3T^cn?r, F =37?T:MSw
36.0 = 3.37• 10.667
■SRTTUT %. ^ pTHc^d tl
'^TIT^-4
1?r?ipra^ % mRwiihI ^KiVi
(Source) ITH.ssdf MS ' F
2 . • 72 36 3.37 Fo5<2.9> = 4.26
F:o,(2.9)= 8.02
% 3T^ 9 96 10.667
16811 15.273
3TT^ F = 3.37 ^ 1TB df(2, 9) ^nsNFBT % .05 ^..01 3TT^?^ FiTPTf ^TIT^: 4.26 ^ 8.02, ^ t, 3TTT: -^T? ^ ^ tl ^3TTTT^ W-^T^W % 3TTUR "CR W :3TT 1T^ t % 1¥*R ’^T’^ % ^wmPTf 3TB^ tl
ti'lMi’i (Steps)
iRTT^ % 'dW ^ ‘4’ ^ ^ -sn ^bbi t-
(i) yikil'chT ^ -qW WtB TT8TT ^ ^ ^ SX ^ ^1
(ii) UMfchT % ^ ^ -^T^tt f¥«TB tT8n ^ TTJJ? % IX^ iTB ^1
(iu) (C) W ^1’ (ii;) ^ ■4‘1TB TT3^ ^ ^ ■*TW (SS^) iim ^1 .
(y) ^ ^ XmR tthi ^ zfW (SSg) W .
(yt) SS.p ^ ^ SSg *Hd|ch< 3TTBf^ ^ ^ (SS^) ^ ^1
(uii) SSg, SS^ ^ SSq, % %q;. ^ ^l
TsgfR 189
'3?f»yqnrV'jf (viii) ^ ■zM'. ^ ^ W\ ^ ■RWT ^ ^ W ^1(ix) MSg ^ MS^ ^ 'TFT ^ F ^ MR^ir^Irt FFT ?RT ^1(x) ^=wiVil ■’TT F ^Kujl T^ ^'■^1
(:>:i) hRmRjii F ■FT8^^F^ W "^1
2.6 lTT?f9RHT T^Ot^f(Post ANOVA Test of Significance)
ofnTTT •'
STOTsfcB' "t ■^^cf 3iq(nl(^a 3RR sR^ ^'ql^iq^fi FFTT ^q>ai "tl
■'TF^ ■t3['T>-31 j^W "FR ■!■ cR f^ff^F=T ^iqcrllRhd 3F^ ^ WqViq^fi■FPFTT t srarfcT 1T«FTFT WF ^ ^ f^F^T "STr f I 1^ 3T«f^ 1■RWFTH %=T ti ■?n«Nfr wn t %FWFTrff "5”^ (At Least One Pair of Means) f ^ F^^hm'I %3RF: tl 3^tT; ^ ^ t ^ fkf^ -FKpTFff ^ #^r ^ ^ -
ff % ■% 3RTT 'tl ■'TF^^ T3:^-3T^ ^ t ^ ^ 3Tm^ tl t ^
TT^Wff i{ f’F^T tl ^ "EF^ ^TTrTT tl 37F: ^«fqr■% •SHO-n "qF 375^FFT fqiqT '^TPTT •dlfeq.l ■FT2^q7 T^ % '^qTRf ^-■q(t^
^ 3rqtq qft 3i^ M«f=t fqW' fi ^ ^-wr ■qft^ (Duncun’s
Multiple Range Test), "qt!^ (Newman-Keul’s Test), ’Tftspq (Dunnett’sTest) 3TTR 3Tfttq? (Accurate) t q^ ‘SISFT 3iqJH qft ^ (Type IJError) ^ qjt ■^F^Tfqqr q>q "t q^ fqfq^ 37rq^ 'sifdci (Complicated) f I i^qr ^8in^ Iq^T amfl^
■q^ tl
• H'^Hli
■SRpqr 3TRTiw FTtq q^f T^tq (MS^y) uhRc % 'SRFqT qq ST^qpT t, ^Riy.■qpTqr W qn^ "% %q; qq^q "^PJ^ w^q (Pooled Variance) % 4’ fqrqr*^ ^T^qrti a^cT: ^ qarqrqf % 3FqT qfl ■qpTqr ^ ^ w qn ^rq^qr t-
r 1= . MSw IN, NJ37q: ^ q«FTHf % 3F?R ^ ^^fq^fT % f^ ^-sT^qm,
M,» M2t =• CTd •
q^ Wrqqr ^ %■ fqf*7Fr ^77^ ^ ^-37^qjq ?M q^r^ •d'i«h1qffeq -^nstta tci^: q^ ■qrqf^ ■gqqf^if {df) % iqqf qF% fqqff^ qft qq^T^tl
■f’tI qq aqqqr '^Fnq ^ m q«FTTqf ^ % f^ qrqqr ■jfe qn qiq wr ?W.5«HRrm. fwfq sFlf^qqr q^Fqq 3iqR (Critical Mean Difference) ^nq qq^ FT^iqi ^ 'FTslq) 3Ftr qr^ q^qqR "5^ q^ qFqrn qi ^TqjqT ■!■, qr % '^rq; '^jqqq 3qq?q^ TrwrqH3RR -JTsrfrf qqRqqr Ftqqrq 3Ftr 1^ ^ ^ ^'%qT qq 'Fq^n t-
“ ^.05 ^ ^D •
D 01 ~ ^ 01 ^
D ,05qaq
190 q7?^7qjtq /q/wqf
TTWTTTR 3FcR (Critical Mean Difference) ^ 3Tf%T^ ^ 3T^' jraw■^TTS^^K^ W ■'TT ■?TT^ ■^sn^ SnPti'h '*TWIITPT 3FrR (CMD)
, TET«Tt TTWmH 31^ ^ ■SIT^I iT%3TT7nf ^ WT. aiftq^ ^ WT ^ ^^ 3Tfti^ 3r^ fe: ^ ti
/r
■a^ig^ui—?5I^ % yRi<^^iT ^ ^TTF cToF '^rf^TR % s«i^ % •an^.i'fl-
rnwiPri 1^ "wi "qf!^ ■qr "3^%■3^
TiT^ fW^^TR’ ^1 ^ FT ^ yfcr^jviT % rwriptI ■4‘ srr
t?■^rflrMl«^<<^f<ich IqlyOMKSMI’l
74 15 176 10 12 12r
1215 8 1516• 8 13141811 17 10
14 17 12 19149 6 16
105 8 14
ewTPti yia<^;jiT pci'll
^ 'sraln #ni 31^ ut#^ w^. 5 ■>¥ ti■Rrnrft—5
TJRTJT f^T^isroT ^ 1^ TnrfwT^ ■r^ftt
^n?T
oqiom fqf^nfcraTf ^
■f^rfsT flrenT-'feiTvf Msr Trfirsr^f '^nrN.
X2 - X2 . X2 X2X X X X
16 74 49 15 225 17 2896 36 10 100 12 144 12 144
22515 225 12 144 8 64 158 64 16 256 14 196 13 16911 121 18 324 17 289 , 10 10014 196 17 289 12 144 19 . 361
819 14 196 6. '36 16 2565 25 8 64 10 100 14 196
IX = 72 1X2 = 764n = 8
IX = 102 1X2 = 1422 M= 12.75
IX =94 1X2= 1198 M= 11.75
IX = 116 1X2= 1740M = 9.00 n = 8 >1 = 8 n = 8 M = 14.50
N = 8 + 8 + 8 = 3215^ = 72+ 102 + 94+ 116 = 384 ■
I X2 = 764 + 1422 + 1198 + 1470 = 5124
f^fk^ 191
X
(0 TR (Correction Term) ^ WTT
(SX)2c =N
'Tt^T 384 X 38432
= 4608(li) ^ (Sums of Squares) ^
SS( = SX2-C = 5124-4608 = 516;0
=JWsrr^^^, ss -c"B rii
72x72 102x 102 94x94 116x 116-46088 8 8 8
= 648.0 + 1300.5 + 1104.5 + 1682.0-4608.0, = 4735.0-4608.0= 127.0 •
31RTf^ ^ SS^ = SSt - SSg= 516.0 -127.0 = 389.0
(lit) ♦jewiVil (Degrees of Freedom) ^ W'H
^ d/T=N-l = 32-1= 31
W ^ titiT % d/g = K - 1 = 4-1= 3
3n^R«h ■qfiT % ^chuVi, dfyf, = N - K
= 32-4= 28
(iv) ^TTSq 'aT’ff (Mean Sums of Squares) ^
SSt ■^ Tjm ^ MS^ =d/j
516.031
= 16.65SSg^ -mm ^ MSg =rf/B127.0
3= 42.33
192 3Wrf< W/fis!/*?«V
SSw3<i^Rvf) Tmxj ^ iitn, MS^ =dfyj
389.028
= 13.89(i;) (F-Ratio) ^ fjm\
F = MS B-MSw.42.33 13.89
= 3.05SPM^Tn Iq^C^^ui % mR«IIh1 ■«KKI f^F)^ ^
^TRtTf^ 6
H«^U| % MRuiililf ^ui?!
44Kui) 'SFT ■RH(Source) df SS MS F
Fo5 (3.28) = 2.95
F.oi (3.28) ' = 4.57 •
% TO 3 127.0 42.33 3.05.05"^ ^% ai^sTT 389.0 13.8928
16.6531 516.0
(vi) ot»isf4i (Interpretation) ebb'lli7T^' ^ wift 1df= 3 ^ 28 "9^ .05 ^ 13:9^ ^ irm 2.95 t
.01 ■?^'q^.4.57 tl -SrRTT^q? TTTT 3.05 t’sj) 2.95 ^ 3Tf^ ^ 4.57 ^ ^ t. W ..05 ■?<R tl 3m: .05 "9^ w ^t % ■4' armr ti(vii) tiHT'n' ^-■'T^V^rn’ (Post ANOVA t-Tests) i
■^F '*T^:2TOFT ■qTFP '^q ■Rim^-■qftrt' ^ 1^ "q^ tl ^PrKVi ■^' 8 t TO MS^ = 13.89, 3m:trwPTmf % 3mR ^ HI’I'b
1 1-‘^D =; MSw
N, NJ
(1 1\= 13.89 - + ~V As 8j ; .= V3.4725
= 1.86^ yPd^Tff % ■qwFfpff %.Wpq "g^* ^ feg qRqPqq ^-argqrdf ^ smrf^ f fro
w tl
"S^^f Tfff^T^^ fotfirm 193
oRTRT
^n)f?iM DWT W tn ^D
8 9.00wTHWin1 aT’BWfe2.021.86 3.7512.758
8 9.00oyiom»i2 3TOTSf5»1.86 2.75 1.488 11.75
oen^in ftrftr 8 9.00 .05 W3 1.86 5.50 2.968 14.50
8 12.754 3T^nsfe1.86 1.00 0.54U4VF4 77^ 8 11.75
12.7585 1.86 1.75 0.9414.508
8 • 11.756 3TOT?fe1.86 2.75 1.4814.508
df =14 m tog = 2.16 (WT toi = 3.01
7 ^ ■PTE2 i % TT^ ^ % Hwmnr .05 ^ ^ ^ ti im% ‘R^hI'iI "4' 3i«^ciliVd ^'d<. ■SIT '««hdi 't^l fVH ^
(M = 9.00) rT«?T (M = 14.50) ^ f I
2;7 5H Hi*<n«hl ^ *:*cich'(
(ANOVA by Subtracting a Constant from the Row Scores)MWJ|
TJ^ ^ WFf UIW ^ f I %ft iFlfiT ■^f ^ mm^ ^f5 (Calculator) 151 ^
eldl "ti dUMi "^rnsf «MUi«bT ngWcti % f^iT et)<-ii ^IcTi "t ^ ipni ^ ^5*^^ ^ ^ ^«?T 9R ^ im 37f?^^ ^FRn ti ^ yikii+T ^ (Constant)
fq-cjeid TW "^iRf ^iTlT l?lRf afl^ 't’t 3TNR tR '^nyikllVl "A" ^ 1T^ ^ -511^ cR .f^*FT M RIr)’ (SS’s) % 111111 ^ HR 1^
■§■1 yi'flf'til % li: IT^ yiVii=t)T ^ TR) fl5Rf^ RHRil HIT fq-qfVici Hi‘<ii«t)T ^mm HI ^ f r5 it^ kjh wn ^ in iT«ft mirth)' srqfti^ iT^ft Mfd<^vif ^
HRI^' ^ ^^H ^ fiaRfH RSTH ^1 HRTH RST^ WT teft TJTRTH H Ih^' MIR^T % fH^%cT IR 5}?nTRTH 37TH cR" 3H% ^ UTR ■^* 1?3H1 HHTT HT^ IhH ITRTT f i HRT^
ft«TlIH RHIHI HRT f^cr)q»i hI mOuhi 3TPff^ ^H51R IW
1TH7TT i fn itl IT^
\
194 cfcWcR
yPa'^JjiT "qr ^IknW fHHc^d '^1 ^ cfNf y|^?f
^ ^ f^ t?- WTrlcf^fl '3T'
¥fd^ 'IFT’
3?/»7^?W Z?? JWM¥CFu
82. 85, 84, 90, 88,
89, 92, 87, 83, 94,
80, 78, 81, 79,. 83,4^11^ yi'^i'tj'l y«<ui
yP^uiiHT TR ^ t W -qrftsTZjqr C¥^ -CR 80) ¥^I^ WTT ^ ^?T7^ #711 3TH: 'R# yikfl=hT (X) ■^’ 80
4tii4i< ■yiRT f^qfcict ■yiRn# (X' = X-80) % TRTZqi ^ ir^zHT ¥># 'TT—
91, 86
90, 88
85, 84
#7
'm ^ WTT qi# STr?r'yi’^iiqi 4)IM’1 f ■qrofcRf
wqft-8
'H
X'^ .X'2 X'^X XX' X' X X'
982 2 4 89 81 80 0 085 25 92 144 785 12 -2 484 16 87 .4 7 49 81 1 1
79 ’90 100 8310 3 9 -1 188 8 64 94 14 196 83 3 . 991 121 90 10011 10 85 25586 6 36 88 64 848 164
606 46 366 623 64363 570 •5610
M = 86.57 M' = 6.57 n. = 7 M = 89.0 M' = 9.00 M = 81.43 M' = 1.43n = 7 n = 7
N =7 + 7 + 7 = 21 XX' = 46 + 63+ 10= 119 .
S X'2 = 366 + 643 + 56 = 1065
(i) (Correction Term) Mum
,(IX')2' q^, c = N.
11921
= 674.33(ii) (Sums of Squares)
¥Tf #IT, SSt. = XX2-C
' = 1065 - 674.33
. = 390.67
(SXi)w SSg = I
46x46 63 x63 10 x 10+ -674.33+7 7 7
= 302.29 + 567.0 + 14 = 883.58-674.33 = 209,25
■aiRTf^ ^ SS^. = SS.J. - SSjj
= 390.64 - 209.25 = 181.42
674.33
(Hi) ^RTRfff (dfs) ^^ , dfrj. = N - 1
= 21-1 = 20
, rf/p = K - 1 ,= 3-1. ■= 2
SlV'flReb 'spf ■% "^RnTT, = N - K ■
= 21-3' ' ■ =18
(iu) TTnzT ^ (Mean Sums of Squares) ^ ttdht
QC^ T^TSq ^ ■qtTT. MS^, = ’’
dfj
390.6720
= 19.53SSbcn^ MSg =
. ^/b
209.252
= 104.63qq
3THTf^ qrszi ^ MSyv =d/w
181.4218
= 10.08(i;) TJTK-aT^ciRf (F-Ratio) ^ WTT
MSBF =MSw
104.63= 10.38
10.08
196 3^W?K
/
'3?f»7^?W W 5?r{t¥
(Source) flKuD TTFTdf ss MS F
% •IT2ZJ 2 209.25 104.63 10.38 F 05 (2, 18) = 3.55F-oj (2, 18) = 6.013T=arr ■10.0818 181.42 .01^
■qr20 390.67 19.53
. (i;/) (Interpretation) «KTHT.01 t. m: .01 m ^ t
TT^qqH WR tl -q? ^ ^ iTttRTRT ^PTf ^ f, ^ehVii ■?)qTi
{vii) WTTJT 1cr?^qui^TITRT=PTff^ yp^sh Vifaq?! 7-7 ■qiqrrq' "t, rf^ MSy, = 10.08 ^ R^4h1 % 31RR ^
Hl'1<* . • .
1 l^MS : +Ns1
II lA= 10.08 - + - Tl l7 l)= Jz^= 1.697
3T^: '31' cr«TT ‘^’ % -qwmHf ^ ^-3T^,
86.57*89.00t = : 1.697= 1.43
*3?' ■a«n ‘r’ % ■q^zRTpfr ^
86.57*81.43t = ■ 1.697= 3.03
3lf^ cI8TT ‘r' % RKTRHf ^ ^ ^-3T^,
89.00*81.43t =' 1.697= 4.46
(M ^ ^'31^ (d/5 12 tl %^ t % f|E-'3^ -qft^ df= 12 % i 05 = 2.18 Hair = 3.06 tl 3m:
0 “ 3.03) ^ RH .05 'RR "^R (t = 4.46) 414 .01 "RR "'Rf -51^ (t - 1.43) Rt RR -q^ RT«mr q^' ti am: -qf^ '3i' w 'r'-%
4/fe/*?«9 197J^a.i
■# .05'm m 3RR t ■5lf^ ‘^' % TT^^f ■^‘ .01 TRR 3RRt ■3^% -'3?' cf81T ^ '*T«?^Tnf ^ SRR ^ ^ '^TT«f^.■^' tl
2.8 ^ 'irirR^f ^ ^w-ii ^
(ANOVA for Comparison of Two Sample Means)TOTtrr t^^<rl^ui%■
■^tipH iraroT yfciHi';.'! 3ifir^ TTizrtn^' % ^ m^fepar^ w «n, ^ ^ ■jpftn ^ ’it '^n
^ «TTat ^ TTT^foRcn ^ "SIRT■f I ^ yfas^iT ■% TjUjTfRf^ ^ ^ t '^3Tt% ^-mt^ % ¥#1 PM ^ tl •^1^: ^ ^rwriiHf cfft.l^siftT
1zt-Pt^ ^ t atf ^Ti-ii (Additional Information) P^tl ■qv^Hi'iT "aft PftPf^ 'an ^irT % sRRT ttar tl 3TcT: ^yPd^viT % MPoiPuid ■^-ai^qFT ^*!n ^-31^ % ^ F = (2 aqgfgpf ^ ^ ^
PTT '««t»ai tl ^ '»lPiC1 tt^ % chK*J|, TM 3<^I6<^I % ^tl , .• ■^^T^TPT—'asiT ^ Pt%^^ %p; thrift Pft^iroT PT PMM f^TRT^^llK «tl
^?t-pt^ ^mr TTp^-pt^ pwmpff pt ^-ai^pra ^81T ■^pt ai^qm %PTT P'KF'H'Jl ^tf^l
2, 5,10, 8,. 11
tNf ■^rijtf ^ p^ppHf Pt ■g^ p^R^ rpv6P«i % imi prt pntpti stet; PRRn ip?^p^ tl PRfippr PPRT pn4 1^ TTROTt part PT
'HKwfl —10 -wratf % % giviiPi^' ^ ppht
6, . 4. • 9W
7. 12
^rar fnj^
X2 X2X X
2 4 10 1005 25 8 646 36 11 1694 9 7 499 81 12 144
SX2 = 162 1X2=478IX = 26 IX = 48
n = 5 M = 5.2 s= 2.315 n = 5 M = 9.6 s= 1.855
(/) ^ ^ TCTCiWRlf Pt '^Frr
* dHildd ■PTipft t TW t ^
N = 5 + 5 = 10 IX = 26 + 48 = 74
198
«’
i
SX2 = 162 + 478 = 640
(SX)^
■3?firqgF?-=>> imw
SS^, = SX2 ^ N
ai^:
74x74= 640 -10
= 640-547.6 = 92.4
(SX)^SS3 = 2NIII
26 X 26 48 X 48 74 x 74+5 5 10
= 135.2 + 460.8 - 547.6 = 48:4 SS^ = SS.p - SSjj
= 92.4-48.4 = 44.0^ ^ ^ rHHlPKfT ^1
■?7Rtift-ll
' wniT ^TJT
^?frT (Source) df ss MS F
% Tm F 05 (1, 8) = 3.55 f'oj (1,9) = 6.01
. 8.801 48.40 48.40
% 31^ 5<508 44.00
9 92.40 10.27
31^ % RtAiHlii jet’ll % gRT ^ 3RT: ^Nt % 3IRR
4 4CTd= —V -1 n2~l
2.315x 2.315 1.855x1.855
V 5-1 5-1
= V1-33981+0.86026
= V2.2OOO7
= 1.48326
^ iT«rTRf ^ 3RR ^ ^
Ml w M2t =
5.2«9.61.48326
dWdI 199
33WW . 4.41.48326
= 2.96644(2 = 2.96644 X 2.96644 = 8.79977 F ^ (2 = p ^ ^ ^ tl
% 6i<.iq< f. 3TcT: '*T«jT7PTf*TPT
2.9 TOTO TTRTfin^ (Assumptions of ANOVA)
fq^c^qwi 3T8T^ i^«F tiiRs^qilq 1^ffv (Parametric StatisticalMethod) 't'l 31^ fqf^ql % ^hii W 'MV Vt Hi'^diSTl’ '’TT 37tM^ 'll 3lcT; y<H<.'j| ,
fVfV 3^5^^ ■% "MW ■'jV ^<Hi 3TI^?^T^ ■?f?n "I MiTRRns^' ^ ^ Tt tl ^ -qi^dlK ^ ■^' f. ^ ^ f w TTSfT3IMt«R1 Hi-qai ^1 MicsqTi ■# ■gfVm ■% ■RP^3Tf ««<.«! 3TT^ '5^=5^■Mqr TfT ■f- •
(i) MV^ 3T8?^ yPd^i^iT "V' 6Hir^^ ^ qqpy 3^8t^ M^itt ^qqMq)’MV V> ^ "fq^n ipTT t'l
(ii) 't^ wiM. ■qqMVf) "t 'sifK^ M t, 'tit ■snMi> ^ini-q 'snfViKn Mtt^ (N.P.C.) % 3i;j<^h fqciRd 13T?t?i^ MV'-i yld'^j'iT t ^rmVt n 1sr^effMi ft<i<«i <^qn 'qVtr^
(Hi) 'MV^ ■?TqMVf 'Mrt ^ft M t, Vr 'srspi <hhm 1378t^i'MV^ yRi's>iT 'V> 'srar^■qqH t y^<.wi <^qci Wql'iqjfi "tl h«tH<. fVl tl
WlPqetxll fVcK^I 't 'fVl t'l
(iu) itlfM twi f ^ wm it qM t Vfei ^rmr t,. MVi trtfh
FlcFl (Independent) 'cM VNfti> yjiPd (Additive Nature) % tl
■^irt ■qpRJTst it't sTiH^: t^'H iroi (Random Selection), ^THTP^I Mirw(Normal Distribution), WH'Jiirtlq (Variance Homogeneity) VrUrHlTcn (Additivity) li^ ifTlT tl "^-SH in IFFTT it qPiR-ddi SFjqiTH STfV'+xrH (Research Design) 't. nTFl Mm 1 TFm Rq jtidlqfli it it iFiiT3t it '^tTMRn fiMt 'Fifenti ■qt^ntt Vi ■sttr'tIIT qViirH^hai it IFHT it ^fwRqadl '5Rm 'M^H stVlMl (Design of ANOVA) 't triffti it. IT 'fiM tl "Mm it ttfut 'Vt 1^ 'MVh ■srfii?f ■ynnit % Mmt it ITT^-lt itm (Chi-Square Test) 3R11T ■%. (K.S Test) % -gro yiPlct^dl nr% 3HR ■'R 'tlR Mt it t«&iRqqi 'Mm't liM t^lT IT ^«t>ai t 'M ITU "Mm TTriFl yiPqobdilaF % 3TT«^R It tiTt 'Mt it "Mm ^ hM IT ‘RiTiT t M w<T "Mm Rrm Mm"t RtVij ^ ^ ■fVl t 3T8T^ itll 3Rm ^mniMlT it qr^m't^ itm (Bartlett's Test)
3T8nT itajl (Hartley's Test) n ■?ntl irM ijfl "iRmriT 'M Hq : ~ “^2n ■'TTfm fq>qi IT ■RIkTT tl
it iMlff it f^fd t’ ■RTIT^ 'Mm 3THT TJRTI eiH'flicilqa! it HV-qaist in '3evMT TRm 'Mim '% hRuiihT it 'tm it 3ifVir ■st^tMt itt mr t, 'ett yfd<?i1 % Mq fi 'ilif Hi'qd[3t' it ■'3^ 1 'ttt ■qr 3Rm M^m't iMm qMi (Doubtful) irt ti n: ityld'^^iT'% '5Rm Mtd'61 n ttift fi iRnrst it ■'3^ 1 "tit it it 'Mtt MrI irfeu^ ■% "Mir
200 3mT Wf&ftfpj MVif
TTTT ^Tf^l3ir*i4>r^HI VWm
W^ f^^fi^cim (Student Activity)
1. T^-I^RTT^ fsrfv fq;f^qw| etOPaii^l
2. (F-Ratio) '^t^^mki W^ ■^fRTT "t?
3. ^i«Kii!fi (Degree of Freedom) 3M ^ f ?
2.10 wm (Summary)
• "Srayfr fq^c^^wi ^ yRinKi fiffe?! TR 'OHI<r^ f'RR (R.A. Fisher) ^^pilT «TTI 13;^ TTI^? 3T^ % iTWIHrilf ^ Weft t fW wft TT’ft
^ -RRfe (3T«T^ -qiSF ^mfeft) TT^ I 3Ii!I^T ■^'i• f^RR ^ "^-ai^qTeT, R^-1WTn fT*!?I RKRTH ^ RTW ‘gfe % RRRllf ^ aift^
■SrftTRf RWRnFft ^ -q^r -RT*? ^ WOT ^ IwftTeT 1^1 "^Tf^feft ^ ^ 3TTW % 3Tft^ Rlrof ^ ^ RWTW #TftTRT ^RftlWT % WOT^ ST^W (Sources) ^ WTPTl
V^-'^ ^ tl ■^' ’HRfe % WOT (ct^)■3TT RWT f 1
wT<3^<*l'<i 201
JR77W 1^ffv 31^ yfa'^^fiT 'SrRT % 37^ ■^TT^J^Tfn qaial "tl ''rft’TpTO
T^-375'TRT 3TOT?j^ ■^tcTT "t ^ 3iqciliqia 37^ cf^ ^THT
■^7^ tl T^^-aTjqr^ ^ iTH ■?^TS^^ t ^ .i7«Jri7Rf % ST^efto
37^ ^ WqViq^i ITPTHT ^if3^
y^<«i lqi?c^q«i ^ ^rpf 37«7^ TT^FfT^ (Calculator) "JT^ .'^ T?T"f, eft «bfd'll^ ^teft "t^l "^TT^ ^i^mI '^TFf W<|wi«tiT «i6iqdl % q^u ^ ci^ Tjtrn ^ ^ ^ ^ -^neft t W 5R ^ wr 37f<7^ tl
Vl(d<riflT % ■^T^hI'iI ^ yy<«i fq^jc^qui '^. 9Ft ‘^TTeft !■ el^ '77T«^^i^ %■^' w^T mRuiih ^ ^ ^ t % Tratn ^ iif^ ^ ti
^ '^' W7FT ftaiui v37«M M«<y| ^H'Jiialqeii ^i'^di37t’ 3ed'MH 'JRTT’q
% mRuihI ■^tTelT ^ 3Tf^q> ■q’TTf^ ■^TCefT yid<^^lT % 1^
^ "^Ht *ii'qdi3^f epft ■’jf^ "q qr yti<.wi ■qfTwq (Doubtful) "st
tl
3T«TRT-W^ (Exercise Questions)
qqt'q fqs^e^q'Ji % ^tef=qqr q^ qq f^vo^q'^i qftf^t
qqr^. fon?^qTq TifqiqT qq opjfq cptf^i
qqt’q 1q?^q'q|q77q ■qr^fqqn q^^ qq fq^qq qftfq^i ■
qq^qr fq^^q^q qft qFqqrsTt qq q^^Ti qftf^i
1. j
2.
3.
4.
TFR^ 13^ (Reference Books)
■qffefqftq M^i slf^ q^-«?7w ^c? IPS’ q?^, 77fe q^?^7/q/ ■^TfRgTqftq t^rrJi st? ^yq/
37f^ "^iq^q-^^Tqq ti^MH ^ qpyq/
qrfqqqn rqd<.«i-*y//?t /7r#f, fwnr qf^T^T/q/?77Jqq, ^Rd4>^Hi
1.2.
■ 3.
4.
5.
• f
202, dWrt< W^^Rfhf /q/vqf
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