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Instructor's Solutions manual
Probability and Statistical Inference
7e
Hogg. Tams
Instructor's SolutIOns manual
Probability and Statistical Inference
7e
Robert u. Hogg Unillersity of Iowa
• Elliot n. Tams
Hope Col/ege
PEARSON
Prentice Hall
Editur-in-( 'hicf: Sally Yagilll EXCl.:utive Editor: George Lohell Editorial Assistant: Jennife r Urhan Excl.:utive Managing Edit or: Kathleen Schiaparclli Assistant Managing Editor: Bc(,:ca Richter Production Editor: Donna Crilly Supplemcllt Cover Manilgcr: Paul (joUl'han Supplcmcl\l Cover Designer: Joanne Alcxandris Manufar.;turing Buyer: Ilene Kahn
PEARSON -Pn'lllll'P Hall
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ISBN 0-13-146412-4
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Contents
Prerace
Proh'lbi li t.y 1. 1 B<L'iic Conccpts 1.2 I'n)pcrtk-:; of ProllUlJilily . 1.:5 t· .. lethods of Enlllllc r flt in u .
I. I Couditiolltll P rohHbilit,y 1.5 ludcpcndeut EVt'lIi"'i 1.6 Baycs's ThrorclII ..
2 Discrete Distl'ibulj rms 2. 1 RalldoJll Variablc:) o f tlw Di.stTNt' Typt· 2.2 Mathematica l Expt'('Wtioli .. 2.3 The r.. ICI) II , Variall ("(I, nnd Stljl ldard Deviat iOIl 2.4 Bernoulli 1'rillls 311d the BiIlOlniai Dist ri hillinil 2.5 TI l{' !\ l ofJ1cllt -Ceucwli llg FUlwt iull 2,(; The Poisson ])is lri bul iO Il
;j CO lltinuous Oist.dhuliolls 3.1 ConlimLolIs-TYl>c Dutil 3.2 HaDdam VariablCl> of lilt' ColltiuliOlIS Typ(' . 3:1 TIl(' Uniform IIlId Expoll(,lltial Distrilnllious :1.4 the- Gnlllllla aliI"! C lu·:Sqllarc DisHibutious . :v, DislrihUl ion~ of F'unctioLL!, of a Ral/dotll VMinlM :t6 Addil iolJ:d ~ 1 ()(1('1'\ ...
' I !\'lult.i v~l .. i8I,e Dis l.l'ibul.iollS 4. 1 DistrihluiollS of Tw() J11\IHlo lli V{\t iablf"S 1.2 The Correlat.ion ('O('ftidcnl, .. 4.:1 COlldi tioll ll1 Distrihlltiolls .. I. I l'rall.sformatiOIlS of HhlldullI Variables 'I.!) ~wr.ll l llde l)f'U(knt Hll.udoUl Variables I.G Distributiolls ()f SUIl~ uf Ind('j)Clldcut 1l1lUcJOUI \ariahll,os .1. 7 ChebYI>I1t'v'~ 11lt..'<l llIIii t.y (llId Cou\'ergPllct' ill rrohnhj l il~
5 The Normal Dis l.l'ihul,ioll 5.1 A Brief History of Probnbility . 5.2 The NOfl lln l Distribu tiOIl .. !i.3 Hruulolll [.\II ICtiOIiS A:;"'I •.. wiated with NOl'lliu l j)islrilJll tiOlls !) .4 1.he C.·lltl'll i I'.illiit The<}fl'1Il 5.5 Approx illlatiOllS for Dil!<'retf' Distrihuliolls rdl The OiW.lfiau' NOflmd DistribllUOll ..
iii
v
1
1
3 'I 5 7
o 9
12 1 :~
'" 19 21
25 25 20 37 39 H
'12
45 45 ·17
19 51 [,!i 57 59
61 61 6 1 6. 67 69 71
r).7 Lim iting ~lonl('IIIrCclleratilig Ftult; Lioll~
6 Estimation 6.1 Samplc C haracteristics . (j.2 Point. Estimatioll . 0.3 S ufficient. Statist.i(::; . . . fi,4 Coufi deuce Intervals fur Mealls G.5 Confide nce Intervals For Differellce of Two Means 6.6 Coufidc llce lutcrwlis For Variances . 6.7 Confide nce lutcrvals For Proportio lll'l G.8 Sample Size 6.9 Order Stntistics 6. 10 Distributioll~Frf.,'e COllfillcm:e lulcrvnl:! for PeTl·enl.i lt.~ . 6. 11 A Si mple Regression Problem 6. 12 More Regression . 0. 13 R.C&unpliug Methods 6.14 ASYUlptotic Distributions of J\'lnxiulUlll
Likelihood Estimators
7 Bayes ian Methods 7.1 Subjoclive P robability 7.2 Day(."Si rul EatilllatiOIl . 7.3 More Duycsiau Conccpts
8 Tests o f Sta tis tical Hypotheses 8.1 T(,,; I~~ »bout Propo rtious . . . .... . ... . 8.2 T(.'S ts <lbout Qne ~ Ieall and 0 ue Variuuce .... . 8.3 'l'cshl of tLc Equulity of Two Norma l Dist.ributions ' .4 T he WilcQXolJ Tests 8.5 Chi.Square Goodness of Fit Tests . 8.6 Contingency Tables . 8.7 Om ..... Faetor Alllllysh; of Variancc . 8.8 Two-Fnctor Aua lysis of Vll.rill.lU:e 1'3.9 Teats Concerning Regression <Uld Correlation 8. 10 Kohuogoru\'~SUlil'llov Goodm:.'SS of Fit 11..'l:It . 8. J 1 HUll TI!/'It Hud Tt'St (or Rn udOLlIut."SS .
9 Theory of Statistical Tests D. l Power of /I !:itlltist.iclIl Tt.'8 t 9.2 lk'St Cri tictd Rl·giolls . 9.:1 Likelihood Ratio Tt.'SI,s .
10 Quality Improve m e nt Through Statistical M e thods 10. 1 Timc St.'<IIICIIC(''S . . .. 1U.2 StlttiHtical Qual ity Control . 10.3 General Factoriru ruul 2k Factorial Dt,osignl> . 10.'1 Mo re 0 11 J:>t."Sigll o r Expcrimcutl'l . .
72
73 73 77 79 80 82 ~J
85 86 87 88 90 95
101
IO:J
105 105 lOG 1U7
109 IOU 11 1 11 ,1 117 120 123 124 126 127 129 1 ~1l
133 1 3:~
137 139
141 141 1~~
147 147
Preface
This solut iLlu :, Ill/Unlal I'1'1,vieit's UlIlm'I'I"S f ... ,r the· l' \ 'cll-IHlllllwl"ed ('xl'rdl><'l'> i ll Pn,/H,bll" U lIm{ !;jrull-l>tif'llf
J,,!cn;/It'I', 7th l'(li l ion. 1,J' Il tlbc.-L V. lI ogs {I ud Elliot. A. TU llis. COII IJl ll'll' :o>oi uLi(lJls un' gh 'PII fflr rJ\o.~ ~ of Ih(~' (·XNl'iS<.'S. YO Il , tlu> instnL<'wr, lIlay d('ddl' bllW ,uany of tht$(' IU IJiWNS you \Wl.LlL to lIlakt· u",ilahh,' 10 ,,'(, lI r :-luill' utS. Note that the UIISWl'rs for tlir ()(ld-lIIlJ ulk'n..'C 1 C'.xC'r<'iSt!S arc g iven ill tht· tt'xll)l.)(lk .
All of t ht, tigH r<.~ iLl this lIlallllld were g'- Ul'rn ll 't lll:;i ll~ M al'/C. 1\ ('OUlpltlt' " nl!;,'hm syst('lII . ~ I ost of tht' iiA"I, rl'S \\'1'1'(' g C'll('raICi I (. , 11 1 111tI1'Y of Il l<.' SOII I' iOI IS. L'S jJI '(' ially , [,nS(' iIIW)]\'i l lg dlll..lI. Wt'l'f' sniV('(i
IIsi ll~ pr<Wt'(luI"I 'S \ llIIl Wf" '(' wriltt'll II,\' Za\', 'U h:llrillll frOIll D('uisoll Uui\"('rsiLY, \\'1.' lImuk him for pro\'idiug tht.'lit" Thl.'St' proct.'(lur('S an' (\\'o illtbJ(' frC'C of charge for your liSt' , TIlC'Y lin' Il\'ni lahl(' Ull 111(' C'J)- n ()~ [ i ll Ilu' ll'XI bonk, ~llort , ll'SI.'ri pt ions of lllt~ prOl'I'd llre;,; a rc prl!virlL"(l 011 lll(' -~ l apll ' CUl"(r ' 011 lbt' CD- n ()~ 1. COIUp[t'tt:' dc:-:cr iptiolls of clit.'SC p l'OtNlurl"l:l ar .. A"iwlI ill PrvlxJiJlllty (wd Slfltl.~t/('.~: 1:;'F/J/QIlII.IOIIS with Mit P/~E, St..'Colid ('( Ii tioll, I !J4J9 , writ I t'll hy ZII\"I.' II i<lIriall Ilncl Elliut Tallis. jJuhlisl ltx l
hr Pn'lIliet' lIall (ISDN O-I: ' -02I f,:Uj-$). O ll r Ilupl.' is til;\! th is SOllll iulis 1II111111U1 will 1)(' Itd l>flll \(I t'1\(:11 o f YOIl ill your telll"hiug. I f you filld IlIll'J"ror III' w ish to HUlkt., /1 SUAAI'slio ll , S(.'l1d Iltl'Sf' 1.0 E lliuL Tall is at t an isillhope.edu
!Iud Ill' will pust (-om,,'1 iOlls UII his w~b llS\gl', IItt 1':1 / www.lIU1th.hopt" t'tlu / 1.i\uisj.
n.v,lI. E.A.1'.
Chapter 1
Probability
1.1 Basic Concepts
1. 1._2 (<I) S = ( tJbb, ghb, bgb. IJbg , I,);,g, gug, ggh, ggg);
(b) S = {fcwale', male}:
(c) S ~ {OUO.O()l . OU2.OU:l .. . ,999).
1. 1-4 (u) Cllllt'h size: ·1 5 6 7 l> 9 10 " 12 I :i 1'1 Fh'fllIC' I I(,Y: " 5 7 27 2G :J7 S 2 0 I I
(b) h(~)
0.30
0.25
0.20
0. 15
:::L_J:LC 2 4 6 B 10 12 14 "
Fip;un' 1.1 " Clllu'h s iZ('S fOI'l he COllllllo n gAlIi uuic
(c) 9.
02005 Paa.SOfl EoucaliOO. Inc .. Upper Saddle RMir. N.J. All righI' res61VOd, 1 his malonalos pror9Cled UI'ldfIf 1M copyright IIIws a& IhllY ClJI(enlty e~lst. No OOflion nllhIR ".a to.lill mIIY bft reoroduOIKl in any 101m 1'1' bv aow I1'\MNI. willlOU' oennl5Sinn in Wfllina lrom tile 1'I11I)U,I'IIiI.
2
1.1-6 (a) No. Boxt.':i: 4 5 (j 7 8 9 Fr(."Clncncy: 10 19 1:1 8 1:1 7
(b) h(x)
0.20
0.18
0.16
0.14
0.12
0. 10
0.08
0.05
10 II 9 5
12 l:i 14 2 ,I 4
0.04
0.02
, 2' , .. c4"~ 7~' , :illIl-,D". ,0- x . 6 8 10 12 14 16 18 20 22 24
F'igll rc I . 1- 6: NUlnbcl' of hoxes of (.:crcni
2 :3 a 2 1.1-8 (a) [(L ) ~ 10 , [ (2) ~ Til' [ (:I) ~ Til,/(4 ) ~ 10 '
1.1-10 Tilis is lUI cx perilllCIlL.
1. 1-12 (a) 50/ 204 = 0.245; 93/ 329 = 0.283;
(b) 124/ 355 = 0,349; 21 / 58 = O.:Ui2;
(e) 174/ 559 = 0.3 11 ; 114/ :187 = 0.295;
15 16 2 2
C/III/l' ('" J
19 24 I I
(d) Alt.hough Jalll(~ ' battiug avcrage is higher !.hnt Hrbek 's 0 11 ooth gruss ami artificial turf, Hrbck 's is hiJ(hcr over nil. N Ol.e the diffcrcut numbers of at. hhL..; 011 grit .... " HUt!
art.i ficial turf llild how tbis nffl.'cts the bLLHing Ilvcragcs.
1.2 Properties of Probability
1.2-2 (a) S ~ (HHHH, 'IlUiT, HHTH, HTHH , T HH H, '1l1IT, H'M 'H, TIIIlI , HT HT , T HTH , T H"'T , HTTT, T HTI, TIHT, TTTH, T"'" '"r) ;
(b) P) 5/ 16, Pi) 0, (W) 'I / IG, (;v) ·' / 16, (v) '1/ ' 6, (v ;) 9/ [(;, (v;; ) 4/ 16.
1.2-4 (a) ' I "~ ;
(b) 1'(8 ) ~ , - 1'(8 ') ~ , - p eA) ~ V I;
(c) Pt A U 11) ~ p (S) ~ I.
1.2-6 (a) PtA U 8 ) ~ 0.4 + 0.5 - 0.:1 ~ 0.6;
(b) A = (A n l1' )U(An l1 ) PtA) PtA n 11') + PtA n 11) 0.4 PtA n 8') + 0.3
p (A n l1 ) 0.' ;
(c) PtA' U 11') = PItA n 11)'[ ~ , - Pt A n 11) = , - 0.;1 = 0.7.
1.2,8 (a) PtA U 11) PtA) + 1'(11) - PtA n 11) 0.7 0.'1 + 0.5 - PtA n 8 )
p (A n 11) 0.2;
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P W/;II /;i/il J'
(b) peA' U B' ) ~ p l(A n B ),I
1.2·1{) A u l1 uC A U( B U G)
1 - P(A n 8 ) [ - 0.2 0.8.
p (A U B UG) p eA) + PCB U G) - P IA n (B U C)I p eA) + PCB) + P(G) - PCB n G) - PI(A n B) U ( II n G)l peA ) + PCB) + p (G) - P(B n G) - p (A nB) - p (A n C) + P(A n B nG).
L 2-1 2 (a) 1/3; (b) 2/ "; (e) 0; (d ) 1/ 2.
L 2- 14 (a) S ~ {(I , 2), (t , a), (I, .1). (I. ,.), (2, 3), (2, 'I) , (2, 5), (3, 4), Ct, 5), (4, 5J);
(b) 0) 1/ 10; (; ;) 5/ 10,
1.2-16 p eA) ~ 21' - ,.( /'l/2)1 ~ I _ vf:j, 21' 2
1.3 M ethods o f E numeration
L :~2 (,1)(:. )( 2) ~ 24.
1.3-4 (a) (4)(5)(2) ~ 4(J; (b) (2)(2)(2) ~ 8.
t. 3-G (a ) 4 G) ~ ' 0;
(b) 4(2") ~ 256,
() (4 - 1 + :J)' ', . c (4 1)!3! - _0.
l. :j.1O S ={ FFF', FFRF. F' RFF', HFF'~"', FFRR.F, FHF'RF, BF'F'RF. rIlRF'F', I1F'BPF, RRF'FF, RRJl , RJ1F'R , IlFRR , FHRR. BBFFR, HrHFR. FRRFR, rWFRR, FRFRR, FFRRR } so tller{' 1"11'(, 20 possibiHtics.
1.3. 12 3·:1·212 = 3(;,8u4.
1. ;'- 14 + (" - I) (" - I) I' ,. - 1
(n - I )! ( II _ I )! r! (11 1 r)l + '(,:C. -"'1")1"(."'--:,")'
(1/ - 1')(11 - 1)1 + "(11 - 1)1 r!(11 r) !
1.:1- 16 0
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'. 1.3- l8
=
=
(b)
'Ill (II - ,,')! Il j !(n - l • .) l · ''' :.I !(1l - 11, 11:!)!
(-/I - II I - 112 ) 1
II! ,r " /I L .112· , •• " ,.
Chl'/Jr.er I
(1/ - 1. , -'11 2 - ... - "8- d! ,, ~ !O !
7,6!J5 = O.()(}(j22. 1, 236, GG4
1.4 Conditiona l Probability
1.4-2 (n) 10'11 1456;
(b) :192 6:I:t ;
(e) ij49
823'
(d ) The: proportiou of wOlilen who favor a gnu haw is grcllLCr Limn the propo rl.iOIl of mell who fnvor a g Ull Iliw.
13 12 I 1.4-4 (a) P(HH) = 52 . 51 = 17;
(b) I'(He) = 1:.' , .'.'! = ..'.'!. ; 52 51 204
(e) PeNon-Ace Ht!urt , Ace) + P (Ace of Hcartll, NOll- Hea rt. Ace)
12 4 1 :.1 5 1 I =-'-+- '-=--=-.
52 5 1 52 51 52·5 1 52
l .4-6 Let A = {:.l or 'I kiug.'l }, 0 = {2, 3, or 4 kiugs }.
I'( AIB)
=
~I'"'i( ACi:n",B:,-,) = N (A) I' (B ) N(B )
1:1 7 5G 1.4-8 (a) 1,1 ' i3 = 182 ;
= 0.170.
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Probability 5
6 5 30 (b) 14 13 182'
(8 6) 96
(c) 2 14' 13 = 182 [ 56 30 ] 96
or 1 - 182 + 182 = 182'
1.4-10 (a) Let A = {2 WIN and 4 LOSE in first 6 selections}, B = {WIN on 7th selection} . peA n B) = P(A)· P(BIA)
(b)
1.4-12
(DC47) 1 1
e60) . 14 = 76 = 0.01316 ;
1 35 . -- = - = 0.4605.
20 - 2k 76
1
5
1
5
1.4- 14 (a) peA) = 52 . 51 . 50 . 49 . 48 . 47 = 8, 808,975 = 0.74141' 52 52 52 52 52 52 l1 , 881, 376 '
(b) peAl ) = 1 - peA) = 0.25859 .
111 1.4-16 (a) It doesn 't matter because P(Bd = 18 ' P(B5) = - , P(BlS) = - ;
18 18
2 1 (b) PCB) = - = - on each draw.
18 9
3 5 2 4 23 1.4-18 + "5 ' "8 "5 ' 8 40
1.4-20 (a) P(Ad = 30/ 100;
(b) P(A3 n B2) = 9/ 100;
(c) P(A2 U B 3 ) = 41/ 100 + 28/100 - 9/ 100 = 60/ 100;
(d) P(A l l B 2 ) = l1 / 41;
(e) P(Bl I A3 ) = 13/ 29 .
1.5 Independent Events
1.5-2 (a) peA n B) peA U B)
P(A)P(B) = (0.3)(0.6) = 0.18 ; peA) + PCB) - peA n B) 0.3 + 0.6 - 0.18 0.72.
(b) P(AI B) = peA n B) = ~ = O. PCB) 0.6
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(j G/wpl cr J
1.5·4 [' ",of of (b) , I'(A' n B) I'(lJ)I'(A'I /J) P(Bli l - P(AIB)I P(B)I I - P(A)I P(B )P (;I' ).
P",of of (c), /,(A' n IJ') = I' [( A u 1i),1 = 1 - P(A u B) = I - P(A)- I'(B)+ p ( ;l n B)
I - P(A) - P(B) + P(A)/'(8 ) fJ - 1'(;I)l fJ - P(8 )1 P(A') 1'(8 ').
1.5· " I'IA n(8 nc)1 = PIA n 8 nC[ P(A)I'(£1)/,(C) P(A)/'(8 nG).
PIA n ( Ii u G)J = PI(A n B) u (A n e)J = P(A n Ii ) + P(A n C) I'(A n B nG)
P(A)P(B) • P(A)I'(C) P(A)I'( Ii)P(G) = I'(A)II'(Ii) + P(G) - l'(li nG)1
P(A)/'(B u C).
PIA' n (/J n G')I p(AlnC'n D ) = P(lJ)I I'(.4' n C') I BI = /,(Ii)[ I - I'(A uGI Ii)1 = P(Ii)l l - I'(A uG)1
/'(Ii)/'I(A u C)'I 1'(1i)/'(A' n G') I'( IJ )P(A')P (G') P( A') 1'( IJ)/'( G') P(A')/'( IJ n G')
PIA' n 8 ' n Gil I'I(A u IJ u G),I l - P(A U B UC) 1- I'(A) - P(IJ ) - P(G) I I'(A)I'(B); I'(A)P(G)+
/' ( IJ )/'(C) - /,A)P(B)/'(G) fJ - I'(A)fI 1 - P(B)fI 1 - I'(C)I
= I'(A') I'(B') P(G').
I 2 " I ,I " 5 2 " 2 1.5-8 -.-.- r-'-'- \--- _. - = -
G G G G G 6 6 " " 9
:s;\ !) U)·IO (n) -" - = -"
4 II 16 ' 1 ;1 ~1 2!)
(b) _. - • _. - = -' 4 '. 4 'I 16 '
(c) 2 I 2 " 10
. - t- -" - = -"
" ;I ,,<I IG
1.5-12 (u) G)'G)'; ( b) GY'Gr (c) GY'Gf
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Pmbability
(d) 3~~ ! (~y (~)2 1.5-14 (a) 1 - (0.4)3 = 1 - 0.064 = 0.936;
(b) 1 - (0.4)8 = 1 - 0.00065536 = 0.99934464.
1.5-16 (a) f ~ (~yk k=O
5
9'
1431432113 (b) 5 + 5 . 4 . :3 + 5 . 4 . :3 . 2 . i = 5·
1.5-18 (a) 7; (b) (1/2f; (c) 63; (d) No! (1/2)63 = 1/9,223,372,036,854,775,808.
1.5-20 n
(a) (b)
3 0.7037 0.6667
6 0.6651 0.6319
9 0.6536 0.6321
12 0.6480 0.6321
15 0.6447 0.6321
(c) Very little when n > 15, sampling with replacement Very li ttle when n > 10, sampling without replacement.
(d) Convergence is faster when sampling with replacement.
1.6 Bayes's Theorem
1.6-2 (a) P( G) peA n G) + PCB n G)
(b) P(AIG)
P(A)P(G I A) + P(B)P(G I B) (0.40) (0.85) + (0.60) (0.75) = 0.79;
peA n G)
P(G)
(0.40)(0 .85) -'-----,-:--'-------'- = 0.43.
0.79
7
1.6-4 Let event B denote an accident and let Al be the event that age of the driver is 16- 25. Then
P(AII B) = (0.1) (0.05)
(0.1)(0.05) + (0 .55)(0.02) + (0.20)(0.03) + (0.15)(0 .04)
---,-_----:-5_0--,--,-_::-::- = _5_0 = 0.179. 50 + llO + 60 + 60 280
1.6-6 Let B be the event that the policyholder dies. Let AI , A2 , A3 be the events that the deceased is standard , preferred and ultra-preferred, respectively. Then
P(AII B) =
P(A2 1 B)
P(A3 1 B)
(0 .60)(0.01)
(0 .60) (0.01) + (0.30) (0.008) + (0.10) (0.007)
60 = 60 = 0.659. 60 + 24 + 7 91 '
24 - = 0.264· 91 '
7 91 = 0.077.
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s
1.6-8 Let;l be the event that the VCR is ullder wnrram.y.
PCB, ' A ) (0.40)(0.10)
(0.40)(0.10) I (0.30)(0.05)+ (0.20)(0.03) I (0. 10)(0.02)
40 40 '3r. 40 + 15+6+2 = 6:1 = O./j;J:
15 = 6-;- = 0.238;
.1
6 63 = O.OIJ5;
.)
== (j-;j = 0.0:12.
l.il- I O (n) P(ltO) = (0.02)(0.92) + (0.'/8)(0.05) = 0.0184 r O.tW)O = 0.0674:
(b) P (NI AO) = ~:~~ = 0.727; P(A I AD} = ~:~~: = 0.273;
() '( , ' 0 ) - (0.98)(0.05) 9'"0 = 0.998' c I N A - (0.02)(0.08) + (0.98)(0."") 16 + 9310 •
P (;I ' NO) = 0.002.
(d ) Yes, partif-ulariy thuse in l)tlrl (b).
Cllllptcr I
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Chapter 2
Discrete Distributions
2.1 Random Variables of the Discre te Type
2. 1-2 (a)
(b)
{
O.Il. f(x ) ~ 0.:1,
0. 1,
0.3
0.2
0. 1
.r = I , I -'" 5, J ' = 10.
2 3 4 ~ 6 7 8 9 Ib ...
Figuft· 2. J 2: A Prohabil iL), lI il>tlJgra lll
1 (t' ) f(.c)=iU' :r =: 0. 1, 2. ···, 1O,
( b) N((U))/ L 50 ~ L1 / 1 ;0 ~ U .07:l ; .>!([1Il/ ".o ~ L ·'/ 150 ~ U.09:1; N({21l/ 150 ~ l :l/ IW ~ 0.087; N({3)) / IW ~ 12/ IW ~ 0.0.0; N({"Il/ IW ~ IG/lW ~ 0.I07 ;
N({5 J )/ 1 50 ~ 13/ l f>O ~ U .O"; N ({GIl/ " O 22/ 1"" ~ 0.117; N«71l/ 15U 16/ IW 0.107; N ({81l/ 150 _ 18/ 150 ~ 0. 120; N({9})/ 15U- 15/ 150 = O. IOO.
02006 PsM&on E(lucalion. IflC., Upper SaOClle RfV6I , NJ All righlH rtllWli'VW 'I his malerial III jl<OIti!;lOO uridei' aU OOjlyrlgtll ~ws 1l5 1hE1y cuul,looy ElIIl!il No 1lOI1'1ur1 III IhiiI !TIiIlaMI ,nav 1)0 UI'lI00ocft(l. ~1 any t()fTll Of bv IIIN mllllrnl. wililo1,1 oermlMlon In wrltinQ hom Ihe 1lUIlIlshef.
I
10 Cllllptcr 2
(e) j(x), 11(.\")
: j .1' ~
. 12
~JJ -r - '-, 8 I-
o
o . o . It;,
o 1 . o 12 ,
~
234 5 6 7 8 9 x
Figure 2.1 - 4: ~ 1i C':hjgnB Daily Lol tery Digit.s
G- 17- <1 2.l-6 (n) f(x) = 36 • x = 2,:i.4.f>, 6,7,8, 9. JO , II , l 2.
(b) 11x)
O. 16 ~ 0.14
0.12
0.10
0 .08
0.06
O.04 j
0.021
3 -, S "6 8 9 1'0 i l x 2 7 12
Figure 2.1 6: Probability histogmlU (or the Slim (If Il pair of diet'
02OO!i PealllOfl EdllcabOn, Inc .. Upper Saddoo RivIN, NJ AlI1ghIs reS6<V8d Thl5 maloM' Ilt protBClOO under IIY oopyrighllaws as UIUY ctlff&m!y e ~lSt No oortion oI lhos "",111M) ITIiIV bII ~ ., anv IO(m or IN IInll mtIIIl\tI. wfthoOJ' oormlsslnn In III'1tIirln (rom ttMI oubIishef
DiSCl'cte Disll'jlJlltiull~
2.1-8 (a) The spaCe of Il' is S' = to. 1. 2. :i , '1,5, G, 7, 8. 9, 10. II J.
I 12 ' llSSuming indepCLUlerl(,p,
( u)
P (lI' = 1) = P(X = 0, \' = 1) = ~. ~ = 12
I CotlliulJ iug l his, we see LILa L f (w) = P (\V = w) = 12 '
fix)
o .0
o .0<
o .0
o .0
1 2 3 4 5 6 7 8 9 10
w E S.
11 .,
Fib"II'C 2. 1- 8 : P robability h iswg mlll of s um of l-wO sp~'dal dice
2.1-10 (co ) G)(~) :19
G~) = 98 ;
(b) t C)(,O '~ ,) .. , C~)
221 24!',
2. 1-12 OC(IJ.O' ) = G) C:) C) (~')
("55) + e:) = 1.000;
OC(O.08) G) ("53) G) (";') en + (";')
O.!JG7;
OC(O. 12) G) C52
) C) C;) cn + e:) = 0.909:
OC{O.lG) wcn G)(~') Cr~) + c:) = O.8:I .. L
II
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12
2.1- 1. I'( X ;' I) = I - P (X = O) = 1 _ 9 1 137 = 1 - 228 = 22 = O.GO.
2 .2 Mathem atica l Expectation
2.2-2
c
E ( PnYlIlc m )
a 9 (' I 1 1 I ') '" / (.r. ) = - + c - + - + - + - + - + _ L.. 10 I 2 :1 'I 5 6 r-O
2
22-'1 £( X ) = (-I)G) + (O)G); (I ) (~) = 0;
E(X ') = (- I)'G) + (O)' G); (I)' (~) = ~;
£(3X' - 2X + 4) = "G) -2(0) +4 = ~~. 2.2-0 E(X ) = S4!J9(O.OOJ) - S 1(O.9!J9) = - SO.50 .
.,... 6 (j "" J 6 11"2 2.2-8 NOl\.' t hnt. L ""t':i = """"i L 2" = 2"- =- 1,:;0 th is j ij a p.d.!.
~ r 11" r 11" 6 r oo ' r E I
... ~ (i GL~ I Il(X) = '\' J . -0- = ., -L.. rr-J'l To- :r 7 _ 1 r u l
nlld it. is well kllOWIi lhat. the SIIIIl of t h is hanuoui(' ~rk.>::; is lIot. {i llite.
2.2-LO ECI .:\' - el) = ~ L II - el , where 5' = 11 ,2,3, 5, 15,25, 50} . z ES
\VbclI C = 5,
Chapff'T 2
I £(1).' - 51) = 7 [(5 - 1)+(5 - 2)+(5 - 3)+(5 - 5)+( lIj - fi) t-{25 - 5) 1-(50 - 5)1.
If c is eithe r ill("rCll.;;;cd or dccrcnst'<l by I , Ihis cx pecllltiOIl is illc rcuscd IIY 1/ 7. Thus c = 5, thl' lIIediAII , wiuilll izes th is cXp(..'cttltiou while b = E(X ) = I' . the mcan , lIlirlillliz~ E[( X - IJPl YOII could aL'iO let. h(c) = £( I X - c I) lind show lhllt. h'(c) = 0 wh(' 11 C = I).
15 2 1 - G - 1 2.2- 12 (1)·:16 ; ( - 1)· 36 - :16 = 6;
(j :~O (j - 1 (4)· 36 +(- 1)·"6 = 36 = 6·
( 16)(25) + (:1)(1 00) + ( I )(3(lO) 2. 2~ 14 (a) Tile lIn-mge class s ize is 20 = &0;
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Discrete Distributions
(b)
{
0.4, f(x) = 0.3,
0.3,
x = 25, x = 100, x = 300,
(c) E(X) = 25(0.4) + 100(0.3) + 300(0.3) = 130.
2.3 The Mean, Variance, and Standard Deviation
2.3-2 (a) E(X) 3 3! (l)X (3) 3- X ~ xX!(3 -X) ! 4 4
(1) 2 2! (l)k (3)2 -k 3 4 L k!(2-k)! 4 4
k=O
(1)(1 3)2 3
3 4 4 + 4 = 4;
E[X(X -1)] 3 3! (1) X(3)3-X ~ x(x-1) x!(3-x)! 4 4
(b) f.-L
(1)23 (1)3 2(3) 4 4 + 6 4
(72 E[X(X - 1)] + E(X) - f.-L2
E(X)
(2) (~)(~) + (~) - (~y
(2)(~)(~) + (~)(~) = 3(~)(~);
~ x x! (44~ x) ! (~r (~y-x
4(~) t k! (33~ k)! (~r (~y-k k=O
4(~)(~ + ~y = 2;
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13
14 Clt8IJtcf 2
u' = (12)G)' + ~ - G)' ~ 1.
2." - 4 EI( X - 1,)/. 1 = ( I /o)/E (X ) - 1,1 ~ (1 / 0 )(,' I') ~ 0;
EII('\" - I, )/ol' } ~ (l / o') EI(X - 1,)' 1 ~ (1/0' )(0' ) = 1.
, 36 " x/(x) = - ~ O.n'7, L 49 ._0
'2 ~ ... 5,547 q.~ ~ (.I' - /1)·/(.1') = 9,604 ._0
"I ax 9~ v'3 = 0.7000;
435 ,4G\ 4 12.542 «) 1(0) ~ 998,844 > 998 ,8,1.1 ~ / ( 1);
(d ) Ti le nllUlbCn! arc rpusolluhle because
(2[>,000 ,000)/ (6) ~ 1.79;
(25 ,000,000)1(5) = 461.25;
(25,000,000)1(4) 24.215,49 ;
7= O, I, 2, J . 4, 5.0:
0.[1776;
x = 0 is most. likely to oec-ur.
(e) The respective ('xpect.ed wllu(.'S, (138)/(x), for T = 0, 1, 2. :1, li re GO. Hi , ~7 .00 , 18.27, ami 2 .44 , SO Lllp results arc rCI\SOJlable. &"(' Figure 2.3-G for II (,·olllp..·uisoll of the theoret.ical probiibili ty histogfllJ11 anr1 tile his togram of the data.
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Discrete Distributions 15
I(x), hex)
Figure 2.3- 6: Empirical (shaded) and theoretical histograms for LOTTO
2.3-8 (a) Out of the 75 numbers, first select x-I of which 23 are selected out of t he 24 good numbers on your card and the remaining x- I - 23 are selected out of the 51 bad numbers . There is now one good number to be selected out of the remaining 75-(x-l).
2.3-10
(b) The mode is 75. 1824
(c) J.1 = ~ = 72.96 .
(d) E[X(X + 1)] = 70~~24 = 5,401.846154.
(e) 0- 2 = 46,512 = 5.724554' 0- = 2.3926. 8,125 '
(f) (i) x = 72.78 , (ii) 8 2 = 8.7187879, (iii) 8 = 2.9528, (iv) 5378.34.
(g) I(x), hex)
0.35
0.30
0.25
0.20
0.15
0.10
0.05
x
Figure 2.3- 8: Bingo "cover-up" comparisons
21 2 (a) P(X ~ 1) = G) = 3;
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, (b ) L P(X ~ !) = P(X = 1)1 2P(X = 2) , ... + 5P(X = 5) = I'; , ..
5. I(iS (c) It :;;; -- = I A !J1 'I!J;
:1, MiS
(d ) In the liulil , II = i' . _ 409
2 .3-12 ;r = 50 = 8. 18.
3 2 :1 2.3- 14 J(I) = S' J (2) = S' J(a) = S
:.I :2 :J /1= 1 ---1 2 ·- t3 ·-= ')
8 8 8-' :J 2 3 , 1
q1. = 12. :; -I 22 . -8 + ~12. - - 2 =-. u 8 'I
2.:1-1 6 (8) X = ~ = 1.3:l3;
(b) 88
s2 = _ = 1.275. 69
2 .3-1 8 (a) [:1, 19, 1U, 9[;
( b ) :r. = 1~275 '" 2 .00, s = 0.87;
(c) h(x)
~:~ j 0.30
0.25
0.20
0 .15
0. 10
0.05 !t 2
-
3
Figure 2.:1 18: Number of pets
4 x
2,4 B ernoulli Trials and the Binomial Distribution 11 7
2.4-2 J( - I ) = Iii , J ( I) = Iii ;
" 7 ,.=(- 1)- , ( 1)- = t ~ 18
.. 18;
( ')'(") ( 4)'( 7) 77 - ll i'S 18+ 1 +18 18 = 81'
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Discrl'i(' D&trihlltiolL:;
2 .4- 4 (a) P(X '$ ,1'1) = 0.5269 ;
(b ) P(X " 0) = 1 - ('(X <; 0) = 0 . .[73 1,
(e) P(X S 7) - P(X S 6) ~ 0 .••• 3 - 0.7:1"3 = 0.1.90,
(d ) 11 = (I2)(O.<\.'j) = fiA. (12 = ( 12)(0.45)(O.[~5) "" 2.97, (1 = J2.97 = 1.72:t.
2.4-6 (a) Xi .. M,7. U. (5);
(b) (i) (,(X " 2) 1 - P( X '$ I ) = I - 0.7166 = 0.283.1 .
(ii ) P(X - 1) = P(X S I) - P(X S O) = 0.7160 0.1200 0.3"(;0,
(iii) P (X S 3) = 0.9.79.
2 .4-8 (a) X is b( I!i,0.2).
( b ) Ii = \5(0.2) = :l, (1'1 = 15(U.2){0.8) = 2.4, (1' = ..12."4 = 1.['·19;
(e) ('(X " 5) = 1 - ('(X S . ) 1 - U.0358 = 0.16·12.
2.4-LO (a) X is b{6, 0.U5);
( b) JI = 6(0.05) -= O.:i ; (12 = 6(0.05)(0.95) '=' O.2SS;
(e) r;) I' (X = O) 0.735 "
(ii) {' (X S I )
(ii i) P(X ;:: 2)
O.9Ci72;
I - P(X <; I) = 0.0328.
2.4- l2 (a) It = 14(0. 55) = 7.7, (11 = l'I{O.5!i)(UA5) = :iA6fi;
(b) {'( X <.) = {'(X '5 7) = 1'(14 - X ;> 14 - 7)
P(I 'I - X ~ 7) = I - 0.5461 = O. l fi:m.
P( X > 6) P( 14 X < 11 -6)
= P ( 14 X S7)= O.741 ·1.
2 .4- 1.4 (a) X is /)(S. 0.90);
(b ) (i) J>(X = 8)
(ii) P (X '5 0) P(8 - X " 2)
I - P(~ X '$ I) = 1 0.81 31 U.186!J ;
(iii) P (X " 0) P(8 X '$ 2) = O.go lfl.
125/216. .r; = I ,
75/2 l6 . r = I,
f(I) = l5/216. J' = 2,
1/ 2 1 G. .t -= :i ;
(b) 125 75 15 I 17
" 1)·216+ ( 1)· 216 1 (2)· 216 +(3)·216 = - 216 '
269 ( 17)' q2 -= E(X :.!) 112 = - - -- = 1.2J!.I2· 21() 216 '
(1 1.11 ;
17
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-
'8
(c)
(d )
(c)
Sl.'(l Figure 2.4· 16 .
.::..!.. = - O.o! · 100 '
s' ' 00(' 29) - (- ' )' 100(99) :;:; 1.:3029;
s = 1.14 .
ft ,x). hex)
0.5
0.4
0.3
0.2
0.1
I - 2
~"'ig llre 2.4 JG: Losses ill chuck·a-Iuck
CJwj)U'r 2
3
2 .4- 18 Let X (Xlutl.1 tbe IIlllllhcr or wiuuiug tickets when II tickets lire plIrdH\S(.,(1. Tbcu
P(X " I) , - P(X ~ 0)
(9)" I - TO .
(n) 1 - (0.9)" 0.50
(0.0)"
II III U.9
" so /I := 7.
0.::;0
In O.5
III 0.5 .= 0.58 lnO.9
(b ) , - (O.!)" 0.%
(0.9)" =- 0.U5
" so II = 29.
III 0.05 -- = 28.4:1 III 0.09
(0. ' )( ' - 0.95') 2.4-20 "('''"'''' '' )''( ,""_-:CO."'J7"''C) -,+-7(0"'.""5)"'(;-' --:;0"'.9;;;8'"")-'+""("0."'" ")(-;-, -_""0".9"'50"') ~ U. '78.
2.4·22 (a) 1 - 0.01 " = O.!)!)!)!)!)!)!)!); (b) 0.90 1 = 0.!)G0590.
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----------------------------~
Discrete Distributions
2.5 The Moment-Generating Function
2.5-2 (a)
(b)
(c)
(d)
(e)
(f)
(i) b(5, 0.7); (ii) J..1 = 3.5, (Y2 = 1.05; (iii) 0.1607;
(i) geometric, p = 0.3; (ii) J..1 = 10/3, (Y2 = 70/9; (iii) 0.51;
(i) Bernoulli , p = 0.55; (ii) J..1 = 0.55 , (Y2 = 0.2475; (iii) 0.55;
(ii) J..1 = 2.1, (Y2 = 0.89; (iii) 0.7;
(i) negative binomial , p = 0.6 , r = 2; (ii) 10/3, (Y2 = 20/9; (iii) 0.36;
(i) discrete uniform on 1,2, .. . , 10; (ii) 5.5 , 8.25; (iii) 0.2 .
2.5-4 (a) f( x) = (~~:y-l (3~5) ' x = 1, 2,3, .. . ,
(b) J..1 1
-1- = 365,
365 364
365 = 132860
(_1 )2 " 365
(y = 364.500;
(c) P(X >400) (~~:yoo = 0.3337,
P(X < 300) = 1 - (~~:y99 = 0.5597 .
2.5-6 P(X 2: 100) = P(X > 99) = (0.99)99 = 0.3697.
126
1024
63
512
2.5-10 (a) Negative binomial with r = 10, p = 0.6 so
10 2 10(0.40) . . J..1 = - = 16.667, (y = ( )2 = 11.111 , (y = 3.333,
0.60 0.60
(b) P(X = 16) = C95) (0.60)10(0.40)6 = 0.1240.
2.5-12 P(X> k + j I X> k) = P(X > k + j)
P(X> k)
00
2.5-14 (b) L f( x) x=2
qk+j . -k = qJ = P(X > j).
q
~ ~ [C+2.rsf- C2.rsf]G") 2 00 (1 + VSY 2 00 (1 - J5)X
VS(l + VS) ~ 4X - J5(1 - VS) ~ 4
x
(fill in missing steps) 1;
19
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(e) E( X) ; Jg [C \VSf' - C -2vsf] U,)
= 2~t. ["C+,vsf -xC-,vsf] 2~ [(1 -(1; VS)/'I)' - (I - (I ~ /sfl)' 1
-= (till ill lIl iStjing SlCp.;) 6;
(d) EIX(X - 1)1 = 00 I 11-/5 1-J5 1 [( )'. ' ( )'.'] ~"("'- I)VS -2- - --Z C,) I ~ I +VS I - VS [( )'.' ( )" '] 2V5 f; J'(x- 1) - 4- - - ,-
= _1_ [I + VS f "Ix _ I) ( I + vs)' , _ 2V5 ·1 ,.._ 2' 4
-- £.£ - 1 - -I - VS 2:00
( ) ( I - VS) '·']
" " ,~,
I 2(¥ ) 2(~) 2VS (I _ I '~I VS)' - (I _ _ 1_-I_VS_5)'
(fill ill Illis.'4iug stl.'lm) 52;
0' EIX (X - 1)1 + SeX ) - I" 52 +6- 36 '),) . . _,
(J m = 4.690.
2.5- 16 (n) 1/ (1/ 6) = 6;
(b) 1 - 15/6)' ~ 11 / 36, 1/( 11 / 36) = :16/ 11 ;
(e) I
(5/6)" S 2,
0.5 S l - (5/ 6)",
(5 /W S 0.5,
" ~ 'I.
51 5t.2 5tl [,( 2 .ft.. l B M (t )= l +-+-+-+"'=e
I ! 21 31 '
/(.r ) = 1, :r = 5.
02005 PeartJOO EducIIllon. h'IC., Uppar ~ Rivof, NJ AlIIighIS fesa<ve!l . This maltKlal ~ proI9Cl1Id under al CDflyrighllltW5 as they CUtllllllly 8lIIst No nortion (11mb rMlaMI IIIiIY bI! rBDrOduced h li llY f<)rm Of bv IIIW ml!olins wltho", oemllssion In wrillnolrom the oubii!lho.
DiSl-'rPte Di:;[ l'iblltiOIlS
2.5-20 (a) R(t) = 111 (1 - IJ+ pe!),
R'(.) [ ,*' ] - " I - p -t pe',o -'
R"(t ) [ ( I - I' + pc' )(7)1 ') - (10(,1 )(PC' )]
11( 1 p) j ( I - fJ + pC')2 1 .. 0
(b) RU) n ln( l p t- pet ),
R'(. ) [ ",,,, ] I - P I IJe
t 1. 0
= TIp ,
H"(t ) [(J-p+pe')(pe'J-(pe')(Fe')] _ ( I ). n ( 1)2 - "11 - P , l - pt-pe h.O
(e) n(l ) = Illp + t - IntI - ( I - ,,)ell,
11' (. ) I -I ::: I + - - = -. [ (1 - ,,),,] 1 - " I I - (I - p)c'l; o 11 P
1I"(t) 1 - IJ
I( - I ){ I - (l - pic' )'( - ( I - p)"}],~" = - ,,-:
(d) R(t) r !ln/J t l - lll{ l -( l - /I)C' }] ,
/(1(1) = [ I ] , l' I - ( I - IJ)I'" t=o=P'
R"(,) = " I(- I ){ I -( I p)c' J ' {-( 1 p)e') I ,~"= ,·(Jp~P)
2.5-22 (0.7)(0.7)(0.3) = 0." '7.
2.5-24 (a) 0.9'2 = 0.2/;)2. Note LilaL "miss" = ~S Il CCess" ;
(h) C~) (0 ())27 (0. 1)2(0. 1) 0.0236.
2.6 T he Poisson D istribu t ion
2.6-2 >. = /' (12;:;: J so P(X = 2) = 0.'1 2:1- 0. 199 = 0.22'1.
2.6-4 >"1"-.\
3--I!
>.2e-).
2! e -). >.(>. - 6) 0 , G
Thull P(X = 4) = 0.285 - 0. 151 = 0.13 .. 1.
2.6-6 ,, = (1 )(50/ 100) = 0.5, so P IX = 0) = , "'lot = 0.G07.
2.6-8 Il l' = lOOO{O.005) = r);
(a) P (X '; I) '" 0.0.10;
(b) Pi X = 4.5, OJ = PIX ,; 6) - I'(X ,; :IJ '" 0.762 0.265 = 0.4~7 .
2 1
lOO5 PeQIPl Educa,loI'I. Ir.::. lJppeI SaOcIe A,V\!f . tiJ .... tI(JIU mMIV9d Thrs malenalls PIOIecled I.n)Ijf ,M coprnghllaws as IhBy curranlty OIOSI oo.1Icn 04 "'" IMletlel "IfIY bflllM'II'OduCed In atlY 101m or bY I!Irw mea",. wiIhouI DOrrni:ssion In Wlillna IIDIJII"- QUbIIsnef
2.6· 10 q v'U = ;~,
P (J < X < IS) = P(X :5 14) - P(X :5 :J) = 0.95D - 0 .021 = 0 .9:1$.
2.G- 12 (a) 1'7, '17.6:1, 6:1, 49,28,21, 11 , IJ; (b) 'f ~ :)0:1/ 100 ~ :1.0:1. 8
2 = 4, 141 / 1,:100 = 3. 193, yc:;;
(e) fix), li(x)
0.25
0.20
0.15
0.10
5 6 7 8
F'ig ur(> 2.6 12: Ilu('kgl"ollud wdillLioll
(d) Thc lit is vcry good [tue! the PoiS$Ou dis tributioll !)('CUlS to (ll"ovidt, 1111 cxcdlc.ut probability ulOdel.
2.6· L4 (a) fix). /'(x)
0.25
0.20
0.15
0. 10
234 5 6 789
Fig ur<.> 2.6 '4 : CrCCIi 1>Cl\lllit JIl&IIl ':;
.,
( b ) The li t is quite good . Also "1; = 4.956 nnd !P = 4. 13/1 (U'l' dose to ClLdl oLlier .
02005 POBrson Education. Inc . Upper S8dcIe AlveI, NJ. III rights r8SefV8C1 . This malerlal. pfOlOClad unoo. 8~ copyrlghllaWS 8. IhOy Cl)rrer1liy 8)uSI No oortlOll ollhls malerial mav 1M! tBOroduce!l if> allY Iom"I 01" bv 81lY mMf"I!I wlrhoot oonnl!;sion in wrillna fmm tna oonItshft.
Di8CI'ete Distributions
2.6-16 (a) j(x), hex)
Figure 2.6- 16: Bad records on a computer tape
(b) Yes. Again note that x = 2.225 and 8 2 = 2.025 are close to each other.
2.6-18 OC(p)
OC(0.002) OC(0.004) OC(0.006) 0C(0.01) OC(0.02)
3 (400PY e-400p
P(X::::: 3) ~ L I ; x=O x.
~ 0.991 ; ~ 0.921; ~ 0.779; ~ 0.433; ~ 0.042.
Oe(p)
1.0
0.9
0.8
0.7
0 .6
0.5
0.4
0.3
0.2
0.1
0.002 0.006 0.010 0.014 0.018 0.022 - P
Figure 2.6- 18: Operating characteristic curve
2.6-20 Since E(X) = 0.2 , the expected loss is (0.02)($10 , 000) = $2 , 000.
23
2.6-22 Use the Poisson approximation. If n = 200 and p = 0.01 , then A = 2. Using Table III , P(X ::::: 5) = 0.983.
If n = 200 and p = 0.05 , then A = 10. Using Ta ble III , P(X ::::: 5) = 0.067.
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2.6-24 Usiug ~ lilliLfl.tJ ,
(a) Pi X ~ 0) ~ 0.082 1;
(b) P(X ~ 3) ~ 0.21:18;
(el P(X ~ :1) ~ 1 - 0.54"8 ~ 0.4562;
(<II Pi X S 3) ~ 0.7570.
Cllapr,cl" 2
0200S Peal'5Ol'1 Education. Inc .• Upp6t Saddle River. NJ All rlgtlls resllMKl. This maHHial is proIeCled lIIldeI aM oopyrighllaws as they a nrenlly slUSl No nortion oIlhis rllI.Ilerh'll rAAV he 1&oI!)ducllld In 8m1 10ffll or bv 8JN n1fI<In8. wittn.1I1lft~ in wri!ina Imm the Dubllshar
Chapter 3
Continuous Distributions
3.1 Continuous-Type Data
3.1- 2 x = 3.58; S = 0.5116.
3.1- 4 (a) The respective class frequencies are 2,8, 15 , 13, 5,6, 1;
(b)
h(x) 1.2
1.0
0 .8
0.6
0.4
0.2
=4="'"'--- x 8.12 8.37 8.62 8.87 9 .1 2 9.37 9.62
Figure 3.1- 4: Weights of nails
(c) X = 8.773, u = 8.785 , S x = 0.365 , Su = 0.352;
(d) 800 * u = 7028, 800 * (u + 2 * su) = 7591.2. The answer depends on the cost of the nails as well as the time and distance required if too few nails are purchased .
25
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26 C/lIlpwr .1
3. t -O (.) CIa..,;; Class Frequency Cln..:;s
Intprvnl Limits / , l\ lurk,uj (93.555 , IOL555) (93.56, 101.55) 5 97.555 (101..'"155.109.55:.) ( IOI.5G, 1 O!).5f1) II lOfl.f.55 (l09.5fI5 , 117.555) (109.56,117.55) 22 J 13.555 ( 117.555, 125.555) ( 1l7 .56, 125.55) 2G 121.55[, (125.55.'"1, 1:1:1.55fl) (I2r...56, 13:\.55) 22 129.:):15 (13:1.555, 141.555) (133.56.141.55) 22 137.55:. (14 L555, 1 19.555) (141.;,6, 14!J .55) 8 145.555 ( 14!1.555, 157.555) (J49.56, 1~7.55) 4 15:i. 555 (157.[.5!), W5.S!)!',) (1n7.56, 16!l.55) 3 1(11.555 ( 1 GS.5!l5. 173.555 ) (l{iS.flG, ln.55) 2 !(i9.55S
(b) h(.t)
n-I '
0.025
0.020
0.015
:: 1 L""",*,".;:;!I.-:;~! -;:;!;~~~~~;+,;-;:;!; ... 105.'555 12 1.555 137.555 153.555 169.555
Figure :\' 1 li: Old l(eJlt. Rjvcr Oallk nun ti lll\.'S
(e) The histognull is ~k('wed !l tiglitiy to the right.
3 . l - 8 (a) With the <"lass wumiMics :1};'005,:t5005. 3.6005, ... , 4.100.5, the rC6pl.'C'li\'c clno'''''! fno'qu{'ucies art~ '1.7.2,1, 23 , 7, ,I, :1, 9, 15,2:1, 18,2.
(b) 10( ... )
35 j 3.0
2.5
2.0
1.5
1.0
0.5
I r
~ ,.,...
~
I ~
r= r= ~
~ I~ 4 .1 .l 3.6 3.7 3.B 3 .9 4.0
Figurc 3. 1 8: Weight.'! of mirror parts
o 200!> PeafSOO Education, Inc., Upper SItddIe R.-- . NJ. All O(JhIs IlIserved. This mataMJ Is ptOIllCtlld under aM copyrlghllaws as Ihoy CU'f~ e>usl No OOIlion oj lhis malorilll m/lW 1M! reorodoced In anv form or twlfllf !Tl6llfIII. willlOUl DefTTllslllon In WTilonu hom IhfI Otlhlo!;hfIf
hex) 0.010
Continuous Distributions 27
(c) This is a bimodal histogram.
3.1- 10 (a) With the class boundaries 0.5 , 5.5, 17.5 , 38 .5, 163.5 ,549.5 , the respective frequencies are 11 , 9, 10, 10,10.
(b) hex)
0 .04
0.03
0.02
Figure 3.1- 10: Mobil home losses
(c) This is a skewed to the right distribution.
3.1- 12 (a)
(b)
Player McGwire Sosa
Means 1998 1999
423.757 415.862 407.485 412.016
St. Devs. 1998 1999
46.409 32.320 38.136 33.197
hex) 0.016
0.014
0.012
0.010
0 .008
0.006
0.004
0.002
L¥:..J'ill¥ill-+-_-- x 549.5
Mark McGwire in 1998 Mark McGwire in 1999
Figure 3.1- 12: Distances Mark McGwire 's home runs traveled
102005 Pearson Education, Inc. , Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No oortion of this material mav be reoroduced. in anv form or bv anv means. without oermission in writino from the oublisher.
28 Clmp, ('( J
h(.I") h(.f) 0.012
0.0 12 r} 7\ 0.010
0.008
0 .006
.',.,fI
~\
-~
rI -"-
0.0 10
0.008
0.006
0.004
0.002
[7 f\ If
/
rl -1-, 1 " J
0.004
0.002
349.5 389.5 429.5 469.5 509.5 549.5 349.5 389.5 429.5 469.5 509.5 549.5
Sa mmy Sosa ill 199$ Stl lIllllY Sosa ill J!)!)!J
Fig UJ"(' J . I 12: DistluWf'!> SUllll llY Sosn 's hOi lie rlLlI :i tn\vcled
3 . L- l 4 (;, ) :r = 1.:tI5, S2 = 0.00:j97 1 ,$ = 0.0630;
(b) M.rJ c-
O
5 f--
4 r- f--
3
,--2
1
I I 1.385
.. 1.225 1.305 1.465
Fig ure 3.1 14: Diameters or graillS or soil
3 . 1 16 (a) St('m:'t L c(\.\"C!:I F'rC<IUCIICY Depths 20f 5 1 1 20" (j 6 7 7 4 f> 20. 88999 f> 10 2 1 ... 000001 7 17 21~ 222223:\3:\J3:J3 1 :1 :10 21f 444 ,1 4 4 455555555 15 (15) 218 6666666666666 7 7 77777177 2:\ a6 2 1. 888888 999!) 10 1;, 22 .. 000
" :J
(~ lul liJlly utlll/bers by 10 I .)
(b ) ( i) ii, = 4(21.2 t 2 1.2) = 21.2; q.2 = 21.5; q3 = 4(21.7 + 21.7) = 21.7;
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..
Com iIlIlOtl.~ Dis. ril:mtious
:tl [8 Stems
u. ,. ,. 1. 2. :, ..
(ii) 1rnlAl = (0.8)21.6 + (u.:1)21.6 = 21.(i;
( iii) iTo. /to =- (0 .7)21.0 ~ (0.:")21.0 = 2 1.0 ;
LCllves FrNIHPllcy
ijl2 1 450 1 560 889 96 I Of)" ·1 06~ 1·12 lSI 172 1!)5 290 f,
rll 0 :-,<1[, 788 8 17 880 !)2 1 9:sR 7 0\1 0-1 l 05 1 060 062 080 {)IJO 7
(i\ h Llti ply mnubers by 10- 2.)
Depths 1 2 (;
12 (7) 7
3.2 Random Variables of the Continuous Type
3.2 2 (n) ( i) J~ ,"j4(U
( .. I/ IG
c 2:
(ii) p(x) J~ ~ JU)dl
ft~ t ;{ I -I dt
,r" l iG ,
- 00 < .1' < O.
{
U,
P(.r)- .1' 1/ 16,
[ ,
n :5 T < 2,
2 ::$£<00.
19
!fx) 2.0
t (.l }
2.0
1.5 1.5
1.0 1.0
j 0.5 0.5 1
~+-~7C-~~::::"~~'~-~ .• 0.4 0.8 1.2 ' .6 2.0
Fih'l/re 3.2 2: (a ) CUUIiUlloJ lIS dbtril .• uLiou p.d .L find c.d .f.
C2\,IOfI PIlllI900 Edo<;.a lion. Inc . lJpp(I< SaC!£lle River. NJ All rights rese,..,Qd. This rllll/erlal Is prOlectlll.l uncial a. copyrighll!lW'l'l as mey curren!ly exisl NI; oort\Oll 01 ' his IMloriA' may be rooroducod. in arw '1)fIl'l 0 1 IlV limo rn8' 111S. w,thoo.n 09nl1tSf;i(Jn In wrirlno lrom!hl! DUbIi.<;he,
30
I
(b) 0) tP/ 16)x'd" eNS
c 2;
F (T) = l [(x)
1.0
0.8
0.6
0.4
0.2
[ ']' :6 - 2
0 , - 00 < 1; < - 2,
.r.3 1 16 + 2'
- 2 ::; x < 2,
1, 2 5X<oo.
F(.f )
1.0
0.8
0.6
0.4
0.2
Figure 3.2- 2: (b) Continllou~ diSl-ribuljon p.d.£. ,HId c.d. f.
Clmpter :1
C 2005 Pearson Education. Inc .. Upper Saddle Rwtn. NJ. All rlghlS reservod. This mBterialls prolllClod uncl6r aH copVrlghllilWS as thay currenlly exist. No 0011100 01 Itlls ma!Orial rM'" 00 reorodur:ad in anv 101m Of bv am means. wilhoul oarmls.'Iioo in wrilino lrorn the oul*SlWlr
Continuous Distributions
f(x)
2.0
1.5
1.0
0.5
(c) (i) j. l c -dx
oVx 1
2c 1
c 1/2.
The p.d.f. in part (c) is unbounded .
(ii) F(x ) .f~oo f(t) dt
-dt lx 1
o 20
{
0,
F( x) = Vx, 1,
- 00 < x < 0,
0::; x < 1,
1::; x < 00.
F(x) 2.0
1.5
1.0
0 .5
31
~ __ ~ ____ ~ ____ ~~ __ ~~ __ ~~ __ ~ ____ ~x ~ ____ ~ __ ~~ __ ~~ __ ~~ __ ~~ __ ~~ __ ~ x -0.2 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.2 0.4 0.6 0.8 1.2
Figure 3.2-2: (c) Continuous distribution p.d.f. and c.d.f.
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32
3.2-4 (a) J1, = E(X)
a2 = Var(X} 12 (x _ ~ y x; dx
(b)
{2 (X5 _ ~X4 + 16 x3 ) dx Jo 4 5 25
[X6 _ 4x5 4x412 24 25 + 25 0
64 128 64 24 - 25 + 25
~ 0.1067,
a v'0.1067 = 0.3266;
J1, = E(X)
a2 = Var(X}
[:4X41~2 48 48 64 - 64 = 0,
[:ox51~2 96 96 80 + 80
12 5'
a = fi252 """ V 5 .- 1.5492;
Chapter 3
<Cl2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No oortion of this material may be reoroduced. in any form or bv any means. without oermisSion in writino from the oublisher.
COlli ;IJU(lIIS Dhlrilml ;ow;
(c) I' := B(X )
1J'l = Var(X)
=
2 0=--
J45 '" :1.2- 6 (a) Af (I)
I =
1., ., [.,
o 2';; ' .
1.] .;; d.l· o 2
[,'V']' I .1 0 = j
£' ( I)' I .1' - - -- d.l· . 0 :\ 2J.C
1., (1 a/ ':!. 2 .] / ':!. ;;:.r - -6' I o -
I -.r 18
[ !.rU/1 ~.r3/2 5 V
t- _J.] / 1 I ]' o 0
·1 4',
0.29$ 1.
J;' - r(l I )
( I - I)'
( I - I)'" I < I ;
(b) Af'(t) " "(l=IjI
!\J"{ I ) := 12
(1 - I)'
I' Af' (O) = ;j
M" (O) _ /1'2 = 12 - !.I = :t
:3.2- 8 (a) ~-... .l.~2 d.1' =:
c = I :
]/2) d.1'
(b) E(X) - J"" ~, do/' = 1111 £1-;--. whidl is IIlIbollllcll,.'(l. ,r
:rl
(.2OOf, P8il/1OIl EoucallOO. Inc, UppeI' saddle RIvet'. NJ. All rights I05eIV!:Id TNs malonalis prolected uncIeillM copyrighl laws as lhey culfom/v eidsl ~DClfIlOI1 OIlh111lMlllIiilllllll~ be f!lnlOOlJOr.oo. in ilfI~ ]nnn OIIW illW mBIIllS Wll houl OIInnission In wrillna ho m lhe OIlh1lsrwlf
;14 'JlIlpr,cr J
:1 .2 10 (nl
F('I ~ {
0, ~< .r <- I , -
(J.3 I 1)/ 2, - I :S .r. < 1,
I , 1 $ .7' <
fi x) P(X)
1.4 1,4
1,2 1.2
1.0 1.0
0,8 0,8
0 ,6 0,6
0,4 0,4
0,2 0.2
--.x "
--=1.0 -0,6 . -<i:2 0:2 0:6 1.0 - 1.0 -0,6 '~,2 0:2 0:6 1:0
Piguro :1.2 10: (a) / (.r.) = (:1f2)..r::l flUd F (J") = (.r:1 + 1)/2
(b )
F(f ) ~ { 0, -00 < .l" < - I ,
(f + 11/ 2, - 1 $ .r < I ,
I , I $ J; <00.
fix ) F(x) 1 0 10
0,8 0,8
0 ,6 0,6
0,4 ,4
0,2 0,2
--=1.0
___ x
·- -<i.6 0:2 0:6 ," - -0,6 -62 ' 0:2 0:6 1 0 - 1.0 -0.2 1.0
Figure :1.2 10, (b l fir) ~ 1/ 2 . "d 1'(. ) ~ (x + 1)/2
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Continuous Distributions
(c)
F(x ) ~ 1 0,
(x + 1)2/2,
1-(1-x?/2,
1,
0.4
0.2
~~~ __ ~~ __ ~ __ ~~ __ ~~ __ ~~~x
-00 < x < - 1,
- 1 s:; x < 0,
O s:; x < 1,
1 s:; x < 00.
F(x) 1.0
0.8
0.2
-1.0 -0.6 -0.2 0 .2 0.6 1.0 -1 .0 -0.6 -0.2
Figure 3.2- 10: (c) f( x ) and F(x) for Exercise 3.2-10(c)
() '() M ' (t) '() M'(O) '( ) 3.2- 12 aRt = M(t); R 0 = M(O) = M 0 = p, ;
(b) RI/(t)
RI/(O)
3.2- 14 M(t)
M(t)MI/(t) - [M' (t)]2
[M(t)J2
MI/(O) - [M'(0 )]2 = 0'2.
100
etX(1/lO) e-x/lO dx = 100
(1/10)e-(x/1O)(1- lOt) dx
(1 - lOt) - l, t < 1/10. R(t) In M(t) = - In(l - lOt) ;
R'(t) 10/(1 - lOt) = 10(1 - 10t) -1;
RI/(t) 100(1 - 10t) -2 .
Thus p, = R'(O) = 10; 0'2 = RI/(O) = 100.
35
0.2 0 .6 1.0
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fix) 1.0
0.8
0.6
0.4
0.2
:1.2- 16 (b) 0, < x ::; 0 ,
x
2'
I'(I) ~
I ,
1.5 2.0
0 < 7 .$ 1.
1 <7:$ 2,
2 :$:r< :1,
:1 :$ 7 < 00;
___ x
2.5 3.0
F(x) 1.0
0.8
0 .6
0.4
0.2
Figure :1.2- 16: J (x ) and F(x) for Exerdsc :1.2- 16(a)
(c) (~ 0.25
ql 0.5,
(d) 1 ::; m :$ 2,
(e) 0.75
'11 5 2 4
'13 5 2
3.2- 18 F (:r ) = (x + 1)'l/ 4, - 1 < r < 1.
(8) F('/I"o&l) == (1I"0M + 1);1/<1 0.64
11"0 (i .\ + 1
1TU M
(h) (1I"0'lr. + 1)2/ 4
;'02&+ 1
71"0 .25
(e) (11"081 -I 1)'/' 11"0.81 + 1
1fU.81
0.25
/ 1.00
O· , 0.81
/3.2' 0.8.
J 2.56
0.6;
Chapter 3
C200S Pea,llOIl Educatiofl, tnc . Upper S/IddIe RivtI< , NJ. All figtlIlI reserved. This matarial Is plOlOldOO uooor aM copyrtglll lliws al they eu' IIIn11y exial . No oonK)fl nllIlis 1T\OI11WiII1 maw he nJOmducod . ... It"" lorm or IN 81J\1 mmll"IS. wUhout IlIInnlssiol'l ~l wmina I,om 1M oubIisho.
Cow i/JUolls DiSrfJbulio lls
3.2- 20 (n) :r = 1. :\13.1:
(b ) ~ O':)22{);
(r.) T ill' rC'SJ)C'C l iH' fn.'<IUt' II('it>'iaH' 1.6. 7, 6,8. 10.0, l:t 20. 20
(d) 1.0
0 .8
0.6
Ok
021 i S--oc~-Ic--~ I-=.L ----'
0.2 0.4 0.6 0.8 ' .0 1.2 1.4 1.6 1.8 2.0
a' [,'m'L' -G)' [~I[ _ ~~ = ~
3.2 22 P(X > 2) = J;"" .1.2/ -.r\Lr = [-(-:r'1: = C 16
3.2- 24 (a) P{X > 20UU) = I:l~_NJ ( 2.': / IOO()'l)(,- \.r / I OOOI J d.r = [_c-V/ lOOlI 12} ;ooo = / --';
(bl t - {Z' IOOO )2 ] ~ ." ,~
( ._("" , •• / lUOO)2
11"07&
(e) Iro 10 :12-1.59;
(d) 11"0.60 = 9f17.2:l.
0.25
10, (0.25)
1177.41;
3.3 The Uniform and Exponential Distributions
3.3- 2 JI = 0, (J 'J = 1/ :). Sf'(' Lhe fij.!; lI rfos fOJ' Ex('rcill" :1.2- 1O(b).
37
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38
:S .3--4 X is U('1,5);
(a) Il = 9/ 2; (b ) 0 2 = l / l2 ; (c) 0.5.
3 .3- 6 (u) p ( 11l < X < :10) = J.:m (~) e - r /,10 dr IU 20
= l _ t>- :r / 20f~ = e.- 1/ 2 _ c- 3/ 'l ;
(b) p (X > 30)
(e) P(X > ,,01 X > 10) = l'(X > 'Ol l'(X > 10)
-, _' ___ - 3/ 2 . c- I {2 - e ,
(d ) 0' = 0' = 400 , M(t. ) = ( I - 20W '· 35
0.383, d ose to I.he rdative fn::q l1ency lOn ' (el P (IO < X < :1O)
P(X > :l0)
'n 0 .223, close to the relaLive frC( luency 100 '
CIIIlp tCT :1
P(X > 40 l X > 10) = 0.22:\, d ose to the relative {requeuey ~: = 0.2-11.
(2) . 3 .3- 8 (a) I (L} = 3" r.; -2:r / J, O ~:r<OO;
(b) P{X > 2) = j,()(l ~('-2"'/~ dx = [_e_'l.r./,J]OC = e-4/ "J . :2 3 2
3.3- 12 lkt. P(x) = P(X $: ..r.) . Then
P(X > x + y \ X > x) = P( X > y)
1 - F (y). l - F(x+y) I - F(T)
Tha t is, wit h g(x) = 1 _ F (x ), y(x -I- y) = y(x)9{Y)· This fl1ucLional equatio ll implies dmt
when' /) = cllI (I. T hat is, F{x) = l _ ct>;r . Since F(oo) = 1. b 11111SI. be negal i\'c . say b = - ). with). > O. Thus F(x) = I _ e.-.\ r, 0 $: :t, t.he dist.rihutioll fllW·t iOll of a ll cspollf' ll t ial
dist ribl ll.iOl I.
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Continuous Distributions
3.3- 14 E [v (T)J
3.3- 16 E(profit)
derivative
n
3.3- 18 (a) P(X > 40)
J03
100(23- t - 1)e-t/ 5 / 5dt
J03 - 20e - t/ 5dt + 100 J03 e(3- t ) In 2e - t / 5 / 5dt
- 100(1 - e- 0.6 ) + 100e3 1n 2 J03 e- t In 2 e - t /5 / 5dt
[
e - (in 2+0.2)t] 3 - 100(1 - e-0 .6 ) + 100e31n 2
In2 + 0.2 0
121.734.
J;' [x - 0.5(n - x )J 260 dx + J~OO [n - 5(x - n)J 260 dx
1 [X2 (n _ x)2] n 1 [ . 5X2] 200
200 ""2 + 4 + 200 6nx - 2 o n
260 [- 3.25n 2 + 1200n - 100000J
260 [- 6.5n + 1200J = 0
1200 -- ~ 185. 6.5
(':>e ~e-3x/100 dx J40 100
[_e-3x / lOO j 00 = e-1.2 . 40 '
(b) Flaws occur randomly so we are observing a Poisson process.
3.3- 20 F(x ) - 00 < x < 00 .
1 0 < y < 1,
the U (O , 1) distribution function.
3.4 The Gamma and Chi-Square Distributions
3.4- 2 Either use integration by parts or
F(x ) P(X ::; x )
Thus, with A = l / B = 1/ 4 and a = 2,
P(X < 5) 1 - e-5/4 _ (~) e-5/4 0.35536.
39
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4U
:1.4- 4 The moment, t1l' l,cmt.ing fnllctioli of X is M(t ) = ( I - Of) - a , t < 1/ 8, Thlls
U(I)
M" (t)
aO(I - Ot) - O- 1
0(0' + 1)02 (1 - Ot) - ,, -2.
Thc mcau aucl varia l!cc are
I' M'(O) = 00
M" (O) _ (0'0)2 = u(o + I}ot - (O'O):l
o·tI'l,
3.4- 6 (&oc Figun' I:UO-2, page :,)79, in tJ.le textbook.)
(a) J(x) 14 .7100 99 _ 101 7"
r ( IOO) " e . , O::S: x < 00,
JJ lOO{I /l '1.7) = 6.80, 0 2 = lOO(i / 14 ,7):l .; 0.4G21'3;
(b) T = 6.7.1, S2 = 0.46 \1 ;
(e ) 9/ 25 = U."6 .
:1.<1- 8 (a) IV !Jas a gamma fiistribut.ion wit.h 0 = 7, 0 = 1/ 16.
(b) Usill~ Table III ill 1.l1t' Apllen<iix ,
6 S"'e-8 I - L IT P(IV ,, 0.5) =
"'=0 1 - 0.313 = 0.687.
bl-'('llI lJ;(: hen: >.w ~ (WHO.5) = 8 .
Clwpter :1
3.4- 12 Since the Ill .g.f. is that of ;\ 'l(24), we have (a) /1 = 24; (b) 0'2 = 48; a.lld (e) o.~m, usillg
Table IV .
:1.4- L4 Notc lbal. ), = 5/ 10 = 1/ 2 is 1.lLe meal! 1111lllber of ll.r riva1ti pe r lIIillute. TlLus 0 = 2 liud t he
p.d.f. of the waitillg tiu\{' bcfo['e the eighth toll i ~
J(x) =
0 < 7' <00,
t.he p.d.f. of f\ cbi_square dlstl' il;utiml wilh ,. = 16 degree; of fr~lom, Using TI)blc IV,
P(X > 26.30) = 0.05.
3.4- Hl P{X > 30.14) = 0.05 wllf' re X denotes a sillglc obse rvation. Let IV e<llIal tbe numher (out of 10 observations thot cx{:et'd 30.14. Th€ll th~ d is tribution or IV is 0(10, 0.05). Thus
P{\V = 2) = 0'!J8~5 - O.!HJ9 = 0.07116,
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Continllolls Distributions 41
3.5 Distributions of Functions of a Random Variable
3.5- 2 Here x = .,fiJ, Dy(x) = 1/ 2.,fiJ and 0 < x < 00 maps onto 0 < y < 00. Thus
g(y) = ,jY 1_1_1 = ~ e-Y/2, 2,jY 2
0 < y < 00 .
3.5- 4 (a)
F(x) ~ 1 0, x < 0,
fo X 2t dt = x2 , 0 ::; x < 1,
1, 1 ::; x,
(b) Let y = x2 ; so X = .,fiJ. Let Y be U(O, 1); then X = .JY has the given x-distribut ion.
(c) Repeat the procedure outlined in part (b) 10 times.
(d) Order the 10 values of x found in part (c), say X l < X2 < .. . < X lO and plot the 10
points (Xi , Ji/ll), i = 1,2, . .. , 10, where 11 = n + l.
3.5- 6 It is easier to note that
3.5- 8
dy dx
and dx dy
Say the solution of X in terms of y is given by x* . Then the p.d.f. of Y is
0 < y < 1,
as -00 < X < 00 maps onto 0 < y < 1. Thus Y is U(O, 1).
x (DlO/
7
170( ~ )3/7 (~) dx dy
f( x) e- X, 0 < x < 00
g(y) e-(y/5)1O/7 (~) (~y/7 y3/7
10/7 3/7 _(y/5) "l/ 7 510/ 7 Y e , 0 < y < 00.
(The reason for writing the p.d.f. in that form is because Y has a Weibull distribution with 0: = 10/7 and f3 = 5. See page 184 in the textbook.)
3.5- 10 Since - 1 < x < 3, we have 0 ::; y < 9.
When 0 < y < 1, t hen
When 1 < y < 9, then
- 1
2.,fiJ'
x= ,jY, dx dy
1
2,jY
1
2.,fiJ
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-42
Thus
g(y) ~ {
I I-I I I ·1 2~1 I O < y < I,
4 2,fij f -
4,fij 4
H2~1 I
~
8,fij J :$; y < 9.
3.5- 12 E( X )
[ I ]" [I . ], lim - In( ] + :r2) + liul - In(1 + .r'l ) ,,- -00 21T I. b_ l",- 21f 0
-21
[ Jim { - IIl (1 + (l'l )} + lim 111 (1 +- &2)]. 11" 0 __ ..., b-+N
E( X ) does Hut cxi!>! bf)(;U Use ueithcr of t hcsc limits ex ists.
3.5-14 Simulate observaTions of t he Ca uchy random variable X tL<; ing
j ' I Y = 2' (Iw
-00 11'(1 + w )
or, l.-'(juivalemly,
L = tlUlj7f(Y - 1/ 2)},
where y is lUI observation from the U(O, 1) dist r ibution.
3.6 Additional Mode ls
3.6- 2 With b = lu 1. 1,
3.6-<1
G(Ii.I)
G(w)
G(6")
a
I - exp [ __ "_ e .. ·dlll . 1 + - "-] III 1.I III 1.1
= O.OJ
O.OOO0264li
P(IV :5 71170 < IV)
:17792. 194 77
P (70 < 1V :oS 71) P(70 < JII )
0.0217 .
A(W) ~ (If}'''' I- r
N(w) Io"'(acW + r) tlt
~ ~ (ell'" _ I ) -I ell! b
G(w) ~ i - ex p [-* (c"'" - 1) - £.,111, U < DO
(/ ( /lUi ) --e - 1 - eft) g(lJI) (at ,It ... + e)1:' b , 0 <00.
Clmpt,(:r .1
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Continuous Distributions
3.6- 6 (a) 1/ 4 - 1/ 8 = 1/8;
(c) 3/4 - 1/4 = 1/2;
(e) 3/4 - 3/4 = 0;
(b) 1/ 4 - 1/4 = 0;
(d) 1 - 1/ 2 = 1/2;
(f) 1 - 3/4 = 1/4.
43
3.6- 8 There is a discrete point of probability at x = 0, P(X = 0) = 1/ 3, and F'(x) = (2 /3)e- X
for 0 < x . Thus
J.L = E(X)
so
(0)(1 /3) + 100
x (2 /3)e- Xdx
(2 /3 )[-xe-X + e-Xl8" = 2/ 3,
(0)2(1/3) + 100
x2(2 /3 )e-Xdx
(2 / 3)[-x2e-X - 2xe-X - 2e-Xl8" = 4/ 3,
(/2 = Var(X) = 4/3 - (2 /3) 2 = 8/9.
3.6- 10 For the uncensored distribution,
3.6- 12
0 < x < 00 .
Thus
E(X) 100
x(3000 )(10 + X)-4 dx
[-1000x(10 + x)-3 - 500(10 + X)-2J: = 5.
For the censored distribution,
E(Y) = 110
y(3000)(10 + y) -4 dy + 10(1 - [1 - 1/ 8])
[- 1000y(1O + y) -3 - 500(10 + y) -2J~O + 10(1/8)
_ 10, 000 _ 500 500 10 = 3.75. 8000 400 + 100 + 8
T= { X , 4,
X :::; 4,
4 < X;
E(T) 14 x (~) e-x/5 dx + 100
4 (~) e- x/5 dx
[-xe-X/5 - 5e-X/5J~ + 4 [- e-X/5J:
5 - 4e-4 / 5 - 5e-4 / 5 + 4e-4 / 5
5 - 5e-4 / 5 ~ 2.753.
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, a.6- 14 (0)
yU )
= IUT
f( , )d:r t -~, e di =ce . -00< 1 <00.
(b) = ft+iJII1U'
dl /l =
w
h(w) , .. , _ Oftll""' (fi) e<> >' " lJJ (' l,. _
W
o <w<oo.
Clwptcr 3
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Chapter 4
Multivariate Distributions
4 .1 Dist ribut io ns o f T wo Random Va riables
'1. 1 2 ,
·1 , .1. ei6 - it: eft " " .1. 3 , .1. eft - /0 e if, '" W
, 2 , .1. eft era , .1.
iii " " , , .1. . ~ eis . ~ " "
2 :1 ., , • , iii " " 16
(c) independent, because h (x)h(Y) f(x , y) .
4. 1- 1 ')51 7!;!d"I! (o .:m}7 (0 . 10)8(0 .20)6(0. 10 ) I 0.00105.
4 . 1- () (n) f (x ,y) 71 (0. 7S)'" (U.O] )"{D.:H ) 7 -7-", x !y!(7 - x - y)!
0 :5 x+y "5 7;
(b) X ;, 0(7.0.78), J' = 0 , 1, .. ,, 7 .
" . l S(n) P (O :5 X :S ~) 111'3 ;- dy dr o ..,2 2
(b) p (l $ 1' $ 1) 1.'1""3 - dxdy t 0 2
J.' 3 -.,;y dy ~ 1
! 2 ( ~)"'. 2 '
'15
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46
(c) P (~ :::; X :::; 1, ~::;}'":::; 1) r r v'i~ tlJ: (lY i t it -~'H /Y- D dy
~ _ (~y/2
(d ) P(X ::::: ~,y ::::: ~) P{~ ::; x ::; 1, ~ :::; 'i' :::; 1)
(c) X llud Yare dc]>t!l1dcnt.
5
8
4 .1 - 10 (a) f,(L) 11 (x + y) fly
[:ry -I :y2] I = x + ~ , 0 5.c ::; 10; 2" -
11 (;I' + y) dx = lJ + ~ , n :::; 'Y :::; 1;
H y " (r+ D (Y + D = f,(r )h(y)
h(y)
/(x, y)
(e) Silllilnrly, q~ = 11414 '
4 .1- 12 The ,Hen of t ile Spl\CC is
Tltus
P('l'l + Tz > to}
II
144
Clutpter .,
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Multivarite Distributions
4.2 The Correlation Coefficient
4.2- 2 (c) /-1 x
(l2 x
Cov(X, Y)
p =
0.5(0) + 0.5(1) = 0.5,
0.2 (0) + 0.6(1) + 0.2(2) = 1,
(0 - 0.5)2(0.5) + (1 - 0.5)2(0.5) = 0.25 ,
(0 - 1?(0.2) + (1 - 1)2(0.6) + (2 - 1)2(0.2) = 0.4,
(0)(0)(0 .2) + (1)(2)(0.2) + (0)(1)(0.3) +
(1)(1)(0.3) - (0.5)(1) = 0.2,
0.2 = JOA. VO.25VOA . ,
(d) y = 1 + VOA ( .~) (x - 0.5) = 0.6 + O.Sx. yO.25
4.2- 4 E[al'Lil(X1, X 2 ) + a2u2(X1 , X 2)]
L L [al'ul(xl, X2) + a2u2(xl, X2)]J(Xl, X2)
4.2- 6 Note that X is b(3, 1/6), Y is b(3, 1/2) so
(a) E(X) = 3(1/6) = 1/2,
(b) E(Y) = 3(1/2) = 3/2,
(c) Var(X) = 3(1/6)(5/6) = 5/12,
(d) Var(Y) = 3(1/2)(1/2) = 3/4;
(e) Cov(X, Y) 0+ (l)f(l, 1) + 2f(1 , 2) + 2f(2, 1) - (1/2)(3/2)
(1)(1/6) + 2(1/S) + 2(1/24) - 3/4
-1/4;
(f) - 1/4 - 1
P = J 152 . ~ = y'5 '
4.2- 8 (b) 1 2 .1 6' 6
2 1 .1 .1 6' 6 6
3 0 .1 .1 .1 6' 6 6 6
0 1 2 3 2 1 6' 6' 6'
(1) (2) (2) 1 4 -5 (c) Cov(X,Y) = (1)(1) 6 - '3 '3 = 6 - 9 = 18;
47
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(d) o~. ~ + ~ - (~y = ~ =0;. - 5/ l H 1
/(5 /9)([, /9) - 'I' p
(e)y ~ - ~J~;~ (X - D , y 1 - '2 L.
4.2- 10 (a) f!{x) 1'" 2dy = 23:, O:S x :S l.
h{Y) i l
2dx = 2(1 - y ), o:s y :S I;
(b)
4.2- 12 (")
(b)
I"
, ax
Cov(X, Y )
p
J,(x)
My)
I"
I' , =
a' , =
11;'
£' 8 .d:l:(l - J?) &1' = -" ,
• • • (y. <l y dy =- , J 3 "
5
Clldpter 4
Q 2005 PealliOfl Ed\JCalloll. lr..e .• Upper Sadd!6 River. N.J. All rights r6SeMld "This malef.aJ is Pf010C1!)d Ufldar a" cq:Iyrlghllaws as 1hay curTlllllly 1l~1 No oortion o t lh\.<; ma1llr\!l! may be rllorodtJ(;ed. 1n anY lorm or bv anv rnearos WllhOu1 oormlsslon In wlil li'oO hom 1M oubhsher
Muitival'ite Distributions
Cov(X, Y) 1111 (x - 8/ 15)(y - 4/5)8xy dy dx = 2~5 '
4/225 2V66 p
J(11/225)(2/75) 33 '
4.3 Conditional Distributions
4.3- 2
2 1
4
3
4 g(x 12) 3 - 21x - yl
equivalently, g(x I y) = 4 '
1 3
4
1 -4
1 2
g(x 11)
h(YI1) h(YI2)
x = 1, 2, for y = 1 or 2;
3 - 21 x - yl equivalently, h(y I x) = 4 '
2
1
1
4
3 -4
3 4
1
4
1 2
/-LxiI = 5/4, f.L x 12 = 7/4, /-LI'II = 5/ 4, /-L1'12 = 7/ 4;
~2 _ ~2 _ ~2 _ ~2 - 3/16 vxl l - vxl2 - vl'll - vl'12 - .
4.3- 4 (a) X is b(400 , 0.75);
(b) E(X) = 300, Var(X) = 75 ;
(c) b(300 , 2/ 3) ;
(d) E(Y) = 200, Var(Y) = 200/ 3.
4.3- 6 (a) P(X = 500) = 0.40, P(Y = 500) = 0.35 ,
Y = 1, 2, for x = 1 or 2;
P(Y = 500 I X = 500) = 0.50, P(Y = 100 I X = 500) = 0.25 ;
(b) /-Lx = 485 , /-LI' = 510, 0"; = 118275, 0"; = 130900;
(c) /-L x ll' = lDO = 2400/ 7, /-Ll'l x=500 = 525;
(d) Cov(X, Y) = 49650;
(e) p = 0.399.
4.3- 8 (a) X and Y have a trinomial distribution with n = 30, PI = 1/ 6, P2 = 1/ 6.
49
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50
ChapIN 'I
(b ) T he condi t ional p.d .f. of X , given Y = !I, is
11 /I. _ y. -- = /;(30 - y, I ;:»). ( ''') I - /~l
( e) Si nn" B(X) = 5 llLld Vit r (X) = 25/ 6 , E(.': 2) = Var( X ) + IE(X )12
= 25/ G -\- 25 = 175/ 6. Siwilarly, E{ )') = 5, Vm(Y) = 25/ 6. £(1'2) = 175/ 6. The ("orrcllltioll
cocilide.nt is
(I =--( I/ H)( I/ G) (5/ 6)(5/ G) ~ - I/f'
E'(X I') ~ - 1/ 5 J(25/G)(25/ 6) + (5)(5) ~ 1.lf, / H.
Thlls
4 .:\- LO (a) f (',,,) ~ 1/[1 0(10 - ")1-T = O, I, . .. ,9. y = X,T+!,···.O;
, I
(b) h(Y) ~ Lo 10(10 - x)' ,< y = O, I, . .. , !J :
(e) E:(\'lx) ~ (x + !) / 2.
I 2 r.'( "') I 4.3- 12 Frolll Exl1mplc4. 1 10. II~ = ~' /l" = ;J' lIud tj ) - = "2'
4 .3- 14 (b )
f, (x) ~
., I (2)' I (Ii"- = "2 - :i = is;
I'j' (')(') I • I Cov(X , n = )u ,I" 2)'Y d y ,Lor - :l j = :j - ~ = 3G '
10,1" 1/8 dy ~ ,)8. o ~ r :5 2.
L~2 !/8 dy ~ 1/ 4, 2 < x < 4.
[~') 1/8 rly = (6 - J·)/a, .1 ~ ). ~ (j;
,,+2 (e) h(Y) = i I/S d.r = 1/ 4, 0 ~ y :::: 4;
(d)
{
1/)', O :5 y :5 :.r , 0 $. r :5 2,
h(y lr)= 1/ 2, )' - 2 < y < x , 2 < ;; <'1,
1/«(, _ r ), x - 2 ~ y :S .1, .\ :S x $ 6;
(e) y(x I Y) ~ 1/ 2, y ~ £ ~ y+2:
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Multivarite Distributions 51
(f)
E(Ylx) = 1x
y. ~ dy x-2 2
14 Y --dy
x-2 6 - x
x 2'
[y2] x - =x- 1 4 '
x-2
[ y2 ] 4
2(6 - x) x-2
x+2 2
1Y+2 1 [ X2 ] y+2
(g) E(X I y) = x· - dx = ~ = y + 1, Y 2 4 Y
0 ::; y ::; 4;
y 4
3
2
0 ::; x::; 2,
2 < x < 4,
4 ::; x < 6;
Figure 4.3- 14: (h) y = E(Y I x) (i)x=E(Xly)
4.3- 16 1
(a) h(yl x) = - , x
o < y < x, 0 < x < 1,
l X y x (b) E(Ylx)= -dy = - ,
o x 2
(c) f( x, y) = h(yl x )JI(x) = (~)(1) = ~ , x X
111
(d) 12 (y) = - dx = - In y, Y x
O < y < l.
o < y < x, 0 < x < 1,
4.4 Transformations of Random Variables
4.4-2 (a) The joint p.d.f. of Xl and X 2 is
f(XI , X2) = C) C 1) X~ I /2- 1X;2/2- 1 e-(XJ+X2)/2, r ~ r ; 2 (1' 1 +r2)/2
o < X l < 00, 0 < X2 < 00.
Let Y I = (XI/T1) / (X2/T2) and Y2 = X 2. The Jacobian of the transformation is
(TI/T2)Y2. Thus
( ) _ 1'IXI X2 ,r2/2-1 -(y2/2)(-rlyl/r2+ I) 1'lY2 1 ()rI/2- 1 ( )
9 YI, y? - (1') (r.) x2 e , - r ~ r ; 2(r , +r2)/2 1'2 1'2
o < YI < 00, 0 < Y2 < 00.
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-52
ClmJ)tel' -,
(b) T he nmrgina! p.d .L of }' , is Y\ (Yl ) = 1000
g(Yt, 'Y'l ) dYt ·
Mak~ the dmugc of variables U' = Y2 (~+ l). Theil 2 1":.!
r(~) (~)ql'l '"1 / 2 - 1 'J , . V,
( ) - ,
", y , ~ ("'l ("'l( ,y",)('o .,>'I' ·1 , r - r - 1+-2 2 "2
(J < 'Yl < 00.
4,.-4 Vac[u(') + u'(')( X - ')1 ~ IH'(')I' Va,(X - ' )
= [u, (>.)]2(>.) = t, which is fret: of >. ,
" u'{).) .fi. ' 2c.J)... u{>')
If we tllkc c = 1/2, WI' lJave u(X) := JX has 1'L variance t.hilt is llimosi. fn,:c of >..
4.4-6 pew)
few ) = F' (w)
4.4-8 (a) E(X )
~
0 < w < 1
I 'l n +p d X2 dx , lro ;''''' xo - I xe- l e-( l, , +;r;~)lq
o ( 1_00) 1" /'" r(a)f(.B)O {'70 - ;r;'i' - I [( 1 _ W):r l /1111.6- 1 e_l:rl+(I_W).r .tu,\/6 ( - I). ..
10 r(fl)r(.B)OO+P w:l ..I,d"l
1 (1 _ W)JJ - l ('0 J~+ O-I (, _ ¥I / (jW
f(a)f({:1) ui:l+ 1 Jo /1O+fl d.:r.1
no + p) {Bw),,+I!J (1 - W)!3 - 1
r (o -+- (J) W.£l+1 00 +;1
~
rca + ~) , . - '(1 - )~- I r (a) r(~) w w,
(' I' (n+ /l) . - 1(1 _ )'- I d i , x r(a) r(~) " X "
O< w < L
.E I - j ' (1-1' i~( t+ mr«(;t + I) 11 r{o + 1 +m u-II - I ( )a- I I.
r(o )r (a + ~ + 1) ' 0 r ea + l ) r(~) .
(a)r(a)r(o + #) (a + ~)r(a + ~)r(")
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No 00f1101'1 ot 1his material milV btl roOrodllf.ed_ 1f'I anv lorm 01 bv anY means. will101J! oormlsslon In wril inO horn Ina oo >bliStlllf
Multivarite Distributions
fCa + ,8)f(a + 2) rl f(a + 2 + ,8) x o+2- 1(l _ X),8-l dx
f(a)f(a + 2 + ,8) io f(a + 2)f(,8)
(a + l)a
(a+ ,8 +1)(a + ,8) ' Thus
a (a+l) a 2 a,8
(a + ,8 + 1)(a + ,8) (a + ,8)2 (a + ,8 + 1) (a + ,8)2 .
(b) f( x ) f(a + ,8) a -l(1 _ , ) ,8- 1 f(a )f(,8) x x
1'(x ) = f(a + ,8) [(a _ 1) x a - 2 (1 - X),8-l - (,8 - 1) x a - l (1 - x),8-2 ] . f(a)f( ,8)
Set l' (x) equal to zero and solve for x :
f(a + ,8) 2 f> ') ---:'-.,---:---:':- x a - (1 - x )JJ- - [(a - 1)(1 - x ) - (,8 - 1) x l 0 f(a )f(,8)
a - a x - 1 + x - ,8x + x 0
4.4-10 Use integra tion by parts two times to show
dWl 4.4-12 (a) Wl = 2Xl and -d = 2. Thus
Xl
f(xt) = 'ir(1 : 4x i) ,
(b) For X2 = Yl - Y2, Xl = Y2, I J I = 1. Thus
(a + ,8- 2) x a- I
a- I x
-00 < Xl < 00 .
- 00 < Yi < 00, i = 1, 2.
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53
54
(d)
=
= 81l"i 1 [I I "7 l6 i· i - YI
lIt t 1 =!h +Yt · Yl + l · i
1 1 [ I I 1 1 I [lIl + i + !Il - i 1 211" . Yl 'Yl - i + 'Yt + i = 211" .; (YI - i)(Yl + oi)
11" ( I + y?),
A Maple soiutiOlJ ror Exercise 4.4-12:
>f := x-> 2/Pi/(1 + 4*x~2); I
f:= .1· - > 211"(1 +'1:r.2)
Chapter 4
>sialplify(int (f(y (2) )*f (y [1) -y(2) ,y(2) __ i nfin i t y . . infini ty)) ;
I
;-r ( l + yf)
A M Utll(;TlWt1,(;fl SOluLiuB fo r )-Jxc rcise 4.-1- 12·
In(1) ;"" f{x _) :'"' 2/(Pi*(1 + 4(x) ~ 2») g(y1 _ ,y2_] := f[y2 ) .f[y1 -y2]
In (3] I ntegrate (g[y l ,y2], {y2. -Infinity. I nf i nity}]
Out [3l"
-------------2
Pi + Pi y1
4.4- 14 The joiUL p.cL f. is
" h{x. y } = i;3c-(:r+,,)/f>, 0 < x < 00, 0 < 'Y < 00; ,
" , '" y y
x = <;W, Y w
T he .) ucobiall is
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Muitivarite Distributions
The joint p .dJ. of Z and W is
o < Z < 00, 0 < W < 00;
The marginal p.dJ. of Z is
h(z) = tx) zw e-(z+ l )w/5w dw Jo 53
f(3) z (_5_)3 roo w3-
J e-w/(5/[z+l]) dw
53 Z + 1 Jo f(3)(5/[z + 1])3
2z (Z+1)3'
0 < z < 00.
4.4-16 Ct = 24, (3 = 6, 'Y = 42 is reasonable , but other answers around this one are acceptable.
4.5 Several Independent Random Variables
4.5- 2 (a) P(XI = 2, X 2 = 4)
15 15 28 256
55
(b) {Xl +X2 = 7} can occur in the two mutually exclusive ways : {Xl = 3, X 2 = 4} and {Xl = 2, X 2 = 5}. The sum of the probabilities of the two latter events is
[ 3! (1)3] [ 5! (1)5] [ 3! (1)3] [ 5! (1)5] 5 + 3 1
3!0! 2" 4!1! 2" + 2!l! 2" 5!0! 2" = 28 = 32'
(b) E(XI) = E(X2) = 0.5,
(e- l _ e- 2 )(e-14 _ e-2.4 )
(0.368 - 0.135)(0.247 - 0.091)
(0.233)(0.156) = 0.036.
E[XI (X2 - 0.5)2] = E(X1 )Var(X2) = (0.5)(0.25) = 0.125.
4.5- 6 Let Y = max(Xl' X2)' Then
G(y) [P(X .:::: y}F
g(y) G'(y)
2 1 -- -( 1)(4) y4 y5' 1 < y < 00;
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56
£(1')
4.5- 8 (a )
(b)
J;,", 8 ['y _4 - 1r lSl dy
;l2 21'
. (:1)[(3)(1)'1(3) 27 P (X, ~ I)P (X , ~ 3)I'(K, ~ I ) ~ - - - - ~-: . :I 4 .1 4102'1
~P(XI = 3,.\"2 = I , X 3 = I ), 3P(X} =- 2,X'l = 2, X;j =- I} =
. ( 27) . ( 27) 162 .
. 1 1024 + ;l lO24 == 1024 '
. , (3 3 I)' (If.), (c) P(l < 2) ~ - + - . - ~ - . - 4 4 4 16
~ 3
4 .5- 10 P{ I < ruill X ,) =- !p( 1 < x ,)jl = (/ ,,-rdI ) == e-!I = 0.05.
4.5- 1.2 P(l' > 1000) P(X\ > lOOO)P{X";! > lOl>O) P(X3 > lOOO}
4. 5- 14 /(x) 2r , 0 < .1,: < 1;
F(r ) ,., .. n <:t < I ;
G(y) P(XI :5 y)f'{X2 :S y)P(Xa S y) P( X~ <:: 'II)
o < y < \:
9(Y) == G'(y) = sl,
£'(}' ) J~ y81}7 (ly
O < y < I;
l~l]: = ~. So Lhe VI:l\UC in dollar:; iii $(8/ 9)(100,000).
4.5- 16 P (iILltX > 8) = 1 - P(IIl(l..X :$ 8)
[t (~) (U7)'(03)'" '1' I _ ( 1 - O . 1 '19:~)3 =- 0.38'14 .
4. 5- 18 G(y) P(l' ,, 1/) ~ P(X, " y) ... P(X, " ,,) ~ IP(X " !I)I'
0 < y < 1;
ChapiN ·1
P{0.!J9U9 < }' < \) = G(1) - G{O.999!») = I - O.99!J9t1\) = 0.008.
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Multivarite Distributions 57
4.6 Distributions of Sums of Independent Random Variables
4.6- 2 E(X)
E(X2)
Thus
/-l x 1 2 3 1 1 2; a x 10 4 20' and
/-l y 1 1 2 1 1 1 2 + 2 = 1; a y = 20 + 20 = 10
4.6- 4 E(Y) E(X1X 2) = E(Xr)E(X2) = /-l1/-l2;
4.6- 6
Var(Y) E(Xf xl) - (/-l1/-l2)2 = E(Xl)E(Xj) - /-lr/-l~
(/-lr + ai)(/-l~ + a~) - /-lr/-l~ = aia~ + /-lia~ + /-l~ar. My(t) E[et(Xl +X2)] = E[etX1 ]E[etX2 ]
(q+pet)nl(q+pet)n2 = (q+pet )n 1 +n2.
Thus Y is b(nl + n2, p).
n n
IIE[etX,] = IIeJ.Li(e' - l) i= l i= l
the moment generating function of a Poisson random variable with mean /-ll + /-l2 + ... + /-In .
[1/(1- Bt)]h = 1/(1 - Bt)h, t < l /B ,
the moment generating function for the galmna distribution with mean hB.
4.6- 12 (a) Mw(t) = Mx(t) . My(t) = fi(e 2t + 2e3t + 3e4t + 3e5t + 2e6t + e7t )
(b) The p.m.f. of W is 1
12' w = 2,7,
peW = w) = 2 w = 3,6, 12 '
3 12 ' w = 4, 5.
1 4.6- 16 (a) g(w) = 12' w = 0,1,2, ... , ll , because, for example,
peW = 3) = P(X = 1 Y = 2) = (~) (~) = ~. , 6 2 12
1 (b) hew) = 36 ' w = 0, 1, 2, ... , 35 , because, for example,
peW = 7) = P(X = 1 Y = 6) = (~) (~) = ~. , 6 6 36
2'
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58 Chapter 4
4.6-18 (a) W = Xl + X2; (b) U=X l +X2 +X3 +X4; (c) V = L:~=l Xi'
W P(W = w) u P(U = u)
2 1/16 4, 16 1/256
3 2/16 5, 15 4/256
4 3/16 6, 14 10/256
(a) 5 4/16 (b) 7, 13 20/256
6 3/16 8, 12 31/256
7 2/16 9,11 40/256
8 1/16 10 44/256
v P(V = v)
8,32 1/4
9, 31 8/48
10,30 36/48
11,29 120/48
12, 28 322/48
(c) 13, 27 728/48
14, 26 1428/48
15, 25 2472/48
16, 24 3823/48
17,23 5328/48
18, 22 6728/48
19,21 7728/48
20 8092/48
4.6-20 Let Xl, X2
, X3 be the number of accidents in weeks 1, 2, and 3, respectively. Then Y = Xl + X2 + X3 is Poisson with mean>. = 6 and
P(Y = 7) = 0.744 - 0.606 = 0.138.
4.6-22 Let Xl,X2,X3,X4 be the number of sick days for employee i, i = 1,2,3,4, respectively.
Then Y = Xl + X2 + X3 + X4 is Poisson with mean>. = 8 and
P(Y > 10) = 1 - P(Y ~ 10) = 1 - 0.0816 = 0.184.
4.6-24 Let Xi equal the number of cracks in mile i, i = 1,2, . . . ,40. Then
is poisson with mean >. = 20.
It follows that 14 20Y -20
P(Y < 15) = P(Y ~ 14) = L e, = 0.1049. y=O y .
The final answer was calculated using Minitab.
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111 III" \7iril .e 1)ist ri IJUtilJ/J:S
4 .6- 26 ~/ = XI + X2 + X3 + Xol I lIiS a guuuna disl riblltiou with 0 - 6 and 0 = 10. So
O.~8 13 - 0.1 1 57 .
TII(' final UlI.'nIJ('r was C(\lc lllllt{'<! using Mini t.uh.
4.7 C he byshev 's Inequali ty a nd Convergence III Proba bili ty
4.12 VHr(X ) = 2!lt'! - 17:! 0.
( il ) P ( lU < X < 21) P ( IO 17 < X 17 < 2,1- 17)
4 .1- 4
P(I X !} 10
171 < 1) ? 1 - 4!J = .19 ' bt.'(: II WW k _ 7/3;
(b) " P{ IX - 111 ? 16):5 162 = 0.035. 1ll.'C11II:-;e k = l tip •.
( I Y I ) (0.5)(0.5) P 100 - 0.5 < O.OH ~ I - IOU(O.OM):.! = 0.609:
bCCIUISt· k = o .o~/ J(O.5)(0.5)j IOO;
( I \. I ) (0.5)(0.") (b ) P 500 - 0.5 < U.OS ? l - GOO(0.08)'l = 0.922;
hl:!('H II!iC k = 0.08/ )(0.:))(0.5)/ [100:
(0.")(U.5) lOoo(0 .08)'
O.!JG I,
4 .1-6 P (75 < X < ~5) =. P (7!l -1'10 < X 80 < h5 80)
= P (l X 801 < 5) 2 1 60/ 15 = (\.~.1. 25 .
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Chapter 5
The Normal Distribution
5.1 A Brief History of Probability
5.2 The Normal Distribution
5.2- 2 (a) 0.3078: (b) 0.4959;
(c) 0.2711 ; (d) 0.1646.
5.2- 4 (a) 1.282; (b) - 1.645;
(c) - 1.66; (d) - 1.82.
5.2- 6 M(t) = e 166t+400t2/2 so
(a ) p, = 166; (b) a 2 = 400;
(c) P(170 < X < 200) = P(0.2 < Z < 1.7) = 0.3761;
(d) P(148 ::; X::; 172) = P( - 0.9 ::; Z ::; 0.3) = 0.4338.
5.2- 8 M(t) = e-6t+64t2 / 2 so X is N(-6 , 64).
(a) P( - 4::; X < 16) = P(0.25 ::; Z < 2.75) = 0.3983;
(b) P( - lO < X ::; 0) = P(- 0.50 < Z::; 0.75) = 0.4649 .
5.2- 10 G(y) P(Y ::; y) = P(aX + b ::; y)
P (X ::; y: b) if a > 0
__ e-(x- J.L) 2 / 20"2 dx j(Y-b)/a 1
-CXJ ay'27r
Let w = ax + b so dw = a dx. Then
G(y) = j Y _ _ 1_ e -(w-b- a!")2/2a 2u 2 dw _ aay'27r
which is the distribution function of the normal distribution N(b + ap" a2a 2) . T he case
when a < 0 can be handled sim.ilarly.
61
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62
CJUlpfcr 5
5 .2- 1.2 (a) Stems Lp;w(!:; Fn'< IILc licics Depths
11 . B I I
12", 03 2 " 12_ :) li 2 5
1:1-"' I :.\ ,I " S
1:\- 55 77 7!J G I'
Ilh 0 02'2 :.\ ·1 ,1 ,1 8 22
14 _ 66777 S !)!) B :\0
15. ' 0000 1 I 1 2:l~\44 12 :lO
15- 55G7 888 !) 8 18
iG. 0OO2 :H 6 10
w- 05 2 <I 2
17* 17- 0
(b) N(O. /) qlwlI/;/es
2
. -- x
-1\ -2 •
..
Figure fl.2 12: 'I_II plo~ of N (O. I) qlllllll.i1es Vf'rSIiS d tl1.fi qURmi les
(c) Ye;.
5.2- 14 (a) P(X > 22 .07) = P(Z > l. 7fl) = 0.0·101 ;
(b) P(X < 20.857) = P (Z < - 1.2825) = 0. 10. 'fblls dlc d istTiblltio ll of l' is b(15. 0.\0)
Hud fro Ul Tnhlt' II ill the Appendix, P(l~ :5. 2) = 0.$ 1:)9 .
5.2- 16 We must soh-c !"{.z:) = O. We \uwe
III f (J')
1'(,) ! (r)
!(,.)I"(') - 11'(,)1' IJV)I'
I"(x)
:,. - I'
=
_ IH(~ O") _ (:r _ p)'l / 20"2 ,
- 2(J· - I' )
- I a"l
2a2
£ = I' ±O".
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No oonion 01 this matertalll)llV bI'I fllOfnducftd. 1n a,w IQfTll or bv anv means wllhou! OIIfl111s.'\lOflln wrilirlO lrom lhe nul)li!;f1ofl' .
Till' NormaJ Di:.tri/)uliflll
r.. 2- 18 X is N(500. IOOOO); MII(X - :KXl)J/lOOj' is ..\:.1( 1) I:lml
[ ( "'0")' 1 P 2.706 ::; ' ;~ :5 5.2U-1 = 0.975 - 0.900 = O.U7fi.
5.220 (a ) (\"1 (0.7) - 1'(- 1.%)=0.7580 - 0.0256 = 0 .732'1:
(b ) Q)( 1.51) = O.OOO(j;
(c) '1)( - u.55) = 0.2912.
5.2- 22 G(.r) P(X ,; x)
p ({1 :o::;.c)
P(Y :o::; lux)
__ e - fll - IO)'/"ldy = i[)( ln :c jIll I 1
-"'<. jfi 10)
g(L) = G'(.r} = _ ' _p_ll" .. _ IOjl / 2.!:.. 0 < :r < 00. J'E I
P( 1 ()OUO < X < 200UO) P (lu 10000 < Y < III 20000)
.:. .to (ln 20000 - 10) - 'Jl (ltllOOOO 10)
5 .2- 24
0. 161557 - 0.211863 = O.24G(i9·1 usiug hlillitah
k Streugths 1' = 1.:/10 Z I _ p k Strcugths p -:- k/ IO
, 7.2 0.10 - 1.282 6 11.7 0.611
2 8.9 0.20 - O.R42 7 12.9 11 m :I 9.7 0 .:)0 - 0.521 8 I:J.9 0 .80
4 lO.5 DAD - 0.253 9 15.:.i 0.90
5 IO.!) 0 .. ;0 U.OOO
2
o --e
- 1
Fig ure 5.2 24: II-II plot of N(O. 1) q lli\lIlilf~ Vl'rSlls dat.u q UlUlliles
It SC('UIS to 1)(' Il ll ('xcl· llell!. fi t.
ZI_"
tI .25:i 0.[.2.1
0.842 1.20:.1
P(X > 120 1X> 105) ~ P(X> 120) P(X> 105)
I - '~(2) t - (l.I( I)
0.0228 := 'Q."i587 = 0. 14:.17.
CIJl;lpter 5
5.3 Random Functions Associated with Normal Distributions
5.3- 2
0.4
0.3 /I = 36
0.2
11 = 9
0. 1 ,,= 1
40 60
Figll1'e 5.3- 2: X is N(5Q. 36), X is N(50, :16/ n) , n :;;::: 9 , 16
5.3- 4 (n) P (X < 6.017 1) ::::::: P(Z < - 1.645) = 0.05; (b ) Let W ccWal the number of boxet t.hat. weigh less t.han 6,017 1 pouuds. Then IV is
b(9, 0.05) and P(W ::; 2) = 0.99 16;
(e) P(X S 6.035) ~ p(z < 6.0:15 - 6.05) - 0.02/ 3
P(Z :5 - 2.25) = 0.0122.
5.3- 6 (a) Using x2(JG), p(~ < E:~ L (.X. - 50)2 < ~) = 0.95 _ 0.05 = 0.00-
100 - 100 - 100 '
(b) USiHg x:.!( l 5), p(72ooti'l < L::: I(X,- XY < 25
0000) = O.9~) - O .05 = O.90.
t - tOO - 1
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Nfl oorLio" (Ii Ihili rnaloMllT\IIV b6 rearodllcEKl 10 any 100001 bv anv means wI,houi oenn~ In wriling 110m IhO oublishe,
The Normal Distribution
0.25
0.20
0.15
0.10
0.05
Figure 5.3-8: N(43.04, 14.89) and N(47.88, 2.19) p.d.f.s
(b) The distribution of Xl - X 2 is N(4.84, 17.08). Thus
( - 484 )
P(XI > X'J) = P(XI - X2 > 0) = P Z> ~ = 0.8790. - V 17.08
5.3- 10 The distribution of Y is N(3.54 , 0.0147). Thus
( -0.32) P(Y > W) = P(Y - W > 0) = P Z > = 0.9830.
VO.0147 + 0.092
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9
Figure 5.3- 10: N(3 .22, 0.092 ) and N(3[1.18], 3[0.072]) p.d.f.s
65
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66 Clw pter 5
5.3- 12 p(X > Y ) ~ P (X - Y > 0) ~ J'(Z > - 55/ 110) ~ 0.69 15.
0.005
0 .004
0 .003
0 .002
0 .001
Figure 5.:3- 12: N( ~I7 'I , 6368) and N {529 , 5(32) p.d. r.s
5.3- 1·1 (n)
(b )
'_) ~ 3.8' b(X = 24.5, Var{ X ) = - :::: 1.805, 8
_ _ 2.72 E( l') ~ 21.3, V.c(Y ) ~ 8 ~ 0.911 ;
N (2,1.5 - 2 1 .~ = 3.2, 1.805 + 0.9 11 = 2.7 16) ;
(e) p (X > Y) _ _ (0 -3.2)
p (X - y > O)~ I - '~ -].648
I _ '~(- 1.94 ) ~ " (1.94 ) ~ 0.9738.
fi. 3- l 6 T ile joim p.d .L is
I(T x , ) ~ _ 1_ e-:4/2 I x r /2- l e-Z2/2
I, '1...,fiir f(1) 2)2" /2 1. '
-00 < XI < 00, 0 < J.'2 < 00;
y, XI/ / f 2/r , Y'J. ~ X,
" ~ VI /Y'!./r, X2 Yz
T he .)acobiull is
J ~
T he joiut. p.d .f. of }~I aud Y2 is
9(1/1 , Y2) = _ ' _e-J/:1n/2r 1 'l /2- 1e- IJ·/2./iii , . J2i. r(r/ 2)2r/2 '}, .;r -00 < Yi < , 0 < Y2 < 00;
T he marginal p.d .r. of 1'1 is
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The Normal Distribution
U clY2 Let u = Y2(1 + y2l / r')' Then Y2 - and
- 1 + YUr du
1 ------,2c-:-. So 1 +ydT
f[(T + 1) / 2] r :o 1 . 'u(r+l) /2- 1e-u/2 J1iTf(r / 2) (1 + yUr)(r+l) /2 Jo f[(r + 1) /2]2(r+l)/2
r[(T + 1)/ 2] J7TrT(r / 2) (1 + yUT)(r+l )/2'
-00 < Yl < 00 .
5.3- 18 Let Y = Xl + X 2 + ... + X n . Then Y is N(SOOn, 1002n). Thus
P(Y 2: 10000)
P (Y - SOOn> 10000 - SOon) 100yn - 100yn
- 1.2S2
SOOn - 12S.2y'n - 10000
0.90
0.90
10000 - SOOn 100y'n
O.
67
Either use the quadratic formula to solve for yn or use Maple to solve for n. We find that yn = 3.617 or n = 13.0S so use n = 14 bulbs.
5.3- 20 Note that Y - X is N(10000 , 50002 + 60002). So the probability that B 's total claims exceed those of A is
(O.SO)(O.lO) + (0 .20)(0.10)P(Y - X > 0) [ (- 10000)]
O.OS + 0.02 1 - <r> 7S10.25
O.OS + 0.02(0.S997) = 0.09S.
5.4 The Central Limit Theorem
5.4- 2 If f( x) - 1 < x < I ,
E(X) [11 x(3/2)x2 dx = 0;
Var(X) = rl (3 /2)x4 clx = [~X5] 1 ~ J-l 10 - 1 5
Thus P( -0.3 ::; Y ::; 1.5) P < < ---=== ( - 0.3 - 0 Y - 0 1.5 - 0 )
V15(3/ 5) - V15(3/ 5) - V15(3/ 5)
;:::::; P( - 0.10 ::; Z ::; 0.50) = 0.2313.
5.4- 4 P(39.75 ::; X ::; 41.25) P (39 .75 - 40 X - 40 41.25 - 40) V(S/32) ::; v(S/32) ::; V(S/ 32)
;:::::; P( -0.50 ::; Z ::; 2.50) = 0.6S53.
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68
5.4- 6 (n) p 1 2X( 1 - X/2) ((x = tx; _T:[ = 2 - ~ = ~;
(J'l. = 1\;'J.(1 _ x/2) fix _ (~) 2
=tx; _:~I ]: _ ~ = ~.
(b) PO~/ :8 ~ J~~;8 ~ ;~/ :8) ;:::;: P (O :S Z :s 1.5) = 0.4332.
5.4- 8 (a) E(X ) = /t = 2<1.'13;
(
-;'J' 112 2.20 <
(b) V.,.X) = -;;- = 30 = O.073.J;
(c) P (24. L7 :S X :S 24.82) ~ p(24 .17 - 24.43 < Z < 24.82 - 2/1..I3)
J O.On3 - - J O.0733
P( - 0.96 ~ Z < 1.44) = 0.7566.
5.4- 10 U:;iug the nOl'lnw approximation, p( 1.7 -2 < ~ < 3.2 - 2) P (1.7 ~ Y ~ 3.2) = } 4/ 12 - } 4/ 12 - }4/ 12
;:::;: P( - 0.52 :£ z :s 2.078) = 0.fi796.
Using the p.d.r. of y , P { 1.7 ::; Y ::; 3.2) = f~'T I( - L/2)y3 -+ 2y2 - 2y -+ (2/ 3)1 dy
-+ J :I( I / 2)y3 - 4y'l + lOy - 22/3] rly
-+ 1;·2[( _ 1/ 6}y3 -+ 2y2 - 8y -+ 32 / 3\ (ly
I( - l / B)y" + (2 / 3)Y' - y' + (2/ 3)yl1,
+ I( l / B}y" - (4 / 3)y' + 5y2 - (22/ 3)uJ!
+ I( - 1/ 24)Y" + (2/3)y' - 4y2 + (32/3}yJ ~"
Q.l!)20 -+ 0 .458:J -+ 0.0246 = O.6i49.
5.4- 12 The distribll tioll ('If X is N(2000 , 5002/ 25). Thus
CllIIpteJ' 5
_ (X -2000 2050 - 2000) P(X > 2050) = p 500/ 5 > 500/5 ;:::;: 1 - 1) (0.50) = O.:m85.
£(X + Y)
Vn.r(X -+ Y}
30+ 50 = SO;
= 52+ 64 +28 = 1.44 ;
" Z L (X. + F.) in approximately N(25· 80,25· 144) . . -, C 2005 Pearson Education, Inc., Upper saddle AiV8f , NJ. All rlrJhlS reseiV9d. This malerlalls protected Uf'1(Je1 all copyrlgl\llaws liS tney Ctllfenl/y lids. JIkJ nortion ol lhis mIIlari&ll1l8v be moro<ttJced in arw lom'I or bY arw means. wilhool oormlssoon IfI writinn from lhe oubIis/'I6r
The N()rIllHI Disrri/)ut joll (i!J
Th us P( !!J70 < Z < 2ltlJO) p(1970 - 2ooo < Z - 2000 < 2090 - 2(00)
GO GO GO
~ (1)( 1.5) - ((.(-0.5)
== O.9a:l2 - O.JOd5 = 0.G247.
5.4- 16 I. .. :l .. \' , {'(jua l tile lilll" l..ctwt..'eu nai('f; (.or lkk('t. ·j - I and i . ror , = 1.2, . ... IU. Eadl X, Il[\l;
a g alllllll i disLrihll1.ion wit.h (} "'" 3. 0 = 2. V == E:! I x. has n gaullUA d istribution with paralllcl('rs Il l' = 3U, 8 .. = 2. Thus
TIIP uorulai nppl'{Jxim.atioll is givell hy
(If_ GO GU GO)
I' roM " """ " <1>(0) ~ 0.5000. yl20 yl20
5.4- 18 WI' a n' givclltliut V = L~l X, hu.'l1l1lcnn 200 aud vnriaucc bOo We wam to find !I so thai
PlY ;, y) < 0.20
p(l'- 200 > 11 - 20°) JSij JSij < 0.21),
We ha\·p 11m!.
y - 200
Jiffi
u
= O.d4:l
207}". T 2~ daytl.
5.5 Approximations for Discrete Distributions
5.5- 2 (a) 1'(2 < X < 9) = 0.!J.5a2 - 0.0982 = u.8550;
(b) P(2 < X < 9) p(2.5 5 < X - 25(0.2) < 8.5 - 5) 2 - J25(O.2)(0.8) - 2
::::::: f'(- L2!i~Z~ L75) 0.81) 1:1.
5.5 4 1'("5 " X $ '10) " p(:34.5 - 3G < Z < '10.5 - 36)
3 - - J
== P( - O.50 ::;: Z :S 1.50 ) = 0.G2·17.
5.5-61', = S·I(O.7) = 5$.8, Var(X ) = 8-1(0.7)(0.3) = 17.G·I,
P(X S. &2.5) ~ (1)CJ2·&.~25It$ ) = '1)( 1.5) = O.OGGS.
&.5 8 (a) P(X < 2U.6f)7) =- 1'( X - 21.:H < 2n.8ft7 - 21.J7) OA 0.·1
P(Z < - 1 282) = tl.lO.
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70
0.12
0 .10
0.08
0.06
0.04
0 .02
CJlllpter 5
(b) Tbe distrib LL t.io lL of}' is b(IOO, O.iO). Thus
PlY < 5) ~ I' ( I' - 100(0.10) < 5.5 - 10) '" P(Z < - 1.50) ~ 0.0668. - ) 100(0.10 )(0.90) 3 -
1'(2 1.:.H < X < 21.39) '" p(21.31 - 21.:n < Z < 21.39 - 21.:17) (e) _ _ OAlIO - - 0.<1 / 10
= P (-1.50 $, Z ~ 0.50) = 0 .62.\7.
5.!>- LO P (477G $, X $, 4S5G) " 1'(4775.5 - 4829 < Z < 4857.5 - LI ~29)
\/ 4829 - - V4829
P (-O. 71 $, Z ~ 0.4 1) = 0.4385.
5.5- 12 T he distribut,ion of Y is b{JOOO, 18/ 38). Thus
P l Y > 500)" p(z > 500.5 - 1000( 18/38) ) ~ P(Z > 1.698) ~ (J.0448 . - )1000( 18/38)(201'18) -
5.5- }') (a) SeX) = 100 (0.1) = 10, Var(X) = 0,
P (11.5< X < 14.5) ::::::: 'll C,,·5;1O) _tj'CI.5;10)
= (N \.5) - i'I'{0.5) = 0.9332 - O.G915 = 0.2417 .
(b) P(X S 14) - P (X S 11) ~ 0.917 - 0.697 ~ 0.220;
(e) f: (I~) (0.1)'(0.9)'00- ' ~ 0.2244 . ..:= 12
5.5- 16 (a) E(Y) ~ 24(:\.[,) ~ 84, V" '(Y) ~ 2,1(35/ 12) ~ 70,
P(Y :?: R5.5);::;: I _ 1, (85.~ 84) = 1 - 4) (0.18) = 0.4286;
(b) P(Y < 85.5) ~ 1 - 0.-1286 = 0.~}7 14 ; (e) P (70.5 < }" < 86.5) ::::: ,j'(0.30) - il){ - 1.(1) = 0.6 179 - O.05:3i = 0.50·12 .
5.5- 18 (u)
12
10
8
6
4
2
o 0.1 0.2 0.3 0.4 0.5 0.6 0 .7 0.8 0.9 1.0 10 20 30 40 50 60 70 80 90 100
Figure 5.5- \8: Normal approxi mations of t.he p .dJ.s of Y !"l.nd Y/ lOO, P = 0. 1,0.5,0.8
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No OOflion oI1hlf; maleno1l mav be reoroducod k. anv loml Of j)\I arw moons 'Mll1OlJl oerm\sSklfl in writlno lrom th9 ouhllshOr
1'/1(' Norlll ll l Db".ilml if)1l
(b)"'ILI'IlJl U. I .
P( _ 1.5 < )' 10 < 1 . !,"~);:::: jl) (1:~5 ) _ (1'( :!1·5
) _ !).(j!Jln U.:UI.'ifl tl .:u.,:jO:
\ \ ' !It'li p = O.!'").
) ( l.5) (1.5) P ( - I .!'I < ) · !',O < I .:) ;:::: .!. T, - <II T = U.1I1 7!J U.: I~:l I = tI.:t: lfl.s :
"'hl'lI I' - O./) .
P (- 1.5 < }' RO < 1.5);:::: (11 ( 1}) _ (II( -I~.!'"') = 0.(i !(j2 U.:I!'",:U) O.2!J2·I.
5.!'i 20 P(X > ;f.') 1'( _,.'\_-_ 1_!'") > _:I :}-"'~' c-'2,--,,) !'", !'"I
;::: 1 (11(2.1) O.Ol7!) .
5.5 22 (n) Y 11I11i II P f)is."iU1i dis ll'ihllljlll1 wil li 1Il(';W :m.
( 1)) 1'(\ ' $ 25) 1'(_l'_ :_'O < 2,::;f" "",,-,;-::::IlJ) J:jjj - - J:jjj
;:::: (1' ( O.ltlJO) = 0.2057.
U~j .. g ~ Ii .. jlab, P(l ' ~ 2:') = 0.20$ 1.
5,6 T he Bivariate orrnal D is t ri b u t ion
5.6- 2 1/(.1' . y) oIY'--.!."C!,_-:l'f(a;.;,,,/.:o"'1)C:('C,' --":.:' ',,)LI2
{.r I' \ }l - :l' I 1
O"~ ( I fr" ) a.\
_ ,_ [( Y - II, ):.! 2p(,)' I, d(y 11 ,) I ,,2 f1 ~ a,a,
+ ,r( ,)' _, I' d 1 of ( ' _ Ii) (.r - ~' d ..! 1
(7, (7,.
= 1 ~ P' [( ~ )' -11'(" ::")(" :," , ) + (Y a,'" )']
r, ("' ) 5.G -4 (tl) E(l ' I X 72) 80 +- ,a W (72 (0) = 8 1:
( t. ) V,u'( l 'I X = 72) 160[1 C~, ) 'l = 1'14:
(e) 1'(\ ' $ . ,11 X 72) I' (z < ~) - 12
5.6 6 (n) P ( IS.!'i < )' < 2!'l.fl) = 'I' ( {I. ~) - .,,( 1.2) 1).linO;
(h) EP' IJ') = 22 .7 + 0.7$(3.!'"1/ 1.2)(J· 22.7) O.G!'"IJ· 1- 7.!)·1.'):
(e) Var(), 1.1') = 12.2:)( 1 O .78~) - ·1.7!)71 :
(d ) p ( aUI < Y < :.!!j .f) I X 2:1) = 11) ( 1.I~9) - .,. ( 2.007) 0.81'l18 0.0221 1).$(;0 1:
(c)P{IS.5 < ) · < 25.5 I X 2!'"1 ) = <11(0.506) .,, ( :l.GU) 0 .721·1 O.OUI7 O.71!J'i' .
7 1
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72
(f)
0.16
0 .12
0.08
0.04
~"'ig lL re &.6- 6: Conditiollal p.d .Ls of Y , given x = 21, 2:3 ,25
5 .6- 8 (a) 1'( 1:1.6 < Y < 17.2) = '.fl{0.55) - (\,(- O.35) = 0.3456;
(b) B( l' 1 x) ~ 15 + 0(4/ 3)(x - 10) ~ 15;
(<0 ) Var(Y 1 x) ~ 16( .\ - 0') ~ 16;
(d) P (13.6 < Y < 17.2 1 X ~ D.l) ~ 0.3456.
5.6- 10 (a) P (2.8U'; Y ,; 5.35) ~ <1> (1.50) - 4> (0) ~ 0.4332;
(b) B(l' 1 X ~ 82.3) ~ 2.80 + (-0.57) C~75) (82.3 - 72.30) ~ 1.877;
Va,( l' 1 X ~ 82.3) ~ 2.89\1 - (- 0.57)'1 ~ 1.9510;
<~ (2.479) - (\) (0.632)
0.99:34 - O.i363 = 0.2571.
5.6- 12 (a) 1'(0.205 S Y S 0.80&) = ~) ( 1.57 ) - (LI ( U 7) % 0.0628;
-1.St. - 0.60 (~) ('20 - 15) = - '2.55 : 4.5
1.5'{1 - (-0.60)'1 ~ 1.44 ;
0 ) Ix=20 1.2;
P (O.2 1 ,; Y ,; 0.81 1 X ~ 20) ~ •• (2.8) - " (2.3) ~ 0.008 1.
5.7 Limiting Moment-Generating Functions
5.7- 2 Using Table III wii,h ..\ = liP = 400{O.U05) = 2, P(X ::; 2) = 0.677.
Clwptcr 5
5.7-4 Let l' = L X;, where X I , X2 ,' . . , Xu arc nmi.mdly indellelldl'nt X2
( I} n\.l1d oJU variahlc:;.
;", 1 T hen It = E (X ;) = 1 and (72 = Vnr(X,) = 2, i = 1, 2, .. . , H . lIence
Y - lll ! }'- n J71q2 = .j2n
has fl liwit:ing distribution that is N(O, 1) .
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Chapter 6
Estimation
6.1 Sample Characteristics 4
6.1- 2 (a) x = - = 1.333; 3
(b) 82 = 88 = 1.275. 69
6.1- 4 (a) x = 1.711 , 8 = 0.486 ;
(b) and (d) graphs.
-[[]-
-OJ I-----
2 3 4 6 Figure 6.1- 4: (b) Female underwater weights , (d) Female (above) and male underwater weights
( c) The fit in the left tail is not the best.
(d) Females generally weigh much less than males underwater.
73
Q2005 Pearson Educa tion, Inc., Upper Saddle River, NJ. All rights reserved . This material is protected under all copyright laws as they currently exist. No oortion of this material mav be reoroduced. in anv form or bv anv means. without oermission in writino from the Dublisher.
74
6.1- 6 (0)
StewS i..cI.i\'C's Frt.'<IUc.ncy Ifi:'pt,hs
I 2
0 2
4 6
!l 15
14 ( 14)
S 16
2 8
5 r.
3* 10 :11 25 :l1 :1.~ 70 70 70 70 :1_ 81 ~3 90 90 90 90 90 00 92 II ... 00 00 00 00 00 02 02 05 10 10 10 l5 15 15
4t 20 20 20 20 20 25 30 30
41 4050 4 .~ 60 63 70 7:) 75
-I _ 80
Ta.ble (i.!- (): Ordered stcIIH).nd -]eaf diagram of Barry nands's hOl1le rim distalU.:t.-'S
(b) mill = :HO, 'iiI = a90, i11 = 405, q3 = 422.5. max = 480;
(c)
Figure 6.1 6: Dist.am·(,-'S uf Barry Bollds's homerull:;
(d ) The illtcrqllar~ ile mllgc is IQR =- 32.5. IHllcr felll':cs cOll ld be draW l! ilL 356.25 lilld .J53.75. Outer renel'S ('Quid be e1mwlI at 307.5 om\ &02 .5.
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&stiwa/ioll 75
6. 1- 8 (a) T ill' order stalisti<..1i Il.f(,: 5.5,5,7,7, 7,7, 7, U. 9. 9, 9 , 9 , 9. I I , II , j I. II. I~ , 1;.\, 13. 13,15.1.1), 17, 19,19, W, 19, 19, 2 1,21, 23, 2:l. 25. 25 , 25, 27 . 27. 33. :1;t :~5. :~.Ij, :n, :39,4 1, 4:1, 4 ~~ , 45. 47 , In, '\9, 51, 53, 57, 57, 01. 6 1. (j: l, Ij: ~ , 6:1, 6:1. G5, (I;), O!). 6!), u7, 71 ,71. 75, 75, 75. 8~1, 8:.1. 85, 87 , 89, 9 1. 93. 95 , 95, 10 1, 10'3, 121, 131 , I:H. 1;' 5 , 117, 2 1:1,241,301, 413,443 , 47 1,507,515, 615,703, 817, 18 1[1;
( b ) VI = .5, ijl = 35/ 2, ill = -1 8. ijJ = 113/ 2, YHl() = 1, 8 15;
(c)
- 1000- ' 15Oc) ..
Figure G.I 8: (c ) BCl ll let te dlHA. (d ) show ing fCllces and ouLlicr!l
(d ) ih ller ff' nceat In / 2 -j 1.5.(j9 I OO,ou t~r rt'IJ ('t'llt In / 2+3+6D = 29J.5:
(e) l' = 112. 12:
( f ) T he lUen ll is iufluClIccd Kreatly hy t he o ut lie r!!.
G. I lO (0)
Stems Leaves Frf'(IU('UC'V
II 805 12 :i8U 523 527 590 GOO 837 9:ro 7
l:l 008 143 112 217 300 :1:25 3 12 3·1:5 J5U .:125 6!l8 710 728 652 9~O 15
III 087 28)) 375 5U5 507 548 {j!j5 G61 801 875 oS!J:] 977 12 15 010 022 062 082 OSf. 11 0 1'13225260290 555 702!)5R 970 !JSO 9!J2 991 17
16 U :J 217 360 545 G23 8 10 gGO 9 17 9!JJ 9
1/ 088 120 :J2~1 122 857 o8:J 6 18 0,15 308 607 6·18 977 5 19 252 71:18 !JbO :1 20 392 1
(~ I u l t ip ly II ll lHll('fS hy III 2.)
Tahle 6. 1 1 U; Ordered .:sll'llI-llud-lf'llf diagram of f8('(' tinws ff)r 125 IIIa lc rUUllCni
(b ) lIlin = 118.05, iii = 1:i7.0J. "~I 150./2, ih = iG7.6:J25. max = 203.92 ;
Depthf;
1
• 2:5
35 (17)
2·1
" 9
4
0. Pe;i,tiOll EducaIlon. IIIC., Upper Saddle Ri_, NJ All rlgllIS reselV9d TlQmatefial is protected undef ... copyI\ghIlaws as they cumN#f eldsl '*' OOIIOnollhlsmaleriai m/!V boIlf!OKld\Jc8d. ... anY lorm or bY itIN mea,. wdhOuI Dltrmlssion In wriIina hom ~ DUbllshAf
76
6.1- 12
Clmpter 6
(c j
---II II f--
160 180
F'igUl'(' 6. 1~ 1U : R)u:c Limes for women (al)O\'e) aud !lIeu .
(uj Stems Leaves Prequellcy Depths
101 7 I
JO'l 000 3 4
103 0 4
104 0 ·1
105 8 9 2 6
106 1 3366778 8 9 {9j
107 3 7 9 3 JO
108 8 7
\09 I :1 9 3 6
11 0 022 , ,
(tlilu it iply !lumbers by 10 ' .j
Ta ble li.t L2: Ordered swm-aud-Ieaf diAgram of weights of indicat.or housiugs
(bj
-----1\ \ ~ Figure G. I~ 12: Weights of indicator housings
nUll = 101.7, iii = 106.0, tn = IOG.7 , if:! = 108.95, mn. ... = 110.2;
(c) The inLcrqua nile range in IQ R = 108.95 - 106.0 = 2.95. Tilt.' iiillC[' feuce is loco-ted fl.!. 106.7 _ U l(2.95) = I 02 .27~ 80 t here are four suspected Qu di t- rll.
02005 pearson Education. Inc.. Upper Sa(kle River. NJ All rlghlS res6fVf:ICI This rnal&rlal!s PlOlectOO..-,de, aM oopyf'igllf laws as they CII~ exI$I No oortton 01 lhls malariBl rnav btl r9Ol'odticf)C\ . .. anY 101m 0< bv anv meaJ\!l· wi!hou1 oermis'liloo In wrlllna hom the l'lI!bM:iIhe!.
EsrinmtiQJJ
6.2 Point Estimation
0.2- 2 The likdihood fU llct.ion is
[ 1 1 "I' [ " /"{O) = 271"0 eXI) - ~ (XI
The logari t hm of Ihe likelihood flUlCt.iou is
In L(O) = - ¥(LI211") - ~ ( l n8)
Sctting the fi rst. derivative e<]ual to 7.ero a.nd :'oh'ing for 0 yields
T hus
d in 1..(0) dO =
o =
6 ~t (X' - lj )2. " ' 00-1
To SPe t hat 0 is a n uubillscd f';5tilllULQr of 0, note llllll
£ (8) = E ~ '" " p = ~ . f) ( , "(X )' ) ,
' 1 L 0'4 1/ .-,
77
s ince (X. - /. )2/02 is:\, 2(1) and hence the expected value of each of t.he 1/ SUlIIIIIU II(h is equal to I.
6.2-4 (n) I = ;}9,1/7 5G.2857; S2 = r.452/~17 - r>G 20r.2;
(b) A = r 39 1/ 7 ~ 56.2"57,
(e) YlOS;
(eI) I. is lienee lhan .i2 OO,;nuse
V ( V ) 56.2857 0 57 3 6 895' 56.285712(56.2857 . <ISH 971 V (S')
ar . \ ::::: ~ = .' ,I < 5. v = 98(97) :::::: IIr .
6.2-6 9, = fi. = :~3A2{)7; 82 = ~ = 5.mJ80.
6.2- 8 (0) L(O)
In L(O)
dill /., (0) dO
- n 1 " 02 III 11 I, = 0 .. , o
1 " - - In n I,
" .. , 1 "
-- Lln J' •. n . ,
0 < 0 < ()..;
,. PMrson EduC:aIUl. Inc .• 1Jppef 5aOcI8 AMI(. NJ AI ngtllS 'IJ3Urved. TNs materi!ll ill protected uoaer a" c:opyngnIl;lws as they cu1f8fll1~ 8.'lIS1 tt,,)~ (Jl1hll malEK\ilI may b!Illln<OdlOO1~_ in 81lY rOlln orbv' IInv me/lnII WI.hOUl oormlllSioo In writ.no lfOnl lhe ouIJIlsh9r.
78 CIJlljlter 6
(b) We first find E(lll X):
E (11i X ) = 101
In x( I / O)x l/O-
1 lb'.
Using iutcgrnLioll by parts , with 1i = lu.e and lit) = (l/O)xI16
-ldx,
E(ln X ) = lilll [XIIO In.r - 0.£1 10] ' = - 0. /1 - 0 "
Thus
6.2- 10 (n) T= I/ psop = I/ K =11 /"5:::=1 X, ; (b) p (,"("\uuls Lue ulIlIlhcr of successe;, '1, divided by the number of I3cruoulli I,rials,
(e) 20/252 ~ 0.0794.
6 .2- 12 (a) E( X) ~ E(Y) / n ~ "p/ n ~ p;
(b) Vl:lr( X ) = Var(l' )/'I12 = np( l - V)/n'l = lJ( I - lJ) / n j
(e) E[X ( I - X )/"[ [E(X) - E(K ')[/" ~ (p _ [1" +p( l - l» / "Il / " ~ U,(I - I/ n) - ,,'(1 _ 1/ " )[/,,
( I _ l / n )lJ(J - p) j 1l. = (11 - i)p(l. - p)/11'!;
(d) Prom pu.rt (c), the CQllstaut c = t /{n - I).
(j.2- 14 (u) £ (cS) = s{~ [tn - ;)S~l 'I'} ~ (7-
~ _ _ _ <Lv Cq 100 ul /'2 u(" - 1) /2- l e- vI 2
.;n=1 0 ,"(II 2 1)2("-1)/2
W J2 f (,,/2) ~ I n - 1 q(" - 1) / 2)[ ,
;n=ll'[(" - 1)/21. so c = J2 r (,, / 2)
(b) When" = 5, c = 8/(:1 ../'5) a nd wlleu 11 = 6, C = 3,ffYi/{S .,j2).
C 2005 Pearson EducaliO!1, Inc .. Upper Saddle RIVEn, flU . All rlgtlt& f8Sefyed This malnlis plOllOCIed IlfJOOf all oopyf\ghllaWS as they CUlfllnlty exeS! No oortiorl 01 IIIIS maleriat may he rfIOfoduced. 1n allY k!fm Of bY!lJl\I m8<l1"lll . wllhouI!l(Ifmission In wrilina 110m Ihe DUblIsha.
E~,jnl8f jOIl
(c) , 1.
181 1.1 6
1. 14
1. 12 1. 10 1.08 1.06 1.04 1.02
1.00 0 .98 0 .96
We se(' t hu !.
. . . .. . . ... . . . . . ...
"--2 ' 4 '-6 B ' 10 ' 12 ' 1 '41s' 1 '8 '2'o ~i224 '26 'i8 '30 /I
Figll re 6.2 14: c ~lS a fli llct ioll of /I
lilll C -=- 1. tI_x.
6. 2- 16 "£ == uf) . I' o.O!;'I(1 timl. 0;::: I)/:i, a = :;:2/ s2. For lh ,~ Kivell data, 0 = 102. 1990,
o = 0.0058. Note that :i = 6. N , I ' = O. 1..J:t? /11 = 0.·16 17.
79
6. 2- 18 TI l{' CXIX'ri lll(, lI t has a hypergcoillclI il' distrill1l t ion wit h " = S ilud N = 6.. . Fl"Clll1 t ill'
MIl/pIe, :i"';' 1.4667. Using this a:s IlIl ('St.illlil lc for I I w (' Inwe
I AG(i7 (N.) 64 im pliL'fi till\!. NL = 11.73.
A gues.'I for t ile vil ille (If N. i>; l hl.'rd orc 12,
6.3 S uffici ent Statistics
6 .3 2 T he d istrihut ion of Y is POif;SOU with IIlt'a n 11>'. T hIlS, sillcl' Y EI"
which doe; n O I dCPl' ud W I A.
6 .3 " (a) ! (J:; 0) = , .(O-l) h, ;r+ Lu 0,
::it) K (.r) = In I ami t hus
x" x"I Y = y ) = (>.r:"'l..-U'\)/(Ji l I l ! . , . .r ,, !)
(1/..\ )11 e' ".\ / y!
y!
O< .r < I , o < o <oc:
" }' ....: L III X , = In(XL ... \ ':I", X n)
(b )
I' I
is II s lI fHc ipllt. stal.istk for O.
L(O)
II) L(B)
d Ill L(O) dO
J,I H (J + (0 - I ) In (J' LJ::I .... 1',,)
1005 PUISOI' EduI:8llon • • rw;; .. UppeI 5a!ldle RIval . NJ . An rights IfI&e<vfIO This ma1611I1L IS prolllCllld ur"ICIUf aM copyrlghl !aWl! U 1118)' culfenltV exl5l. "DOttM oIlhIs ma.eM! mav be reorDlJO(:ed In an\llonn Of bv flrIV fTl6lllIS. wImoIJ1 oennlssIon In writin() from The OUblilllWf
.....
80
t 6.3- 8
CII8]Jter 6
Hence
which is u fUlIct.ion of Y. (c) Since n is ~ siugle valued function of Y wit.h !\ single \1l.llICtJ inverse , knowing t.he val ue
of 0 is eqniVl.d\!ut 1.0 kuowiug t.he value of lr, !U\d hel l(;C it. is sufficicut.
( :1"1:1"2 • . • X .. )'t - 1,.- 1: r ./0
If ( <> lI" 0""
(;'~ /O) CXIXI~(~~::)· - ') .
The se(;c,ud factor is free of O. Tile firSt, f,,!"tor is u funct.ion of the .r ,8 th rough :[:'",,1 .J:,
un ly. so L:::"I X. is f.l sufli cient sLllListic for 8.
(b) III L(O) In(:r ,x2" . x,,)o - I - L::''''I :r.;/O - In\r(a)]'' - cOlln()
d in L(O) dJJ
t\ uO
",,, :£ /02 _ o-n / O = 0 L..,..= t I
,~" ~'=IX;
-' t,X, o-n .. ,
.. Y = L Xi hM a gamma distribut.ion with p..'tNI.1J1Cters (1'11 Ilnd O. Hence
. = 1
B(c'Z) = J~ . - 00
- , £(0) = -(<>" 0) = O.
on
Le t l·,/ .,fO = y, . i = .1,2 , ... ,n. T he .I ucobiall is (JO )n . H~J lcc
which is fn .. 'e of (J . Sincc t.he dist.r ibutioJl of Z is fn..'C of 0, Z and )'
sunic-ient statistics, are indcpcmlellt ..
"," X ·' I ~,= l ;, llC
6.4 Confidence Intervals for Means
6.4- 2 (3) 177.272 , 92.7281; (b) [79.12 , 90.88[; (c) [80.065 , 89.935[; (d) [8 L15" , 88.8461·
6.4-<1 (a):r. = 56.8;
(b ) [56.8 _ 1.96(2/ Jlo ), 56.8 + 1.96(2/ Jlo I = [55.56 , 58.04\;
(c) P( X < 52) = p( z < 52 - 256
.8
) = P(Z < - 2.4) = 0.0082.
Q 2005 PI.I81SQ11 EdllJCll1lon. 11"1(; .. Upper 5aQ(Ma RI~. NJ. AU rIghIs reserved This mal(l1.11$ prolectea undo. ali copyrig/lllaWS as they CUlIllIlIIy IIxIsI No 00<1100 ot this malmial may b6 roomtl'J(',ad.. in IIIlV lorm or bv snv 11\00116. w\tho<Jl oermi!lSlQ(l ln wrilloo lrom Iha ollbllsher
E SfjllWL iou
6.4- 6 [ (".80) 11.% - 1.96 .jTi . (11.80)] I L!:)r) + 1.96 v'37 ..: {H. I!"). 1~ . 7!)1·
If rll(lfC pxtl'usiw' t-tahk-s I.trl' nVl:li lolJlc or if fl cOInj)utl' l' program is 11 :>1'<1. we have
[11 .% - 2.U28( I~) , 11 .% + 2.02S( I~U)} .;: 18.0 16. 15.$M-l j.
l1.<I- 8 (8) :r '"" 'I(iA2; (b ) 46.72 ± 2.1a2s/ V5 or pO.2li. ::i2.nSJ.
[ (0.31)) (j,4 - 10 2 1.'15 - 1.:l\o.I J2S .()... ]2 1.373. oc).
6 .4- 12 (1I) :r = 3.5hO;
(b) , ~ 0.512;
(e) [0. 3.580+ 1.83:J(D.512( JiO [ ~ [0, :1.877[.
6.4 l 4 (0) 1':::: 2'1[').80,.;0 = 23.64 , so 11 !J5'X cOlllidt'llCC intervlll for II is
{245.80 - 2. 145(2:1.6 1)/ Ji5, 2¥).lSO + 2. 14!"1(2:l.6/1)/ Ji5 ] = ]232.707, 258,,,,93!;
(b)
-1 11--1 220 240 260 280
Figure 6.4 \ ·1: Box-aud-whi.skcr d illgnull o f sigmlis (rom d('t4..'ctorb
(c) 'fbe .stalJdurd dC\'iat iou is qui t£' large.
6 .4- 16 (a) (:f + 1.960/ J5) - (:r - U)GI1/ .J5) = :1.92(7/ .j5 = 1.7530';
(b ) (1' + 2.7708/ ./5) - (x - 2.77G.t;/ .;5) = S}i52s/ .j5.
81
From EXt'rci~' 6.2 14 wit h " ~ 5. £(8) _ J2 r (.'( 2)o ~ Arli( 2)
O.!J·lo . :10 I hat
E{5.552S/ v'5 1 = 2.3:\4(7.
6 .4- 18 Ii .U5 :L2.576{OJJ2)/ J I2J9 or [(;.049 , G.051j .
6.4- 20 (u) T:::: ,1.48:1, .9'l = 0 . 1719, .:-; ....: O.'ll4(;;
(b) 1 " " 8~ - 1.714(D.4I4o)( v':i4), 00) ~ [4 .:138, <>0);
!OO!.PNrlOll Educ81ioo. 1nc_. lJR_ 5atXIe AIveo-. NJ All figlws fese<wd Has malerial ifi prolQCI9il unci. all copWrlght laws a s lhoy currendw elllsi DOftI(Jn or dUlTllllenel fMV be reorodU/:ed In IInv form Of tw allY moe ...... ""rlhOIlI oeflTllS!lofo in ....-nuna hom ihIIllIlI:*IIhar
ChapIn li 82
(c) yf!S; cons truct a q_(/ plOL or compare empirical o.nd theoretical distribution fu ud ious.
N(4.48, 0. /7/9) q(Wllliles 1.0
521 5.0
4 .8
4 .6
4.4
4 .2
4.0
3.8
3.6
0.8
, 0.6
0.4
. / . , 0.2
, • ,-..---.---.--, ,_,--.--~ . --.-- ,..-, -,-- x 3.6 3.8 4.0 4.24.4 4.6 4 .85.05.2 Figure 6.4-20: 1/-(1 plot ami a com parison of empirical f\.nd t heoretical distrihuLiol1 fu nd iou.::;
6 ". il Confide nce Intervals For Difference of Two M eans
6 .5- 2 x = 530.2 , .~; = 4.9-18.7 , Y = 544.625, s; = 4, :\27.982 , .if" = G7,481 , 10.05(11 ) = 1.79G,
so the confidence inlcrvtll iR \- 74 .517, 63.607\.
6.5- 4 (n) x_ Y = 15lL7\ ,I - 1118.400 = 393.3\ ,j; ( b ) $; = 49, 669.905 , ,~; = 15, 297.(}00, ,. = L8.599J = 8, 10015(8) = 2.:m6,';0 the
(;(lIlJi.denec interval is \179.148, 607.480J.
6.5- 6 (a ) 7 = 7t2.25, y = 705.4375, s; = 29 , 957.8,109. $; = 20 , 082.1292. sp = 155.7572. tOO'l6(26) = 2.056. Thus I). 95% confidence illterval for Jt .~ - II.,. i!:i \- 115.480, 129.105].
(b)
x
y
900 labo figure 6.5- 6: BOX_find_whisker diagrams for hutterfat pnxiw:':Lion
(c) No.
6 .5- 8 (a) x = 2.5.8:1. Y = l.oG4, .~i = 0. 1042, s~ = 0.0'128 , sp = 0.27 1 L t oo~!i(18) = 2.101-Tilms 11 95% coufidcilce intCn'al for 1.1 . ..: - ILr is IO.7G5:!, 1.27'17].
02005 pearson Eth.lCluion. Inc. lJppfIf SaCIdIe RiYEIf. NJ. Afl righls res&rVOO. This material jB prorecled undor aU copyroght laws as \hey cuf'l'tll'lll)' uisl No OIlflion oI1hls mamrlill ma~ be rfK'l(Odllc9(l in 81l\1 100n Olb~ <trW means. wiThout OIIrmlsllion in wri\ino fmm the oubllsher
EsliJIIll , joIJ
(b)
x 111--1 -
yID-
Figure (to 8: Ilox~ruld-whisk(' r diagrams, w(·dge on (X ) fi nd wedge o ff (l")
te) Yes.
6.&- 10 From (a), (h), a ud (e). w(' klLow
x - Y - (Jl ~ - II,) . (d) ' c
do? o~ -+-[
(U - I )S~ I ' " , a,
(m - I)S' 1 / 0; I (II + m - 2)
" m lIa ... a t (1I HII- 2) distributiou. Clearly. luis ra lio does 1I0t dcpeud upou 0:: 90
(11 - l).'<!ld I- em - I):;; (d I) -+~
11 + lfI 2 I, 111
provides it 100( 1 - 0)% confidence iUlen-a] for 11 .\ - ll" .
6.5- 12 (u) d ::..: 0.07875:
(b) [d - 1.71 .IO.25.192/ v'f.i, 00) ~ [-0.010' .00); (e) not. lLeccs.<;llril,\,.
6.6 Confidence Intervals For Variances
6.6- 2 ror I hf!SC !) weights, 7. = 20.90, s = 1.858. (a) A lX)illt estimate" for (7 i.i .~ = 1.d!.>8.
(b ) [1.858Jil 1..58Jil ] ~ [1.255.3.5tm[ J I7.5.1 ' J 2.1BU
[1.1l-'>ilJil 1.858Jil ] _ [ .. ? ~ j' =' """'" - 1.131. ,L·I, , v 2 1.595 v 2.62:S
[ 1."58Jil 1.85"Jil ] . . (c) ,,",,' "'""" ~ jI..l34 ,.1.179J v 15.5 1 v2.7:l3
0'
$:S
_. PMIson EcU:abOn. Inc • Upper ScKkIe An/'EII . NJ. AI righIs reserved. 1m matena •• proMded mOer .1 CXJPYI1Ijf1I a- iI$ .hey cureody eJlSt. lfDeortion oIlhia mII leriai may lie flIClIllduced. ~ arwkwm orbv all\l rna;l ns. wrthouI oemltS9ion '" wrilino'rom IhillllIhIIsher
84
6.6-4 ( ) [ 1l(0.2372) 11 (0.2372)] ~ [0 19 0 ' }.
n 2 L92 , :1.816 . 1 . > .6<:>4 ,
(b) [J{ITl9, VO.684 ) ~ [0.345, 0.827[;
(e) [ 11 (0.2372) 25.(;48
11 (0.2:172) 1 ~ 10.320 , 0.772}. 4.377
6.6-----6 (a) S ince E (e'x ) = (1 - Ot)- I ,
Ele"'S/8l} ~ [1- O(2, / OW' ~ ( I - 2, )-2/',
t he snou'Wut. gellcfUling fUllctioll for X'2(2) . Thus W is t he s um of )'I illdcpendell t ,\ 2(2)
variables and so IV is ,\ 2(271).
( 2"" X ) (?"" :( .,,,,, X) (b) P X2 (2n) < L..=I ' < \ 2 . (2/i) = p · L,. ;- L" , < 0 < . L.. , I • .
l- u/ 2 0 - a /2 v2 (2n) - - , '2 (2,,) '\.0 / '2 .... 1- 0,;2
Thus, l\ 100(1 - a )% confidence interval for 0 is
[2L::'. , x, 2L:: ,x, 1 X~ /2( 2'/'t) ' xLo:/:l2n) .
( ) [2(7)(93.6) 2(7)(93.6)] ~ [5534 19942}
c 2:t68 , 6.57 1 . ,"
6.6-8 (a) s;ls~ = 0.0040/ 0.0076 = 0.526:3;
(b) [" \ 8) s; , Fo..,o(8, 9d] ~ [(~6) (O.5263), 4 . 10(O.526J)] ~ 10.121 ,2. 15A}. 00'2& 9, '~y .'f l/ 4.3
(j.6- tO A 90% coufidence interval for a;' /a; is
So a BO% ('onl1dcucc interval for a,, !u\. is given hy t.he squa. re roots of t hese values, uamely
10.383, 0.1J76}.
6.6-12 (a) [_l_(~) 3. 115(~) ] = [0.589 5.719]· 3.115 329.258 ' 329.258 "
(b) 10.77, 2.39}.
6.6- 14 From the resLrict.iou, tJ"~~lt.iug b as 8 function of 0 , we have
or, tXllIivnient iy,
Thus
f('quires d Ull
or , e<jlli va\PIILly,
db g(l, ) -I - g(a ) ~ 0,
, a
db 9(a) da ~ y(W
dk r-c-7 ( - 1/ 2 - 1/ 2 g(a») d(t = s vn - 1 a3/2 - 1J3/ 'l f} ( b) = 0
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£sfiulllfioll
[ 1 (2Y T957.X.tI ) . ' (29, !l57.MI)] _[ , . , [.
O.O- i6 (n) a.o1 20.082 .129 , .l.J5 20, 0$2.129 - OAJ(i , 1.!l97 .
The F \'nhK'S were roulld using Table VII a.nd liuf'M illtf'rpolutioJl . 'fhe rigll t cudpoiut is 4.%8 i f FU02S( lf, . II ) J .:l:\ i:. "sed (roll nd lI!'iug i\liuitab).
(b) [2~2(~~~~~) ' 2'2G~~~~)] ~[O.894 , "6801; U · I' I ' 0 ( 9 9) 4(2.46) + 2.76 2 'I . 1 slIlg Hlf'llr iuterpo a t loll ; 1"0.021> I ·, It -::::: 5 - .52; IIs illg 1\ illltfl);
FU 02::;( I!J, J!I) = 2.[,26;;.
[ 1 (0. 10.116) (0.1().j16)]
(c) 4.0:1 iJ.04283 , 4.0:i 0.0,1283 = [O.60:i , !.I .BOI ].
6 .7 Confidence Intervals For Proportions
6.7 2 [u.7I 1.645 (0.7~~. 29 ), 0.71 + 1.645 (0.7~~.29) 1 = /0.66 , 0.76/.
6.7-4
6.7-6
[0.70 - 1.96 /(0.70)(0.30) 0.70+ 1.96 /(0.70)(0.30) 1 ~ /0.674 0.726/.
V 12:\1 ' V 1234 '
[0.26 _ 2.:126 (0.26)(0.74 ). 0.26 + 2.326
f)757 (0.26)(0.74 ) 1 ~ /0.2<7 1).27'1/
5757 "
6.7- 8 (a ) fj = 3880
? = 0.:.1796; 1 _2
6.7 to
6.7- 12
(b) 03796 ± 1 61' (0.:m6)(0.6204 ) • . j i) 1022 or [0 .3546, DAO.WI.
. (0.'8)(0.'12) (a ) U.58..1: 1./.i45 00 or [U.5 1.1, 0.(; 161;
5 0.0·15
(b) = 2.0·, ('orrcfipo llcls to nil I)pprox imatc 96% ("oufidC'IICc h.'wi. J (0 .58:'42)
(u) PI = 206/ 37'1 = 0.551. fi2 = :n S/42(i = U.793:
(b ) (0.5'>1)(0.449) (0.79")(0.207)
O},{i l 0.79;1 ~ 1.96 :174 t- 426
- 0.242 ± O.(}(j;J or [-D.aD5 , - 0.179]. II
! 6.7- i4 (1I) iii = 2M/1 9,1 = O.1 'I"{ ;
~~~~ (b ) O.144± 1.96 / (0. 144)(0.856)/ 19.1 or /0.0%, 0.193/;
(c) fit - fi2 = 2~/ 194 - ll / 162 = 0.076;
( I) [0076 - I " 'I" «1.144)(0.8.;6) + (0.008)(0.9:12) ,1 0' [0.044 , 1[. ( . .v " 19,1 162 '
6.7- L6 PI = 520/ 1300 = 0. 10, ih = aH5/ ll00 = 0.35 ,
0.40 - 0. :~5 ± 1.9G (0.'10)(0.60) (0.35)(0.65)
1300 + 1100 or [0.01 1, 0.0891·
1005 P1IiIf9OI"l Educ8\.On. Illc .. Upper 5add:e AI~eI. t<il. All righU III6ef\lOO. This tllBt8f\a115 PlQtaalKl uncI&r at oopyrigl_ 18_ a. ~ CUiIBNty elClll IICIIIPl oI\tli1 mII\8fIaIlT\iI~ b8 reoroducII!Id. In _ klIm 01" bv I!Ifl\I moan:ll. wIthouI OBITIlIlI:IIion 11'1 WrilIIlO 110m the oubIisho.
86 Chapter 6
6.7-18 (a) fiA = 170/460 = 0.37, fiB = 141/440 = 0.32,
0.37 - 0.32 ± 1.96 (0.37)(0.63) (0.32)(0.68)
460 + 440 or [-0.012,0.112];
(b) yes, the interval includes zero.
6.8 Sample Size
-2 = (1.96)2(169) = 288 5 hid d' 289 6.8 n (1.5)2 . so t e samp e size nee e IS .
(1.96)2(34.9) 6.8-4 n = (0.5)2 = 537, rounded up to the nearest integer.
(1 96)2(337)2 6.8-6 n = . 52' = 175, rounded up to the nearest integer.
30 1.962(0.08)(0.92) 6.8-8 If we let p* = 375 = 0.08, then n = 0.0252 = 453, rounded up.
(1.645)2(0.394)(0.606) . 6.8-10 n = ()2 = 404, rounded up to the nearest mteger.
0.04
(1.645)2 (0.80) (0.20) 6.8-12 n = ()2 = 482, rounded up to the nearest integer.
0.03
686 2.3262(0.6799)(0.3201) 6.8-14 If we let p* = 1009 = 0.6799, then n = 0.0252 = 1884, rounded up.
(1.96)2(0.5)(0.5) . 6.8-16 m = ( )2 = 601, rounded up to the nearest mteger.
0.04
601 (a) n = 1 + 600/1500 = 430;
601 (b) n = = 578'
1 + 600/15,000 '
601 (c) n = 1 + 600/25,000 = 587.
6.8-18 For the difference of two proportions with equal sample sizes
p*(l - p*) p*(l - p*) E = Za/2 1 1 + 2 2
n n
or
For unknown p* ,
1.2822
So n = 2(0.05)2 = 329, rounded up.
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8lillla rioll
6.9 Order Statistics
6.9~2 (a) The locntion of the lIIediun is (U.5)(17 + I ) == 9, thtb the IIIc(HtUJ is
m =5.2.
Tilt" loca t ion of the rust qua rtile is (0.25)( 17 + I) = 4.5. T hu$l il{, fiN! (IUurtil l' is
ii, = (0.5)(4.3) + (0.5)(<.7) = "a. The ioC'-atioJl of t il l' t hird quartill' is (0.75)(17 ,. I) = 13.5. Thus t li(> third quanil t, is
q, = (0.5)(5.6) + (0.5)(5.7) = 5.65.
( b ) T he )oca t io ll of tht' 351h pe rc('I)tile is (O.;j5)( IS) = 11.3. Thus
"" = (0.7)(4.8) ,. (0.3)(4.0) = ·1.83.
The 1000atiOli of til(' 65e/. pprcellLi le is (0.65)( 18) = 11.7. T hus
'0" = (0.:1)(5.6) + (O.7)(f>.O) = 5.G.
0.9-4 gig) = t {k!(G6~ k) ! (kllFly)I'- ' f ly)1 1 - p (y)/6-' l'",, 3
+ k'(66~ k)! IP(y)I'(6 - k)(I - F(y)/6-'-' 1-JlyJl } + 6IPly)/' fig)
6' :.! I ;j 6!1 3 , 2!3! (P(y)1 Jly) I - Ply)/ - :1!2! P(y)1 P - F(y)1 f ly)
+ 3~~! IP(y) /' f (y)P - P(y)I' - 4~ : ! (Plg)/'P - Ply)I' fly)
+ 4~:! (Ply)I'flg)(I - Ply)I' - ,,~~! (PlgJl'P - PlyJl' fly) + GIPly)I' fig)
= 2~~ ! (PlyJl' P - Fly)I' Jly). a < y < b.
6 .9 (1 (a) I, o < .r < I. Thus
0 < w < I;
"(Wl" - I( l ). 0 < IV < I.
(b ) E( IV,J l ' (W)(1I )(1 - u,),, - I dw
6 .9- 8 (a )
(b)
= [ W(I_w)n __ ' _( I _ W)n+ lj' =_1-. 11+ 1 0 u+1
EI IV,,) 11 (W)(1I) ll,n - ' (/", = [-"-w"+'j' = -"-. {j fl+ l 0 11+ 1
£( 1";) 11 II! = (1).1 ( ) ' ( ) ' wr- I(I _ w),, - rdw
o r - i .rI-r,
r(,'+ l) /. ' _0'("""+,,2,,,):;.!,, ""'I )"-" " w --. l. - flO (/w (fI+2)(1If- l ) 0 (r + l )!(lI-r)!
r(r + I )
(11 -+ 2 )(/1 + I ) ~in('e t he integrand is Uk(' that of a lul .f. of t il\' (I' + 2)tJI order statbti<' (.f a $.lIl1ple of sizt, Ii 1-- 2 a nd hCIl<'C the iuttogrnl LIlu:;t l,<\unl one.
Var(Wr)= r(r+ 1) /,2 r(l, - r+ I) ..
(II + 2)(11 + I ) (II + If (/I + 2){u + I f
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88 Clwptcr 6
6.10 Distribution-Free Confidence In tervals for Percentiles
O.LO- 2 (a) (Y3 = 5.2, !flO = G.6);
(b ) (YI = 4.9 , Y7 = 6.2 );
P(Y1 < 71"0 .3 < )'7) = t C:)(O.3)k{O.7)11- k II "", .
= 0.96 14 - O.O l :j8 = 0.9476.
using Table II with /1 = 12 aud p = 0.30. The imcrVllI is (YI = 4.9 ,117 = 6.2).
6.1.0- 4 ( a ) (Y.I = 80.28. Ull = 80.51) is a 9 . .1.26% confidence iu te.n:a t for I/l. .
( b ) (YG = 80.32, Yl:l = 80.5:j) ;
f= C:)CO G)'COA)'H = t C:)t°4)'COG)"-' 11=0 k=3
0.9417 - 0.0398 = 0.90lD.
T be inLcrw l is ('lJij = 80 .32, YI2 = 80.53).
6 . LG-6 (a) We first lind J llnd j so t hat pet . < 11"0.:1& < V)~ ~ 0.95. Let ~he d is tribution of \V be
b(S I , 0.25}. Then P(Yi < 7Tu.25 < Y,) P (i $. IV :S; .1 - 1)
If we lei.
(i - 0.5 - 20.25 Z ) - 1 + O.f) - 20.25)
p ~«~. v 15. 1875 - - v 15. 1875
-i. - 20.75 ~ = - 1.96 tmd
,, 15. U375
j - 20.75 6 = 1.9 J 15. 1875
w(: fiud that i ~ 13 and j F:;: 28. F'u r tllerillorc P ( I:l :S; W $. 28 - I) ~ 0.9453. Also
note that. tbe poi nl. estimate of 1T0 .25,
falls Ilear the ccut.er of t his illM:!r\'o.l. So 1.\ tl" .5:~% confidence interval for "11"0.25 is
b/l:l = 2 1.0, Y28 = 21.3).
(b) L~L the dist. ribl ltioll of \\' be &(8 1, 0 .5). Then
P(Y; < rrO.f> < l's2-i) P(i ~ HI :$ 81 - i)
p(1 - 0.5 - 40.5 < Z < $ 1 - 1 + 0.5 - 40.5). J20 .25 - - J20.25
If i - 41 __ = - 1.96, 4.5
lhen I = 32.18 SO let i = :!2. Also
:8,,-1 _-..c',-. -...;.:' 0 = \.96 4.5
il11plies that i = 32. Fu rthermore
P(l'n < "11"0 .6 < Y50 ) = P(J2 :$ \11 $. .\9) ~ 0.95·14.
So lill approxi uul.t(' 95.44% confidence ime rvu.1 for 11"o.s is (Y32 = 21.<1, YbU = 21.(; ).
(c) Sim.i lar to part (a), P(Yf,4 < 1I"07S < ~'69) ~ O.94 f>:1. T llusn 94 .f>3% confidence iutervlll
ror 11"0 7[0 is (YM = 2J .6 , Yfi9 = 2 1.8).
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Estimation
6.10-8 A 95.86% confidence interval for m is (Y6 = 14.60, Y15 = 16.20).
6.10-10
6.10-12
(a) A point estimate for the medium is in = (Y8 + Y9)/2 = (23.3 + 23.4)/2 = 23.35.
(b) A 92.32% confidence interval for m is (Y5 = 22.8, Y12 = 23.7).
(a) Stems Leaves Frequency Depths 3 80 1 1 4 74 1 2 5 205173 73 92 5 7 6 01 3132525758 71 74849295 11 18 7 08 22 36 42 46 57 70 80 8 26 8 03 11 49 51 57 71 82 92 93 93 10 (10) 9 334061 3 24
10 07 09 10 30 31 40 58 75 8 21 11 16 38 41 43 51 55 66 7 13 12 10 22 78 3 6 13 344450 3 3
(b) A point estimate for the median is in = (Y30 + Y3t)/2 = (8.51 + 8.57)/2 = 8.54.
(c) Let the distribution of W be b(60, 0.5). Then P(Y; < 71"0.5 < Y61 - i ) P(i ::; W ::; 60 - i)
If
then i ~ 23. So
~ P <Z< . (i - 0.5 - 30 60 - i + 0.5 - 30)
v'l5 - - v'l5
i - 30.5 = -1.96 v'l5
P(Y23 < 71"0.5 < Y38 ) = P(23 ::; W ::; 37) ~ 0.9472.
So an approximate 94.72% confidence interval for 71"0.5 is
(Y23 = 7.46, Y38 = 9.40) .
(d) 7r0.40 = Y24 + 0.4(Y25 - Y24) = 7.57 + 0.4(7.70 - 7.57) = 7.622.
( e ) Let the distribution of W be b( 60, 0.40) then
P(Y; < 71"0.40 < lj) P(i::; W::; j -1)
~ p(i-O.5-24<Z<j-l+0.5-24). v'14.4 - - v'14.4
89
i - 24.5 j - 24.5 . If we let f1':4"A = -1.645 and f1':4"A = 1.645 then l ~ 18 and j ~ 31. Also
v 14.4 v 14.4 P(18 ::; W ::; 31 - 1) = 0.9133. So an approximate 91.33% confidence interval for 71"0.4 is (Y18 = 6.95, Y31 = 8.57).
6.10-14 (a) P(Y7 < 71"0.70) = t, G) (0.7)k(0.3)8-k = 0.2553;
(b) P(Y5 < 71"0.70 < Y8) = t. G) (0.7)k(0.3)8-k = 0.7483.
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00
G.ll 0. 11- 2
A Simple R egression Problem
(0)
(b)
x
2.0 3.3
:n 2.0
2.3 2.7
4.0 ~t7
:l.O 2.3
29.0
ii
y :r2 Ty y' (y - iiJ'
1.3 '.00 2.60 1.(;9 0.:16 1716
::t3 10.89 10.89 10.89 0.040701
3.3 13.69 12.21 10.89 0.027725
2.0 '.00 4.00 4.00 0.009716
1.7 5 .29 3.9 1 2.89 0.228120
3.0 7.29 8. 10 9.00 0.206231
11.0 16.00 16.00 16.00 0.006204
:1.0 13.69 11.10 !.l.OO 0.2 17630
2.7 9.00 8.10 7.29 0.014900
3.0 5.29 6.90 9.00 0.G763\0
27.3 89. 14 8:.t8 1 80.65 1.8'19254
y = 27.3/ lO = 2.73;
8:1.8 1 - (29.0)(27.3)/ 10 ~ 4.64 ~ 0 9200' 89. 14 - (29.0)(29.0)/10 50·1 . ,
2.73 + (4.64/ 5.04)(f - 2.90)
y
4.0
3.5
3 .0
2.5
2.0
1.5
Figure 6.11 - 2: Enrnt.·d grade (y) veniUS pn .. >dict{.>( l gmdc (x)
(c) /1:1 1.8~~254 = 0.184925.
Glmpter 6
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08
0.6
0.'
02
-0.2
-0.'
-0.6
E. .. , hUll I iOIl
6. 11- -& (a )
(b)
y = 0.9810 + O.0249x;
y 7
6
5
4
3
2
.~ ... ~ ............. ,........ ... ~.....-.-.... -. ", .~ .-~. , ~ ... ,., ,\ 20 40 60 80 100 120 140 160 160 200
9 1
Fig!!!'!! 6.U 'I: ( b ) Millivolts (y) versus knowLi cQlweulrSl..tions i ll ppm (x)
(e)
7
6
5
•
,.
3
21-_-
2o~40 '6'tfoo ';()(YT2014o'ioo'1'so-'200 Figure 6.11 4: (c) 1\ J'~'S iduaJ pint along witll fl quadl'ntic regret>.<;iou liuc plot ( Exl'n:ise 6 .12 15)
TbC' ('qualiou of til(' quadml ic regression liuc is
y = 1.7350-.1 - O.OOO377x..f 0.000 1 24x .l.
~ PearSOll [ aucalloll, 11lC .. Upper ~ RWM. NJ All rig/lIS rllS6<'Wd ThIS material III prolllCtOO unoer .. 1 oop~ IIIws.,!hey culfllf1lly 81U51 oo.uon 01 .. maluriallTlllv be llIOfoducild ... anY lonn or bv lillY means withouI OI'IITOIS.<Uon In wntIna hom the nublillher
"
92 CJJl:lpter 6
" L I(a - o} + (ii - IJ](x. - x } " 6.11- 6 L It'; - (I - p(x; - xW ;=1 . = 1
" + (Y. - 0 - ii L (x. - xlii'
" n(ii - 0)2 + (jj _ Ii)".! L (7i - :r)'1 ;=)
" + L[Y' - & -.8(X, _ :;;)]2 +0 . • = 1
T he + 0 ill the above expressioll is for the three cross prod uct terllls and we must Rt ill
nrguc that (lacta of these is indeed O. We havl."
" 2(ii - n)(il - IiI L (x, - xl ~ 0,
1=1
~ O.
siu(;e
6.11-8 P [.i- .. n(,,- 2) ., ";,' ., '~/,("- 21] ~ I - "
,[ ,,;;'i , ,,;;'i]_ J :.! . :5 0 S :2 - I - o.
An /:2(n-2) XI _ n/ :2(n 2)
G.ll - IO Recall i.hat a = 2.73, jj = 4.64 / 5.04, 0 2 = 0. 184925, '1 = 10. T he (!uclpuiut8 for the 95% CQllfidc lWC inlervnl ore
)0.1 4925 [ I 2.73 ± 2.:106 8 or 2.a79,3.08 1 for 0:;
1.84925 4.64 / fo.04 ± 2.~lO(i - (-- or [0.42(;8, 1.414~1 for {ii
8 • . 041
[1.84925 1084925] _ Ie ' I r '
17.54 ' 2.11'30 _ 0.10", 0.48 oro .
Cl2005 Pearson EducarJoo, Inc .. u~, Sad<Ie AlVeI, NJ. All rights IfIOOfV'ed This mal(!liall9 prolOClOO llnda, an copyrlghllil_ as they eurren~ylW51 No 00fI1Dn of this maIO""! rmv bII reoroduced. In IIOYIoml or bv /lnv mBllrt!l. wilhcxn oeflllls5iOn 1n WTiIlna trom 11111 oubIl&her.
Es,illlll,jU/I
6.11 12 (n) ,1 ( 12!).I ) ( 110)( 121)/ 12 IS.t .S:\:\ , - --- - DIS I !J·
(12:1.J ) ( 110)2/ 12 - 22S.(i(ji - , "
(b)
(c)
" 121
12 I S·I ,g:r\ (
10.uttl ! 22~d.i67 J '
::: UJsHb: I 2I.7!):
18
16
14
12
10
)'
~) 12
, • r •• , , , , , • , , , ••• , • .l
2 4 6 8 10 12 14 16 18
Figlln' G.II 12: CO (y) "Crs'l~ lar (.r) fur 12 Ilt'a llds of d J!:ItI'l'tlt'S
I/;ii
W.Os:\. ,1 = D.xl!).
I ·HI
~\!).r)2.s0
12
O.S I !Jll~I ( 1 2U I ) I U.8 1!J05( 11lI)( 121)/ 12 :\U.52h!';
:t2!J I.
(d ) TIl\' ('udpoillt!'> for ~J :'X ('ou lid('I1I'1:' illWI'Vlth, arl'
J:t2/j I IO .lIl'1:~ ± 2.228 IU 01' [8.S0.1. 11.:m:lJ for II :
:1!),!)2XU il.oS1!.) ± 2.221'\ 17",,:ii:O;~ 1)1' [0.52.1. 1.114\ for ,1:
IO(225.G67 )
[:m.52$!J
20.48 ' :\0.5289] --- = 11.9:10. 12.17-1j for :t2.Ji
020D5 PNI30Il EducallOl'1. In<: . Uppar SacX.IIlI River . NJ All nglllS nll5ervlld. This '!\aUlnal ~ pI'OIecliId undeI •• c:opyrigIlllaWS as llley CUlI6f1l1y e:o&l itlOOl1lllol 0I1his mlllllriill may be 11IOfI'XIue8d. II'IIIIN IOfm Of by IInv maal1!l WI'haul oermis.<;onn III wnllOll trom lile n.lhlisher
94
6. Ll- 14 (8) (; :$95 Tn = Z6.:1:n,
(b )
jj = 9292 -(346)(J95)/ 15 = ~ = 0 6 8338 (346)'/15 ' 56.933 .50,
Y~ ?G 333 1 SO.667 ( _ 346) _ . + 356.93J x 15
O.5()(ix + 14.657;
y
32
30
28
26
24
22
20
18
16 .----r- , ~.-,--,~_, .........-.--.- .----. ' .--- x
16 18 20 22 24 26 28 30 32
Clillpter 6
Figmc 6.11 14: ACT nat.ural science (y) "er:loUS ACT so('.iui science (J:) scores
(c) 26.3:1, Jj = 0.506,
3952 10,705 - _ _ - 0.506 1636(9292) + 0.5061636(346)(395) / 15
15 = 211.8861 ,
211.886 1 15 = 14.12G.
(d) T he cudpoints for 95% confidence intervals arc
. ') (i4.l2i3 I ,? 26.333±w.160V~ or 24.081 , _8.5851 for 0;
0.506 ± 2. l60 211.886 1 I J"
[3("6.933) 00' 0.044, 0.968 [0< e'
. = 18.566, 42 .301j for (72. [211.8861 21 1 886 1]
24.74 > 5.009
02005 Pearson E(lucalion. IOC .. Uppo! Sadd1fl River, NJ. AM ngnls rtlSelvod. This mllierialis prolected under a" oopyngll1l3ws as 1twy Q.lner.ilyelO5l No OOflion oIlt,15 malerilll mav be reoroducfId . in anv tom> 01 IN (.rw meal16 . ..... ttno.Jl oemrisslorlln Wlilino hom lhe oubIlshel .
3.S·
I 3.0'
",I
'0'
EstilJlllliou
6.12 More R egression
0.12- 2 (a) III EXI'ITisc 6. 11 2 we foullt! t hat
2~0 -
00
iJ = -I .6-1/ f,.0.1, Ii~ = 1.8 H12-1. L (I, - :r l ~ = .'l.W. .-, So t he I'Hdpoilitli fur th{' l'oufiJcnl'c illt.I'r\'al an! gi\'t'u by
4.6·1 2.73+ -(r - 2.00)
5.0·1
L = :2 : 11.335. 2.'1GB}.
I = 3 ; [2.4GS. :.t 17G1.
1 ~ ." ['1.096. 4.'1.9[.
f. 2.:ID6 / 1 .8-S9~' / -.!.. 8 ' 0
(b ) The cudpoillts for the predicli,)Jl illt('f\~.-\I nrl' gil'ell by
(I - 2.90)2
5.W
,I.G4 2.73 I- -(I - 2.90)
5.0-1 ...I:; 2 'J()6/ 1J3.19:1 . -s-
I (.1' 2.!)0):.I
" -'0 + .'i.O·'
J ' = 2 : [0.657. 3.1,161.
:r = 3: [1.658. :t!J8G}.
I = 4 : [2 .459, n.02ul.
- 3:5 - ....... - .1 4.0
! '
4.5
4.0
3.5
3.0
2.5
2.0
1.5
10j
2.0 ,i:5 Figure U.12 2: A 95% ('Qufidf' II CC iutcr\'a l for Jt(J') and II !J5% prCl lit· tiou bnud for }1
2OO!:i P8atson EckJcallOn. lnc.. Uppet S&cI!Ie AoveJ. NJ All righl$ retillMld Tt.s rn.t8flill1S protected Lrder .. oopynghI ~ as !hey currerwty eoust DOITlOn()lIhis IMlerlilJ 1M" tift raorudl.oc&d. In arl'llorm Of tw _ fT"Ml'IIrlII w;IhOUloermlSlloOl'l In ~ lromtne ollbl;sh8f
)'
52.0 ]
5 1.5
51 .0
50.5
50.0
49.5
49.0
96
54
6 .12- 4 (H) III Exercise 6.11 11 , we [ouml t hllt
'" f' ~ 24.8
40 ' 1/;;2 = 5. IS!)!), L ('" - x)' ~ .10,
1=1
So the cndpuil.l! ~ for the coufidcncc i lltl;:fVlll arc giveu by
)0. [895 [lOA 15 + 0.62( .(' - 56) 1: l.n4 - '-8-
1 (x - 56F 20 + 40 .
x ;::: 54 : {48.814. 49 .fl:l6 j.
x ~ 56 , \50.207 , 5O .62:l\,
x = 58: {51.294, 52.UW].
( b ) The eudpoims for the predic.tion inwl'va i are given by
Chllpter 6
5U.41 5+ O.62(x - 56) ± L734 /5. ~~95 I (:r - ( 0)2
1 .~ 20 + 40 .
:r=54: {-IS.177,5O.17:11,
.£ = 56: \49 .4 61 , 51.369J.
l' = 58: !50.(i57, 52.653].
_ x
56 5'7 58
)'
52.5
52.0
5 1.5
5 1,0
50.5
50.0
49.5
49.0
48.5
54 55 56- - 57 55 Figure G.12 ·4 : A 05% confiden~i: int.erval for II(X) and II 9&% pn .·d if"t ioll band fij I" Y
58 x
02005 Pea~ Education. Inc .. Oppel Saddle River, NJ. All rigtllS raselVSd. Th15 malEtriaI hi pTOloctlid undIIl 911 oopyrighI laW5 as they col'lerlllv ex .. '!. No 0011100 ol lhi!l malerlal ma~ be raoroduced. ill anY lQffl1 Of bY fUN means wUhoul OIIf1l11ssion In wrill00 from IN! oublishllf
ESlillJIIliolJ
6.12- 6 (~, ) For ,be,.' data ,
(" )
" L .r, = .')5, .. , "
10
LY' = !Jhll , .. , '\' . ~ J',y. = li;'. 550. ,~ , ,.,
.. ,
Thu~ S = Hbll / 1O = !J81.J aud
~ _ 65,550 -(55)(98 11) / 10 = I l!'18!J.5 = 1.10 , ~ IJ :1.% -(5Sr.!/ 1O R2.S · , 17 .
The leabt squares fegr(.."jSi<)1l li ne is
y :=:: 9S1 .1 + 140.4788(" - 5,5) = 20S..I67 + 140,479;'.
)'
1600
1400
1200
Figure 6. 12 6: NlLInher of progntJlls (y) "S. yem' (.c)
(c) 17S:l.7:I:3 ± ltiO.:36R or {l ,'i9:1.36ri. lYI<I ,HJl j.
"7
2IXl5 Pear90ll EUuc.;u,on , Inc .. Upper Saddle RIve< , NJ All ngtlltl IB$I:IrWd This malBrlalls Pf01QCIBd uooe' aU copyrig/lllilW$ a9 they cur renlly 8~'SI JIolIlQr\01 011115 matorlallT\llv tl8 roorod~ 10\ allY torm or bvllfJ'l mea"",. Wlthc)ul OfIrmossiof> in wrlllf1(l trom !lWlWbh5tWI,
8
4
o -4
-a
98 Cllllptcr 6
" 6 .12- 8 Lei, K (fJI ' Ih {j:I ) = L (y, - fil - fh:J;h - (3;j X2.)2. Theil .-.
.. DI< D~.
Bf(
8 {h
oK 8~3
:0: L 2(y, -I3I - th:Cl ,-.a..JX2i )(- 1) = 0; ,=1 " L.: 2(y, - !3J -132Th - tJ.Jx';II )( - J; ., ) = U;
.~ .
" L 2(y, - fh - Ihx., - !3ax2 .)( - x:.!,) = O. ,=1
Thus, we Jnll i> t solve s imll li,<l.llcollsly th~ three eq uat.iolls
We have
so that _ 4 ~.173
(3 \ = r;!)56 - O.n4 ,
12/31 + 4{h + 4fj.J = 23
'WI + 26P-l + 5rh = 75
4!3\ + 5/h + 22/3:. :0: 37
_ :i8!'i2 fh - 1489 :0: 2.!lR7.
10
5
o
-5
_ 1430 (h = - - = 0.£.160 .
1489
- 10 --_2~:,-, --6------ 1 ' ~-:r~3""··, .... x 0""
y
Figure 6. 12 8: Two views of t he l)Oiuts (l.ud the regressio l! phtue
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E .. t i IIIflt io ll
6. 12- 10 (a ) /Iud ( b )
24.5
24.5
23.5
23.5
22.5
22.5
21.5
,.
. .
L ~ ,_, , , " , _ , .,. 2 1.522.522.523.523.524.524.525.5
LO
0. '
-<l.'
I Fig ure Ii. 1:1 10: Swiuull{'r's U1l'Ct tinl(' (y) versus l>esl yeal" li llie (J") lind l"l-sid ual p lr't
)'
2<0
23.5
23.0
22.'
22 0
".
(e) /lIlt! d)
24.5
24.0
23.5
23.0
,.
22..
22.0 j •
2 1.5
. .
21.0r .. ~. ,_c. _ .•. 2 1.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0
Figure 0.12 10: A 90% ('ollfidf' IICC iut('l"val for '1 (.1') "ud a 90% pr«lIttioli l)Wld for Y
(e) Point. COllfitiCllCC C Olllid{, I]('t'
Pnrlu llt'l.er E."I tilllllll!$ Level I Ulf'r\'l\.!
" 22.S2!1I 0.95 122. :i217, 22.TiG5!
,J U.(i70r) U.!JCo [0. 1577 , O.b8:1:l1
u' 0. 1 !J7(j 0.% [U. 1272. o. '''''·'1
200!0 P-ea..on Educallon. Inc. . Upper StlOOIe AIv(lf , NJ All ngll\lI nlsetV8d this II1.illelllllla protected undo' 1IH OOP~I IaWS as 11lev WlnlllIly 8JU&1 MDDOIIO'I oIlhis malarial mav 1M! fBOfOduC9d in AnV fOtm 0.- tw IIIW maalW;. wllllOUI oennisJlon In wri\tna lrom IhI! ouh/IshIIl
y
24 1 22 20 18 16
" 12 10 8 6
• 2
y
~:~ 20 18 16 ,. 12 10
8 6
• 2
100 Clwptcr (;
6.12- 12 (c) (lml (d)
8
6 .. • ..
2 . . 0 2 3 • 5 6 7 8 9 .. -2
.. -4 .. -6
-6
·-i------r'3~ T' 5----.-6~·89 x
Pigure 6.12 12: (y) versus (x) wit h linear regression line aud resid uul plot
(e) Linca r regression is not appropriate. Finding t he leust~sqlll\rcs quadratic regression litle llsing Lhe raw data yields Ii = - 1.895 + 9.867 J' - O.f.l!J6:c2.
(r) .,," (g)
1.8
1.'
1.0
0.6
0.2
-0.2 2·--- - --,,-- "6
.. -0.6
-1.0
'- 1- ' 2~T' 5 ' 6 ~7 8 -'-9 "0 x -1.4
Figmc 6. 12- l2: (y) versus (x) with quadm.t.ic regn.:ssion curve and rt'Sidual plot
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BOO
700
600
500
400
300
200
100
BOO
700
600
500
400
J'
\'
Eslirultrioll 101
G.12- L4 (a)
Figllrf' 6, 12 1·1: NUllllx-r of I)r<wedure- (y) \cr&us )Nir (.r), lincnr regression a nd residllll i pl()t
(b) Without plotting the dl\ta nllci tht> r~idllal plOl, liut"nr regression set'IlLS W 1)(' appropriate. Howf'ver. it is clear that 'iOIllC 01 her l)OiYllollliai should 1)(> uscO . (e) 3ud (d)
::1 . J ~ __ . ___ . __ . : __ . ~ ,-"-, . ~
- 10 300
2OOl--~-
1
2 4 6 8 10
i -
201 100
12 3"- 4 5 6- 7 89 l011_ x
Figure 6.12 14: Number of procedures (y) \'er.$us ycar (x). "lillie rr-grcssioll and n-:;idllai plot
The lefl.'lL t>(ltulres cubic regression curve b
fj = 209.81138 - 21.3090.r + 16.2631x2 - O,b32:\x3.
Note that. the years arc 0, 1.2 ..... II rather them 19,so, 1981 , .. _. 199 1.
6.13 Resampling M ethods
6.L3- 2 IIc((' is fl Maple program that. givcs a solution of Exercise 6.1:\ 2. It use:; p r ' )('l>dufI"S that wert' wriLlCU hy Zo.",PII Kl)riall . These procedures are available frolll him or f.foUi Elliol l'aui:i. They Hrc abo on the CD·RO~I.
02005 PMI80n EclucaOOn.Inc., lJppef 5acXIe River. NJ. M rights I8SefV8d rm malere11. proteCted...oor a l OIlP)'rlght I.1wa al they C\m~ elUJI .liD IIOf\IIlfI 01 .... _rial me", be rtIOfOduoed_ In JI"" IOfT1"I or IW a ..... meana. ....-tt1louI ~ In wriIino htm !he ~
-
> read 'e:chap03.txt'; > X Example_3_1_3;
X : _ (2. 2, 2, 2, 2, 2, 2, 2 . 2, 2, 2 , 2, 3 , 3, 3, 3, 4, 4 , 4, 5, 5, 5.5.6 ,6,6,6,8.8,9. 15,17,22,23,24,24,25.27,32,
43J > randomize(): > tor k from 1 to 1000 do
L := Oie(40 , 40): # Simulates rolling a 40-sided die 40 times XX :'" [seq(X[L(j]], j .. 1 40) ]: Tr[k] := evalf«Mean(XX) - 5)/(StOev( XX )/sqrt(40»):
ad: T := [seq(TI[k] . k = 1 .. 1000»): UistogramFill (T , Mi n (T) . . Hax(T) , 10); # l-iistogramFill provides
a tilled hi stogram
> Mean (T); 2.588634568
Here i$ <I relative fre(.IIH!IICY histogram for SOUle data gcnemlco by t he (llwVIJ program. Note that this was ri m using N = 1000 rathe r than N = 100 n. .. asked for in thl' exo.!fcisc.
Fo r these d/;l.ta, 1 = 2.5886.
0.5
04
0.3
0.2
0. 1
~. 2 3 4 5
Figure 6. 1:i- 2: lOOO obsen'1LtiollS of T = ( X - 5)/(S'/J40)
C 200S P6il18Ol1 EdtlCil tlofl, Inc ., Upj)6' Saddle Ai~el , NJ All figl~S ,e~od This maleflill ls p lOl9CIOO under all copyr1ghl Laws 11$ tOOy C'IrIOOltly &~hiI.. No 0011100 01 this malenal ""''' be rtKJroduced In anY lon n or bv I\tlV moons. will1Ol.>t ot!ffTlisSlon In wr~illO l rom the OtJI)Ij~he,
Esr;JlJIl(;uu
6.14 Asymptotic Distributions of Maximum Likelihood Estimatol·s
6. 14- 2 (a)
III f(.£. : JJ)
dlu/{r: p ) a,l
fP III f(J': JJ)
iJjJ2
r = 0. 1
.r 1l11H (1 - .r) lu ( l - p)
.r .1' - 1 - +--P I - p
, ,I
- j?+(l - p)"-
[X X- I] P p - I 1
E ," -(I - P)' = ", - (1 - 1')' = 1'(1 - 1')"
Rlio-Cnllller low.~ r hUlluJ = II{ I - /I) . n
(h ) p(J - p)/n = \. p(l ,,)/ u
6.14- 4 (a)
(b)
(c)
I" 1(" 9)
i) In /(:1': 8)
au iF In /(.1:; 0)
=
- 2IuO+ lu.r - r j 8
2 , . - 0-+01-2 2.1:
of)? 02 - 01
[ 2 2X] 2 2(20) 2
E - (}2+(p - O'l+fiJ = 02
8' HnQ-Crruui'r 10wI:r bOllud = ;::-.
;tIl
0' VI'ry :;imilar to (11); ulIswcr = ;- .
• ~II
lu/(.r:: 0)
0111 f(r: 0)
au iP In / (.r:; 0)
00'1
- 11l9 r C;0)IIIX I I
---- 1117 o 0"1. I 2
(f! .;- (p. Ll r
E[ I"X ] 11 III.r (I-0J/B I -0 r d::r. Lt'l y = lu.£..dY=- lu " 1 -1'X j 1.- II{ I - fJ )/(lp -I/(ly = - 0 f(2 ) = - 0
"
~2OQ!, PuIlOn Educanon. k'Ic .. Upper S&do:kI RiwK, MJ AlIi(tU ~ Thos INIlerlillOS proIecled UIlOef al copynghIlaws as Ih8y CUflenlly eXIIit Il1o oortIOO oIlhIs malenal maw be reoroduced.. ., IInv tnrm Of by anv m8/l.fJS. wittlOIIIOItrmissoon WI wnllnO tromltwt ~
Chapter 7
Bayesian Methods
7.1 Subject ive Probability
7. 1- 2 No III1SW('r nl'(.'(If'd .
7. 1-<1 Oil" scllutioll is 1 1.0 7 for a bet 011 A aud !) to I for 8 hN 011 B.
A be ls: for n 7 dollnr liN, the IJO.)l\ie gin.'S DIn' back: 30000/ 7 x I = ,J2Sfl.71. Su thl' hookil' J!,iVf>:,l Oll t -I2ti5 .71 + J()(X)[J = :S 1 2~. 71.
B hets: for It I dollar hel, the bookie g ivt-'s live b,U'k: f>()OO/ l x 5 = 25000. So tlw bookil' giv"S Oll t 2GOOU + !">I)OO = :n)OOO.
7. 1 G Followinv, HI NT: before ullythin~, ,h\, person hn:'!
d rI (d) 'J<l :\(1 d IJI+--tP'l+ -- V'I -- = 1'1 r /~l +-- "-I= -d ... =--: <1 4 4 .j " <I
that i;o;, t llf' pf'r:l()11 is dowu d/ 4 heron:! t he :'Itllrt.
1. If At Ql'curs, hoth win aud liJ('Y c·xchaugc· IlIIit ...
2. If;l 2 happells. agllih they exdl1tuge tlUits.
:}. If lIf'iT/lt'r At nul' A1 occurs, bOlh rC(:t' iw ZNO: am i tilt' pl' rsou b; s lill ([0\\,11 til I ill .. 11 t hn~ ('n.'>(':,.
Thus it is hAd for lIuu, pt'rSOIl tol hclicvc lhut 113 > PI t- I~l for it CIl Il !£our! to a DUidl hook.
7. 1- 8 P(A U A') = P(A) r P(A') fmnl Theorem 7.1 I. FroHI Exercise 7. 1 7. P(S} == 1 Su l.lltlt,
I = peA ) + P(A'). Thu" "(;\I) = I - P(A).
02005 PjlilfaIlf1 Ec.!ucalloo. IOC . Upper 5aQc.IIu RIve< , N.J. All nglllS l&S8rvllc.l. This malilllalill PfQIadec.l unc.Ier ,I oopynghllBWS 85 itlEI)' cunendy 8_181 No 0CIftI0n at !his maumallllllV be rlII"IfIXluced. in InY lonn or bv IInY rna!lflS wllllOUl OIIflflISllIon In wrirln(r lrom IIMI outlIi!;lIIlr
-
7.2 B ayesirul Estima tio n "£'£'/(1 / ')
7. 2- 2 (a ) ! r(£~)l" r(oo )8~O
7 .2- 4
(b)
IX .,.o n +ao- Ie -( 1/ 00 +:£.1', ).,.
wil ieh is r (Ila + 00, (Jon ~ ). I + O£'XI
00 E(T l xl ,X2, .. . , .L'n) = (na+Qo}"----70""--
1+ 00X,l,
o ollo + OItOo 1 + OOIlX l +nOoX
no + 0'0
IjfJo + n X
(e) T hC' 1)()Stcrior distribut.ion is r(30 + 10, 1/ {I / 2 + 10:;;1). Select. (I and b so t.hat P(a < T < b) = 0.95 wit.h equal tail proLabilities. Theil
_+ .., 40- 1 - w( 1/ 2 + 'O:l) / 39 -,, / l' (I / ? I O ~)'O 1 11(1 /2+101' ) I
'W t' C 111 = - -z c ( Z, II f (40) u\I/2+ 10:;') f(40)
makillg the clUlll )l;C of variablC8 'w( I / 2 -I- lOx) = z. Let. V0 0 25 a nd 'I'O H75 be t.he q ua nti lcs for the [(,10, 1) dist.ribu t.ion. T hen
b _ '/10.975
- 1/ 2 + lO y"
It follows t.hftt Pea < T < b) = 0.95.
(30)ll (X I J";! ••• Xn)2 e-OEx~ .0'1 - 1 c- 40 {X 011 -I- 3~ -{.J + E1'~ )O
wh icb is r (II + 3, 4 + ~ :r~ ). Thlls
'/I + J J:n ) = ~ JO
4 +L. ,T,
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IU7
7.3 More Bayesian Concepts
7 3~2 k(,' 0) ~ (")0'(1 _ o)"-r. 1"(0 < Ii) O"~'(I - O)"~ ' .• . r f(o )f (J) .
.,. = ll. l. .. .. u. 0 < 0 < I.
,",(x) ~ (' (")9'(1 -0)"'. 1"(0 +11) O" ~ '(I - O)'-'dD Jo x fCu)f(d)
J' = 0. 1.2 ..... 11.
7.3~4 k(r.O) U < J <X
7.3 6
n,.- I r(a t- I )
r (fI )rI(> (.:rr + l /;1)A+l'
<,<:IT IT I
(llrT -t 1) ... + 1'
0' 11 (01,01
) - !O~ t'(3-0\)21101+(7- 0I FI
The 5('('olld lig llf": s h" .... '); t\ cOll t~llIr plot.
Figun' 7.:1 6: Gral)h:; to help to S('(' when' O. uud ()~ maximize tilt' poslt'rior p.d .f.
lJ'iiug Mapll , II solulioll is 0 , = 5 anti (J2 = 2. Other solutil)JI:; S<l tis rv
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Chapter 8
Tests of Statistical Hypotheses
8.1 Tests about Proport ions
8.1 2 (0) C = (b) n =
= , j =
on (n') C = (hi ) (I
, j
l.r : x =o, 1.21: P(X = O. I . 2; l' = D.H) (0..1)' + -' (O,6){0.4):} -I- 6(U.6f(OAf l
'- O.52'lhi
P(X = :I. 4, P = 0.4) .1(O. I):I(U'£i) + (OA)' = 0. 1792.
l.r: .r = O.I) ; P(X = 0, I : 1) = 0.6) (0 .. 1)" + 4(O.6)(O..l )J = 0.17!)2 :
P(X = 2, :I , 4; /J = 0..1 ) " (0.4)'(0.6)' + 4(U.4)'(0.6) + (0. J) ' = U.',248.
8.1 - 4 Usiug Table n ill till' AI>!>cudix .
(a) n
(b) J
8.1-6 (0) ,
(b) ,
Pc}' ~ 1:1: p = 0..10) "'" J - O.,s-lt;2 = 0. 15:18:
pcr ~ 12; I) = OJiO) P(2!) - }' ;?: 2:'1 - 12) where 25 - l ' is 11(25. O.4U) I - O.8-1G2 = 0.15:18.
Y/T' - 1/ 6 < - 1.645; ')(1 / 6)(5/6)/ " -
J265/ "000 - J/ G --j~~;;e;;';';i!;;~ :::: - 2.0!'. < - 1.6,15. rejcct Ho. .)( J/ G)(5f!;)/ 8000
(e) [O.jJ+ l.GM1JP(1 - ji)/ g{)()()) = [0, 0.16.18\ , 1/ 6 = O.]{jGi is not ill this jllll'n'al . This is ,'olls isleut. witb the condmsioll to rl 'jec" Ho.
8.1 - 8 The \"aIlle of Ihe tt.'St statistic is
0.70 - 0.75 = - 2.28U . .)(0.75)(0.25)/ '90
(.,) Si llcc': = - 2.2gU < - 1.6.}fl . r~j('Ct 1111 .
(b) Since.: = - 2.28U > - 2.:l2(j. du not. rf'jl'Ct. fl o·
(e) ,}-\'it.iut! ~ P(Z ~ - 2.280) = 0.0 11;1. Note that 0.01 < 1}-\'lIJ1W < U.O!}.
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llU
8.1 - 10 (a) Ho:p = D.14 i Ih 1»0.14;
y/ n - O.14 (b) C = (z: z ~ 2.320) where z = I¥P,,~';';'T
)(0.1 '1)(0.86)/n'
(e) z= 10-1/ 590 - 0. 14 = 2.53!l > 2.32G ) (0.14)(0.86)/ 500
tiO lIo is rejected and COlldlidc that t he campaign was stlccessfuL
8. L- 12 (a) z = yi n - 0.65 > 1 .9~} )(0.65)(0.35)/ n - ,
114/ 600 0.65 . (b) z = -iict.ifIi,,"~iM = 2.054 > 1.96, reject flo at /)' = 0.025.
J(0.65)(0.35)/6oo
C}w jJwr 8
(e) Since Lite p-vnJue ::::: P(Z ~ 2.054) = 0.0200 < 0.0250, rejeC't Ho nL lill {t = 0.025 s ignificance level;
(d) A 95% one-sided (:oll fidence interva l for J) iii
{0.6!) - 1.6.)5)(0.69)(0.31)/ 600 , Ii ~ {0.659, I{.
8. 1- 14 We I$ha ll lCSl 110: P = 0.20 against. H I: P < 0.20. Wilh a !>aulplc size of 15, if the critical region is C = {:r : x :5 [} , the sign i fica llce level is 0 = 0.1671. i3t.'CIlU5C x = 2, Dr. X bas not demonstrated significa nt improvcl llc ll t with these r(>w data.
S.l- Hj I;; - 0.20 1
(a) I' I ~ )(0.2O)(0.80)/ n 2 I.nG;
(b) Only 5/ 54 for whicll Z = - 1.97:lleuds to rejectioll of No, so!l% reject 110.
(c) 5%.
(d ) !.I5%.
( ) 2J!J/ J 124 - 0 .20
e z = = - U.4:J , .riO fnil to reject J-/o. )(0.20)(0.80)/ 1124
8.1- 18 (u) Under No, ji = (35\ + III )/800 = OA9;
I , I ~ _-F1~3~."=/~60~5~-74",,1=/~19~5=1 ~ ~ I 0.580 - 0.21 0 I ~ 8.90.
( )
0.0-112 (0.<19)(0.5 1) G~5 + 1~5
Si neI' 8.99 > L9(j, reject, Ho.
( b) 0.58 - 0.2 1 ± 1.9G (0.58)(0.42) (0.21)(0.79)
605 + 195
0.37 ± 1.96 v O.OOO40:1 + 0.000851
0.37 ± 0.07 or [o.aU,0.44J. II, is ill agreelllellt with (8).
(c) 0.49 .i 1.96)(0.49)(0.51)/800
0.'19 ± 0.035 or [0.1155,0.5251.
C 2005 Pearson Educa liofl. 11'1(:" Upper Sacldle RMI!. NJ .... 11 ri9hlS resetV8d This mal(11ial II pnllected UOOOf al oopyrigtlt 1aW$ as !hey CUffeotly Ild6I No 00f1I0r1 of Ihss malaMl mav bit fGnl'OduCed. III ilOV Iomt orIN' 8nv m&aflll withoul oerml:wlon In wOlil>O from II IB oubli~
8 .1- 20 (ta)
<b)
(e) z = 2.:~-l1 > 2.:~2(l, rt'ject 1-10.
(d ) Tllf' p-VU.IUf'::::: P (Z ~ 2.;.J.l 1) = 0.0096.
8.2 Tests a bout One M ean and One Variance
8.2 2 (0) Till' eriticul I'Pgiull is
The (lbserved vU] Ut' of t. f = 10...1 - 10.1 :to.
0.'1 /' is KreMer thnt 1.753 ~ WI' rcj.·ct /10-
(b) Si ll('{' 1000'; (1 5) = 2.!i·17, tilt.! approximutc jr\'alll1' of til is \f'St is o.om),
I" - 7 51 It I = ' ~ ~ luox.(9) = 2.262. .~ / v 10
8.2- 4 <,,)
012::: 0.02. 2 =- 0.025
Fig ul'e ~.2 4: 1'111' critical n·gion is I t I ~ 2.262
(b) 17.5;' - 7.5 1 I'l = r.;; = I .... , 1< 2.202. do not ft>ject. 1-10-O. ' 027/ vlO
(c) A !V',llt ('oli lideu('p iutcrvll l fur 11 is
[ (0.'027)
7.55 - 2.262 jiQ , (U.W27) ] 7.55+2.2G2 .JiO - li AR.7.1;2J.
'"
HI,IIt,p, If = 7.50 is eOlltllim.'t1 ill t h is iuterV[ll. \V(, (-, ould hovO' oiJtllillN till' :<l illIe
('ouclu:;ioll from 0111' answer III part (h).
CI~ PMr.JOO [ducaUQO, IrIC" \JWef 5a<kfIe AlVOr, NJ. All rigllrs rtlllllMld ThIs materlill l5 proreclad unOOl a~ copynghllaws as Ih6y alnenlty oXlSt No DI)f1IOn oIltlis marIlla! ",",v be ,1!DtIlCIllCed II ilr'N 'n<m 01 bv env means. -MthnoJllMIfIrns.sion In wrillnn ',orn rhe oubII/II"w!l
11 2 CllllplC'T 8
8 .2- 6 (n) Ho: J1 = 3.4;
(b) H I: JJ > :1.4:
(c) ' ~(x - 3.4)1( .• / 3) ;
(d) I ", 1.860,
0.4
0 .1
« = 0.05
Figure 8.2 t): 'flit' nitkal region 6 t ~ 1.860
:t556 - :t-l (e)' ~ ~ 2.802 ;
0.167/ 3
(f ) 2.802 > 1 .~tiU, reject //0;
(g) 0.0 1 < I)-value < 0.025, p-val ue = 0.QI 16 .
.,... 3315 8.2- 8 (.\) t = x - v'IT > 2.764; iiI 11 -
3385.9 1 - 33 15 (b) , = r.o = (J.609 < 2.764, do 1I0t. rcje<::t. J-Ju;
:136.321 v 11
(e) Irvu lue ~ 0.25 beca use t o . 2~ (lO) = 0.700;
(d)
(e)
(f)
:.! = I 0,~2 < 3.940' X 5252 - ,
:I 1U{3J6.316:l) \' = 5252 4. 10-1 > 3.940, do not. reject J-JOi
0.05 < p .. \'aluc < 0.10.
02005 PSQI&OfI Eol.IC8lion, Inc., Up!)&! Saddle A1v8r. N.J All rlghiS r6!kNVOO Th/!i nWl IlMlal1ll P<D1~1Id lJOtIer aN aopyrtgtlllaW9 88 1hey corretdly- e:Ml No oaortlon oj Ihis rmlarlal rmv be fllOfodtlOOd III BrN form o. bv alIV meIH .... wIthoul oennl'lsion In .... rillno trom lhe oublishef
Tests of Statistical Hypotheses
/x -125/ 8.2-10 (a) / t / = s/V15 ~ to.Q25(14) = 2.145.
0.4
T,r=14dj.
0.1 al2 =0.025
al2 =0.025
-3 -2 -1 2 3 Figure 8.2-10: The critical region is / t / ~ 2.145
(b) / t / = /127.667 ~5/ = 1.076 < 2.145, do not reject Ho.
9.597/ 15
8.2-12 (a) The critical region is
19s2
X2 = (0.095)2 :S 10.12.
The observed value of the test statistic,
2 = 19(0.065)2 = 8.895 X (0.095)2 ,
is less than 10.12, so the company was successful.
(b) Since X5.975(19) = 8.907, p-value ~ 0.025.
( 100 2282) (100)2 8.2-14 Var(8
2) = Var 22' 100 = 22 (2)(22) = 10,000/11.
8.2-16 (a) The critical region is given by
(b) j3
18s2
X2
= -- > 28.87 or S2_ > 48.117. 30 -
P(82 :S 48.117; (T2 = 80)
(188
2 )
P 8():S 10.826; (T2 = 80 ~ 0.10.
8.2-18 The critical region is
d-O t= .(1;;>1.746.
Sd/vl7 -
Since d = 4.765 and Sd = 9.087, t = 2.162 > 1.746 and we reject Ho.
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113
114 Cl lflPIN 8
8.2- 20 (0) The c ritical region is
0 .1
a = 0.05
-~_"'3'-L_'2;;-~_;-' ~ ·- ;·--';2c--'i;3~
Figurc 8.2 20: T llc critind rCJI,iol1 is t :S - 1.729
(b) Siuce d = - 0.290. Sd = 0.650·1, t .= - 1.99,\ < - 1.729, so we rej('("1. Ho.
(c) Siuce t = - 1.99-4 > - 1.5:\9 , we would fail to rC'jcct Ho.
(d ) From 'Iable VI, 0.025 < p-vaille < 0.0.5 . In fact, p-va iue = 0.030-1.
8.3 Tests of the Equality of Two Normal Distributions
(b) t ..Jl5.16 - ;\4 7 AO
-;=~~~;;;;;;;;~;;;;;;;;;;;;,,=~~~ = 5.570 > 2.4n, r .. jt>d. fl o. 15(1356.75) + 12(692 .2 1) (-'- -'-)
27 J(j + l:l
(e) Tilt· cl' il. ic'81 regioll is
, s~ ;?: FO IY'..b( 15, 12) = :U 8 or
" .' ~ ~ Fu O',lt.( 12, 15) = 2.96. '.
... : 1356.75 Siuct' '2 = ~~- = 1.90 < ;1.1 8 a nd .' ; = 0.51 < 2.96, we accept 110.
(d ) c
,. ,.
'.v 692.21
1350.75 =~i::-~?= ~ 0.662, J:l56.75 + 69_.2 1
0.6622 0.3382 , -- + --? - = O.D.SS7, ] Il L
25.
"
Til{' nificlll I't'gioll i:-; l her'crort> t ~ to.lH (25) =- 2.-185. Siuec t .0.... 5.570 > 2,485, we ag<liu rejcct 110.
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8.:1- 4 (1:1) t r - y
to m,(27) :=0 - 1.7113; 12si + If.~; (~ ..!..)
27 1:1+16
<b) t 7:l.!) - 81.7
-:F===C~~~~;;"':=:====:= = - O.,s69 > - 1.i03. do lIul rejl,,(·t 110: (12)(2'''0)' + (1'.)(2'.' )' ( I I )
(e)
(d)
27 I:l..j Hi
0. 10 < / ... m lue < (I 25;
, (?I": ")' ~~ 0:: (~:j.~):1 = O.1:'UR < 2.!)6 = FG02~,( 12, I:'.). il
il _8.:1
" ~ = L.222 < :.us = Fo.o2r;( 15. 12},
" du BOt. rt'jcct ('<lunlit~· of \·a riallr(':o,.
8.3 - 6 (3) ASfiUluiug o-~ = (1~.
( b ) I 2.151 1> 2.101. rcjCt·L No;
(r) O.oJ < p-mluC' < O.U:.:
(d)
~ too~(l8 l = 2.101:
y --'----------'-----'~
(e) I.:i 18 < ..J .m = 1-"0 02::.(9, !.I), O.7!ill < ·I.U:\ = Fo O'15(U. !I), do uot rej(,(,1 n; ~ 0-; .
8.3- 8 (a) ~ ~ ~ = :t247 < 4.:t~ = FOO"~5({U1 J, ! = O.:l08 < 5.52 = F002!:.(!J,(j )·
SII ~
do 1101 r('j(,( · I . "~ = o~;
(b ) -;=~J;I J':.,-=)}~I== 6s; + 9s~ (~ ..!..)
15 7 -I 10
(r) I IU ollfl ycs so Ihnt the answers an' (:/)Inpatible>.
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116 Cllltpter 8
oS; '1.88 8 .3- 10 (a) "2 = - 0.8·, < 2.f)3 -= Fo 01 (24 , 2i!).
fill f).R I
.'i~ _ 5.81 _ I
.~; - .J .SS - .J!) < 2.0 1 = Foo1(2g, 2'1).
do 1101. rcj('('t C1~ =- u;; x - y
(b) --;:c='~=:C~=~~= = 3.402 > 2.326 = ZOOI.
24s; -I- 28s; (~ ~) 1)2 25 + 29
rt'j l''t: t Jl \' = J-I \ .
$.0-189 - H.0700 8 .3 1 2 (a ) f = -- IA6.Siuc-e - I.:' 37 <- IAG <- 1.7..JG,
(8(0.001 39) + "(0.00050) f0 V \6 Vg-lg
0.05 < Ji-vaiul' < O. LO. III fac t , p-vaille = 0.082. We wou ld fa il to f('jt'd flo Ilt. Ull i t = 0.05 significnllce [("wI.
(b) F = 0 .OU 130 = 2.78 < ,1..13 = Fo 025(8. 8 ) l!O we fail to I't'jt'l't at Ull £'I = U.05 0.000[.0
s il:!,ll ifiCllIlCC level.
(e) TIl{' (olJowilig figure C'oufirulii our 1111l:1WCrs.
x
y
Figure ~.:~-t2: Do..'C-lllld-whiskcr djngram for leugths of colulIllls
.I.lG:J3 5.1050 8.:1- 1.1 t -= J""",,;;,;,;i.;';;~7ii;~icr;=~ = - 1.(i-l8. Sille(' - 1.330 < L.6Jd < - 1.n.l ,
11(0.0142(;) I 7(2.5!H49) ~ 18 V 12 + 8
D.Of) < II-value < 0.10. III fa(·t , p.vnluc = 0.058. We would f,d l to f(·jt..~· I , N o at, an a = 0.05
s iguifi<-auc{' lev('t.
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Tests of Statistical Hypotheses
8.3-16 (a) y-x -r=== > 1.96;
s; s; 30 + 30
(b) 8.98> 1.96, reject J-lx = J-ly .
(c) Yes.
x-[J-
y -----II II t-----
5 6 7 8 9 10
Figure 8.3-16: Lengths of male (X) and female (Y) green lynx spiders
s; 9.88 D ( ) c .1 . 8.3-18 "2 = -0 = 2.42 < 3.28 = rO.05 12,8 ,so 1a] to reject Ho, Sy 4. 8
8.4 The Wilcoxon Tests
8.4-2 In the following display, those observations that were negative are underlined.
Ixl: 1 ~ ~ ~ 2 .3 1 1 1 Q Q 6 Ranks: 1 3.5 3.5 3.5 3.5 6 8 8 8 10 12 12
Ixl: 6 7 7 8 11 12 13 14 14 17 18 21 Ranks: 12 14.5 14.5 16 17 18 19 20.5 20.5 22 23 24
The value of the Wilcoxon statistic is
w = -1 - 3.5 - 3.5 - 3.5 + 3.5 - 6 - 8 - 8 - 8 - 10 - 12 + 12 + 12 +
14.5 + 14.5 + 16 + 17 + 18 + 19 - 20.5 + 20.5 + 22 + 23 + 24
132.
For a one-sided alternative, the approximate p-value is, using the one-unit correction,
( W - 0 131- 0) peW 2: 132) P > __ _
J24(25)(49)/6 - 70
~ P(Z 2: 1.871) = 0.03064.
For a two-sided alternative, p-value = 2(0.03064) = 0.0613.
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117
11 8 CII8.1ltCr 8
8.4- 4 III the fo llowil.lg d isplay, those observatious LllIl ~ were negative nre ullderlined .
I:rl: O.07YO 0.5901 0.7757 1.0962
Huuks : 2 3 4
Ixl: :t0678 3.8,')45 ;).9848 9.3820 74 .0216
Ranks: 6 7 8 9 10
T he vnlue of the Wilcoxon statistic is
w = - 1+2 - 3 - 4 - 5 - 6 +7 - 8+9 - 10 = - l9.
Siuel'
I - 19 I Izi = . = 0.968 < l.9G,
V 10( II )(2 1)/ 6
we do not reject Ilo·
8.4- 6 (a) The <: riti(;a l region is given by
w ~ 1.645/15( 16)(31 )/6 - 57.9.
(b ) In the following displny, those differences tho.!. were uegMivtl arc uudcrli ucd.
lx, - 001 ' 2 ~ 2.5 3 4 1 4.5 !l 7
Ranks : 1.5 1.5 3 '1 5.5 5.5 7 8 9
Ix. - 501 ' 7.5 8 8 14 .!"J 15.5 21
rw.uk!i : to 1l.5 L 1.5 13 14 15
The vallie of t he Wilcoxon statistic is
'W = 1.5 - 1.5 + 3+ 4 + 5';) - 5.5 - 7 - 8+ 9 + 10 + 11.5 + 11.5 - 13 -1- 1-' + 15
50.
8illcc 50 z ~ ,~;;;e"",,,,,, = 1 .'120 < 1.645,
V I5( 16)(:l1 )/6
01' s ince to = 50 < 57.9 , we do 1I0t reject flo.
(c) The opproximate I)-value is, IIRiug t he one-unit. correction.
p-vaJllc ::= P(W 2: 50)
P Z > ~ P(Z > 1.:\915) ~ 0 0820. ( '") - V I5( 16)(3 1)/6 -
8.4- 8 The 24 ordered oh:«:rvflLions, with tlw .t:-values lInderli ned and t h e milks gillcll uuder endl
obserwtioll n l'e:
0.779,1 0. 7M!) 0.7f165 0. 761:1 O.7fil5 0.770 1
Ranks: 2 :\ , 5 6
0,7712 0.7719 0.77 19 0.7720 0.7720 0.77:3 1
Ranks: 7 8.5 8.5 10.5 10.5 12
0.7741 0.7750 0.7750 O.777(i 0 .7795 0. 7811
Ranks: 13 14.5 14.5 16 17 18
0.78 15 0.7816 0.7851 0.7870 0.7876 0.7972
Ranks : 19 20 21 22 23 2'1
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Te.ts uf Slllljsti(;nl llypolll~"S lW
(~, ) The value of lhe Wikoxnu statistic is
Ul 4! 10.5+ 12+ 14.5 1-17+ 18 1-19+20+21 +22+23+2 1
= 20fi.
Thus
( _..;20:;.':;.5:;,,;-;'~5;;0=7) • p-valuc = P(H' ~ 2U5) :::::: P Z ~ = P(Z > :U5) < 0.001 /12( 12)(25)/ 12 -
so that. we c1cl1riy reject. Ho .
(b) l' 0.80
0.79 .' 0.78
,
0.77
0.76
0 .75 ._ ._ .. _ .-.,--------..._ ....... _. _ _ , _ _ _ , x 0.75 0.76 0.77 0.78 0.79 0.80
Figure.s A $; 11-(1 plot of pill weighu,;, (good , defecti ve) = (y, y)
8. -1 10 'I' lle (.rdef1::-d comllined lI,ample with the.L observations ullderlilwd Ill'l';
117.4 69.3 72.7 73.1 7fi.fl 77.2 lli 78,!)
Ilank:;; 2 3 4 5 a , 8
82.5 8~t2 8:q .4.0 "' <1 ,7 86.5 87.5
HAnks; 9 10 11 12 ,,) 14 15
87.6 b8,j 88.a !.IO.2 1&.1 90A 92.7 911.11 95.0
Hiluks: 16 17 1. 1" 20.5 20,5 22 2:) 2·1
The vRlllP of the Wilcoxon iltfltislic b;
111 = 4 +· 8+9+· .. +23+24 = 187.5.
Siucc 187 5 - 12(25)/"
Z = . - = 2.165 > 1.(;'15 /12( 12)(25)/ 12 '
we reject. /-10,
.2006 PeiIlOfl Educallon. lnc_. lJppef Saddle Ri ...... . NJ All riglU rewrvlld. This mal;>riallB proIOCIed under iI. copyrlghr laws as they cu,u.lrMIV tUfal NorlOlbM of thlt !Mlari.-... l rna" 00 rfilltOduclld in a ll'll" 101m Of b\l81lV m&afl!.. wllhDul oormilllllon in wrilloa lrom 11>11 oulllisha<
120 C/lIlpter 8
8.4- 12 The ordered combined Stunplc with the 4&'passellger uus \'O.lucs lIuderlinl>O lift':
8.4- 1"
104 IB4 196 197 248 ~ 2GO 279
Rllllk.s: 2 , 4 5 6 7 8
:mo aM ill :la l :\55 386 393 39G
Illlllks: 9 10 II 12 " 14 15 16
ill 432 450 452 llauks: 17 18 19 20
T he vnlue of the \ ViJct))WH s tatistic is
It}.;;: 2 + :J + 4 + 5 + 7 + 8 + 13 + 14 ·1 15 + 18 + 19 =- lOS.
!::iillce
we do lIot rejcct lio.
lOB - II (21)/ 2 = - 0.570 > - 1.645 V 9(1l)(2J)/ 12 '
'The ordered combined 8llmple with I.he x-vnlues underliued ar(':
.LQ 2.2 2.3 2.3 2.4 2.' 3.9 3.9
HJIIlks : 2 3.5 3.5 5.5 5.5 7.5 7.5
4,3 ,I.{i 4.9 5.0 ".0 5.3 5.1 6.6
RllU k!l : 9 to JJ 12.5 12.5 1·1 15 16
Ll 7.9 B.5 8.9 9.t1 9.6 9.9 12.1
Ranks: 17 I. 19 20 21 22 2:1 24
13.8 15.2 154 16 .2
Ranks: 25 26 27 28
The vulue of the Wilcoxon statistic is
w = 3.5+ f).5 + 9 t- J 1 + 12.5 + 15+ 18 + 20+ 21 + 22 + 24 + 26 + 27+ 28 ::. 2<1 2.5.
Since
we rcjL'Ct lin.
-,';;,.4;;2;;,' ':;.,- ,;2;;0,;;""" Z = V 14 ( J4 )(29)/ 12
1.8 15 > 1.645.
8.4- 16 C =- {w: w > 174}, 0: R: 0.05·15, w = 187,]J..Valuc ::::::: 0.0067, rejcct 110-
8.5 Ch i-Square Goodness of Fit Tests
8.5~2 "I = (22,1- 2~~2f ( ll!-.l - 116)2 (130 - U6):! (48 - 58)2
232 + 11 6 + lllJ + 58
= 3.78<1 .
(59 - 58)"
58
T he mdl hypothesis will not be rejf'f'tt'(l ilL allY reasonable siguificall(~ ]c\'cL Nol.C that. E(Q,, ) = 4 wlwu flo is t.rue.
8.5- 4 q2 = 4.[.1;'; < 5.99\ = .\505(2), so ti~ not. rejcct fi (j ~H (I = 0.05.
02005 Pearson E<Juc:a!bl, Inc., Upper Sadi;Ie ftivor, NJ All rights /'8SeI'YIId ThIs malerial is prOll:ldild UOCIer al copyrl(;1llilW5 as lhoy CUlrertIy 6JOS1 No oortJon al it. malerial maw be reDrouuCllld in anY Io!m Of IN _nil /TIIIIInB ....t!houI oem'II5o!IIorIlI'I WriIjno lromlhe ~
']'':-1'8 of Sr;lI i.sticllJ HYlJothe."Of.'S
( 12 1 117)1 {:.ro - :J9):t (1:1 :l9f ( ll - I:I) l 8.5- 6 f{J 117 -I :.m I :19 + J:}
:::: OA1!) t 2.07i -I U..IiO -I O.:WS = 3.2 14 < 7.H l & == \B os( J )· Thill'! Wi' do 11m rt'j(,(,1 the ~h-Jldt'liull tl!l.'ory with I,\WS(' dUlll.
121
8.5 - 8 We iinJl fllld tlUlL Ii = '174 / -125 :"; 0.6-117. Usillg Tllblc II with I' :::: n.1i5 tlw hYl'oth\.':'i il.(~1
prolmbiliti('l> arc JlI = P(X ::; I ) = 0,05-10, I)~ = P(X = 2) = O. II:'l.I 2, 1' :1 = P(X -= ;\) = OXf6.J , P I = P(X = 4 ) = 0 .312:1. />5 = P(X = 5) = 0.1 160. TIllis tilt' 1'('I>IX'Cti\'(:O {'x p(''Clcd ",.111('5 arl' .1. :)00, 1:,..102, 2$.59-1, 26.1;54, flud 9.860. 011(' drgre{> of fn'i'do lll is 1.)S1 hecull~'
/1 W(\S estimated. Tlu' val ll C' of lhc chi-square !/,fHxiuC'S:> of fit statisti!' i<;:
(6 - 1.59U)2 ( 1:3 - 15..102)2 (:1U _ 28.fi!J.l)'2 (28 - 2lL .... )M )2 (S - !.I .8GO).l 1.590 I- 1~.402 + 28.59.1 + 26 .......... ,·, + 9.860
1. :\O(j[1 < 7.6 15 - '\~. 05(:1)
Do lIot l"(>jP('t till;' ilypotlle:>is Iltut X j" /,(S, /»). The 95% ('oufidc lI{'(> illU'rvul fell" p i .,
0 .6-147 ± 1.!lli J(O.G447)(O.:155:~)1 42[. OJ' [O.!";!)!) . O.GOO].
TIll' pcnnie:. Ihat weTI;' 1ISf'd wen' mimed 199$ 0 1" clU"J icl". RC>(lNII this t'!XpCriHll"1I1 ..... ith s illlilaT p('IIHi~::s or wi t,lI ncwcr pennies unci COIIIplll"f' ,Your r('sulls wit II lho:w ohtailll .. ,<1 hy tJICS(' slllI lpI II:3.
036
0:321 028
1 0.24
0.201
0. 16 --~ ~: I I- -
r-0.04
1 --...""""'"'"- ....... ........-..-~..-..-~ 2 4 5 3
Figlln' ~.5 t\: TIlt' b(5, 0.(5) probabi l ity histov,n\ln and till' H'lat.i\"(' fr~'<iu('lwy hisLf)gl"lUIl (:o;h(ukd)
8.5 10 'flIP re.j:x.'Clh·c (lflJbuhi lit,j<,£ and ("xpet." tlXl fl"t..'<p lellcies lHf' 0.050, 0. 1 'I!), o.n,1. 0.22--1. 0.1 ti~ . 0 .101 , U.Or,O.IJ.022, 0.0 12 (lIl" I~) ,O , 44.7, 67.2, 07.2 . . '}OA. :-In.:!, 15.U, (j.G, :3.0. TILl:' last two (" 'lis cuuld i){' colllhilled 1.0 gh'e an ('XliCCt.cd ft"(,{)lIclLcy of 10.2. Proul Excrd .sf> :15 12. til('
rt"!SIX'f"th'c fn.'<IW'IIf"iCS an' 17. 17 , (;:1, {j:.i , 49. 28, 2 1, ll11d 12 givillg
( 17 - la.Of! (.J7 - 11.7)..! ( 12 - 1O.2f . f/1 = lfi.O + -14 .7 + ... + 10.2 = .tS-11.
Siuc'p 3.811 < 1--1 .07 """' ,~ (X;(7), do lIot r('j('Ct. TIll' sulllplf' 1Il('tl!1 i" r = :tu:\ ~ lId III(' Sluupl{' \'i\ri(u(('t· is $1 -= 3. 19 whieh also lj llppOI·ts thl' h),l)Qth(',.;i~ . 'fllt' f()lIo\yilll!, 1l)..!, IIt"f' ('omj)lIrt>R t il(' prohnbility hj~togn\lll with tiL<' !"elali,,!' fn:.'<l llCIlQ' hi;;togrulJl of IIi(' dUI II.
02OO!i P8a'SQn EducCllIOtl. Inc. Uppef 5.a(J(Ie R,vOi, NJ AN rightS '9S1lI'.lUd. ltD matOilill1S prol8Cled under aM oopvrigtlllllW$ as lhey CU!1"ently UXliOt. NO OIl<tll)rl oj lhis malBli.:IlllIiI" bfI reoroduced. In arll/lOIm Of f)" 31'" means. WlthoulllBrmrsIIOOn III .... rblm !1U1lI Ihe oublishOf
1
122
0.20
0. 15
0. 10
o.
2 3 4 5 6
Figure S.!") I t): TIll' Pois.--;ou pmhabi l il.y hiSl.ognull.). = :S, IIml H'IMi\"(' fl"l 'fIUf'IIC), hill tolgl"n ll'l (I'l lladed)
8 .5- 12 We shall lISC 10 SCt s of (!(11I1I1 prohllhilil.y.
A, Observed Expected q
( 0.00, '1.45) " 1/ 9 [ 4,4 5, 9.'12) 10 9 1/ 9 [ 9.41 , 15.05) 9 9 DI U [15.05, 21.5(l) 8 9 1/ 9 [2UiG, 29.25) 7 " 'II !) [1U.25, ;$8.(7) 11 " 4/ 0 [38.G7 , 50.8 1) 8 0 I/O [50.8 1, G7.92) 12 9 fJ I D [67.92. ~JI.I 7) 10 9 1/ 9
[DUr. (0) 7 9 4/ 9
!J(I 90 26I D- 2.89
Siucc 2.$!J < 15.51 = ~ fi. or. ( 8). wc acccpt. the hypoLlK'l>is Llwt t he distritmtioll of X is ,·Xpt"IIlcut ial. Note I.hat. ouc degree of fl"l'fXlolfI is lost h L'CHIl1iC we lUll! 1.0 L'SIi lll llh' O.
[ 0.020
0.015
0.010
0.005
Figure 8.5 12: Exponential p.d .f . • 0 = 42.2, nud rdllt ivc rrCtlllclicy histogrmu (slmdNI)
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8.5 I . Wt· s lla ll usc 10 sets (If ('( 111:'1.1 prub .. lbililY·
A. 01):i1'1'\'/'(1 E:>:JX'I 'lCd if
( -0...., Jf.l!JAO) III 0 I /~)
1:I!J9AO . • 1:17.92) 7 " '1/ 9 H;n. ~J 2. ,Wa.7 1) " 9 OJ!) 1'](;:1.71. -I S!). I.I} " 9 O/ !) / IH!JA'I. :,11 ,6:1) 1:1 9 IG/ 9 [!'i I l,Ga. fl:3:1$2) • 9 1/ 9 [fl:l:ut~. ;:';J7.M } 7 9 .1/ 9 /!ifI7.&rl. ;:'8;:' .:\.1) (; 9 9/ 9 /f,,sr,,:I_I. li2:1.8fi) " 9 ,I/ !)
[(;2:1.6(;. :x) III 9 1/ 9
90 90 ' lOj !J =·1. II
Sill('t' -1,,] '1 < 1-1 .07 _ \iior, (7) , WI' 1I('('l'PI. 11m hypotbl'sis t lmt 111(> di" t rihutiou tlf S i,.. N (/I . o'l ). Notl' 111/11 2 dq~r('('S of (1'(:'( ... 10111 III', ' losl bt-CIlIISC 2 pal'lllll{'ll'rs we're l'Stil l mtPd.
0.006
0.005
0.004
0.003
0.002 1
0.0011 3fOO;---~4~0~0-L-L~500~~L-'~~~~--~7~00~
8.6 Cont ingency Tab les
8.6 2 10.1$ < 20..11') = \~O:lr,( JO ) , I.W"" pl. 110.
8.G 'I lu tilt· ('Ulubiu('(1 SlIIUpl(, of I:", OI>:,t.·I'\'llt ilJllS, t.hl' 11,\\'pr I hinl illl·llldes I hoSt· wilh S(' ,,(I >:. tOr iii (lr lo\\·t' r . the lIliddlt' third hun' SCQrt':'i fro lll 02 th rough 7S. /Iud lht' higlwl' l ldnl lll'i' tilOS('
w it h S<'Orl'S of 7!) /Iud nbo\"·.
low lll iddlt· high Tota ls
Clll .... " U 9 , '] ].'')
(" ) ("J ( " ) Clu ....... " V " " " Hi
(f' ) (" ) (f))
CIHs.'l \V I G " 15 (f) ) (f.) (" )
Tota ls 15 15 l!i .If.
0200S PeaJ'9Of1 Educa11on. In!;: .• Upper SOOdle River . NJ All rig/lIS reserved. Thos I113teriat ~ protOCled under al OOPYrighrlaws as lhey alllently .,.ISt toooorllOl'l oi1h11i material may be ,entOdlJced on anv lorm Of fly <lnv m&IIr'18 willlOl.l oermisslon In wrillfllJ hom lhe nublisna<
124
Thus (J = 3.2 -I 0.2 + 1.8 + 0 + 0 I 0 I :3.2 + 0.2 + 1.8 = IDA.
Since IJ = 10.4 > 9.'188 = \~06(4),
we rejet"!.. the (.'( IUillity of t hese lhr('t' distriLulioIlS. (p-val\lc = U.034.)
8.6-li q = 1'1 .'110 < !).48.8 = ;X~ 06 - fa il to reject flo. (p-val ll~ = 0.078.)
8.6- 8 q = 4.268 > :l.S41 = \,605(1), reject flo. u>-value = 0.039.)
8.6 - 10 q = 7 .6~tl < 9.210 = ~ ijOI' fnil to reject 110 , "./-Vallie = 0.021.)
8.6- 12 (8) q = B.()()(; > 7.8 15 = \~.&(J), rejccL 1-10•
(b) II = 8.000 < 9.348 = \ 6.025(3). fail to reject 110. (p-vnluc = 0.0-16.)
8.6 1.4 (I = 8.792 > 7.37$ = X~ O:l[j(2), !,<'jC(' t 110. (v-valuc = U.012 .)
8.6- 16 (I = 4.2-12 < ·1.605 = ",310(2), fail to rcjt'CL fl o_ (p-vnluc = 0.120.)
8 .7 One-Factor A naJysis of Va ria nce
8.7 2 Source SS OF MS F
Treutment :li!8 .2~05 3 12u.4268 4.0078
Error :.116.4597 12 26.3716
Tott\.l 704.7402 15
F = 4.0078 > J.49 = ' '-'0.06(3, 12) , rcjt.'Ct. 110.
8.7- 4 SQurcc SS OF MS F 1~ valuc
Trcu tlllc n t 150 2 75 75 0.00006 Error 6 6
Tota l 156 8
8.7- 6 Source SS OF ~ I S F p-vnIIlC
'I)·(,fltulcnt. 184. 2 92..1 15.4 0.000 15 Error 102.0 17 6.0
Towl 286.8 19
F """ 15.4 > 3.59 = Foo:;(2, 17), reject. flo.
8.7- 8 (a) F 2: Fo.o~(:I , 24) = J .0 1;
(b) Source 5S OF ~I S
IJ.-\'aluc
0.0 188
F p-Wllu('
Trea t.lJIell t. 12,280.86 J 4,093.62 ~ . 455 0.0323 Error 2RA:\4.57 24 1,18477
Tot.al 40 ,715.4:1 27
F = 3 .455 > 3.01. rejeet. flo;
CJwpwr 1:1
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(c) 0 .025 < I)-value < 0.05.
(d ) X, --1 l-
X,
X, -I ~
X, I--
Figure 8.7 8: Dox-sud-whisker diugrn.rns for C'holeslcro l lf'vds
8.7 ]0 (nJ F ;:::: Po 0:;( 4, :'l0) = 2.69;
(1)) Source SS 01' MS " p-value
Tn:,l luumt O.Ollll 2 • 0.00111 2.85 0.0·103 Error 0.011 57 :10 0.000:19
' [utttl 0.01 5!)9 ;\ 1
F = 2.81) > 2.1)9. rcjcct flo;
(c) X, '--__ --'--'f--
X,
x,-I
X, -I
X, -I ---
1.02 1.04 1.06 1.08 1.10
Figure 8.7 10: Ilox-and-whisker diagrams rur Ilail wrights
r====9~2~.~II~"~-~1~O~~~.OOO~====== 8.7- 12 (n) t = - - - 2.55 < - 2.179, r('ji'1,: ~ 110,
G(6!U3!}):26(57.669) (~ I ~)
11 2)'>17 F = ;:---; = 6.507 > 4.75, rcjl."Cl 110.
v3.40·I;o;
T he F' 11 11£1 t hl' t t.c:o; ts g ive tllf' sam(' n!ljuiL'i sinn.' (l = F.
( ) r:> S6.:la:J6 ~
b r = 0 = 0.7;d 5 < :t55 , do lIo t r('j ()('t 1-10 . 114 .880:>9
1'15
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126 ( 'Jmp,eI' S
8. 1- 14 (0 ) Source S5 Of' MS F p-vn luc
Tre1:l fmenl 122. 1956 2 fil .097S 2. 1 :~ 0. 1:\6
Error 800.4799 30 28.6827
Tota l 982.6755 32
F = 2. 1:ill < :\.32 - FO.05(2, :.i0), fai l LO reject. lin;
( b)
0 6
0 7 I I
022--{~:=J--
165 170 1 B'"0;--~l1i5
FigUfP 8.7- 14: Box-8.ud-whisker diagra ms for re3ista ncet; 0 11 t hree dll)'S
8 .8 T wo-Factor An alysis of Varian ce
8.8- 2 JI -f 0 ; ij " 7 8 6
10 7 11 12 10 8 ., n 10 8
II + {JJ 8 5 9 10 J! - S
'0 (11 = - 2, 0"2 = 2, 03 = 0 {lud {:Jl = 0, fh. = :}, {33 = I , (J4 = 2. . , 8. 8-4 LL( X , - X . )(X,,- X , - X .,+ X .)
, X ) L [( X" .Y ,.) ( X , X .)]
. _ 1 ) - 1
t (X, -X) {t (X" -X,) - t (X, -x.) } 1"' 1 J "'" I J--I
" L eX,. - X .. j(O 0) ~ 0;
" . L L ( X J - K )(x .] - X , - X J + x .) = 0 , s imilll.r1y;
" , { " LL( X , - x )( x.,- x . ) ~ L ( X, , I ) 0>. 1 1 I
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Test.." of SIMi:_ ' k"ll IIy potlll'scs 127
8.8 6 11 +(\ , U 7 7 12 •
10 3 II " 8 ~ 5 'J 10 8
I' +- /"I) • " 9 10 I' 8
So 0 I = Ot = 0 3 = {) anu {j] O. :h =- -:1, ;h = I , 1'14 = 2 liS in Excn'i,.,.' d . 7 2 . HOWl'\·CI ,
8.8 -8
8.8- 10
/'11 = - 2 becausc 8 j. 0 + 0 -+ (-2) = 6. ~i lllil ltrly Wl' oiJlIIiu the .A1wr "t,} 's:
- 2 2 -2 2 2 - 2 2 - 2 000 U
SOllrce 5S Dr ~ I S F p-" 1\111('
Row (A) !lO.78OS 3 '1!J .kOO:i ·1.807 0.021
Col (0 ) 70 . W5!i I 70.H)!)5 6. 76~1 0.018 lut.( AO) 202.0$27 2 101.4!Jl4 !J .n/'! 0.001
Error U)(i.S306 18 W.:.J79!i
Totnl [,.59.789·1 2:3
~ilJ('I' F ..... = 9.778 > :t57, n ~II is fl·jt·<: l ed . r-. lost stalislicinn.s would probaul.\' U{)I pnJl"t'('t1
to tl':>t // . Illld 1/1<'
Source S5
n (/w (A) 5.1 ua.OUOO
Col (B) 6,121.28;:'7
Jm (A I3) 1.056.571 -1
Error 28,4;'4.571-1
TIML1 40,715 .. 12S6
DF
21
27
j\jS
5, 10:1 .0000
6 , 121.2/'!!j7
1.0['0 .[1711
1. 11'14 .77:.18
" 1"'\"a11l~
4.:i07 O.fW)
5. 107 u.0:12
0.!'~92 O.3!i4
(a) Si llce F = 0.892 < Ftl.Ob( I, 2<1 ) = 4.26 , do IJOt rej(,(,L H Ili:
(h) Siuce F = ,1.:107 > Po 06( 1. 24} .1.26. rej\'(" H ,;
(c) Siucc P fi.167 > PU05(l.Z4) '1.26. rejC't: 't /I ".
8.9 Tests Concerning Regression and Correlation
8.9 2 Tile critieal fl'g ioll i!:lll ? lo :,!,,{R) = ~ .:306 . F'rom Ex(' rC'iS(' 7.1$ 2 , >0
;J -I .ft 1/ 5.0.1 ami 1/;;2 =- I .R.1!)24; (tI:iO L (L, ~ :1" )2 = "', .U4. ,;0
( , i.(j..j / 5.0·1 O.!J2Uu ) ~ 0.21"2' = 4.:'1)9.
1.8'1!J2"
«(5.0' )
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128 Chapter 8
8.9-4 The:! criticlll region ili il ~ 10.01 (18) = 2.552. Siu(;c
- 24.8 -{3 = 40' /lq2 = 5. 1895. " and L (XI - X) 2 .-.
it follows t hat 2' .8/40 ..
II = = 7.:10.\. 5. I R%
18( 10)
Siun: 'I = 7.:SO:j > 2.552, we rejl'Cl 1-10 . We could a lso COllst.rlJe L the fo llowing whle. Ou t put like t hi., is given by Minitab.
Source SS DF MS F p-valU(l
Regression ) fi.3760 15.3760 5~1 .3323 0.0000 Error 5.1895 18 0.2883
Tota l 20.56!'i5 19
Note tlmt lr = 7.:.w32 = 5:1,3338 ~ F = 53.3323.
8.9- 6 For these d(lll\ , r = - 0 .413. Since Irl = 0.413 < 0.7292, do lIot J'cjCCL No.
8.9- 8 Followi ng: the sugg .... stiOIl givcu ill the him , f.he exp l'e>sioll C<Juals
n2S;s;.(ll - 1)5;
" (1/ !)S~ (I _ 2n2+ R2)
(n - l )S;(i - 11').
8.9 10 u(1l) ~ u(p ) + ( ll - p)u.' (p),
Vur[" (,,) r (11 p)u'(p)[ [u' (p)['''or(l/)
. (i - p')' [u' (p)]:l = c, which is free of p,
" k/ 2 k / 2 --+--I - p I +p'
u'(p)
11 (p) - - In( l - p) I- - ln(l+p) =- ln --. ~: ~~ ~. ( I + II) 2 2 2 I - p
T hIlS, tllkiug k = I ,
has /I variflllt.:c uilnost free of p.
8.9- 12 (a) r = O.-l906,l r l= U.tl906 > 0.4258, rcj(''C1 J-lo8t.tt = O.IO;
( b ) I r I = 0.4906 < 0.:1!J73 , fail La rejC(;t Ho ut 0 = 0.05.
B.9 14 (n ) r = 0.339, I r I = 0.:139 < 0.5325 = " 0.02~( 12), fa il to I'cject flo H,t 0 = 0.05;
( b ) ,. = - 0.82 1 < - 0.6613 = "0.005( 12), rejcct Ho at ( l = 0.005;
(c) ,. = 0.149. 1 "1 = 0. 14!) < 0.1',:\25 = "o. o2~( 1 2), fai l to reject flo nt (\ = 0.05.
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Tcsts of Sf 11/ ist irll l Hypo' Itcses 12fJ
8.10 Kolmogorov-Smirnov Goodness of Fit Test
8. 10- 4 (a)
8.10 (l
0.2
FiglJl'l' 8. 10 ·1: 110: .'( IIll:. n CllIlf'hy d istri buLioll
(b) clIO :: O.:~IOO at..r = - 0,77,1)7. S ill ("1' (J.3 1 < 0.37. W(' clo not rcjl-'ct the hYPllrllf'1>i,; that tile:;(' arc o~r\'ntjoJls of II Cauchy r8udom \'ariablc.
y = 1[Jx)
0 .6
0.4
0.2
20 40
Figlll·p $. 10 0: A !JU% t'olifidellCC baud fl)r F(.r)
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130 Cl ll1pter 8
8. LO- 8 The value of the Kolmogorov-Srnimov s tatistic iM 0.0587 which OCClini at x =: 21. We clearly accept the null hYI>othesis.
1.0
0.8
50 100 '1 50--200-~ 250
Figure 8. 10 8: No : X h»..'i all t1xpollelltial d istriblltioll
8. JO- LO c4;2 = 0.068 tl~.r -= 4 so we accept the hypothesis that X has a Poi:;SOll distrioution.
8.10- 12
1.0
0.8
0.6
0.4
0.2
Pigure 8. 10- 12: I-lo: X is N( 15.3, 0.62)
d l6 = 0. 1835 nt x = 15.6 so we do not reject. the hYPOlhl'>iis that the dbtriblil iOLl of I)CHum weights if> N ( 15.:l , 0.62).
Cl2005 Pearson EduCahon. Inc , lJppe< &tckte AMM. NJ All nghla resewed TIlia marttrialla prol8CTed under aM copyright laws 5!1lhey currently m:iSl No IlI'If1ion ot tl1ItI malerlal ma~ be MOI'oducod. In &"" IOfm or bv anll mlllllIlI WlII'Q 1I oennill$loo In wntlnn from 1M 0IJbIishe<.
'n~I S /Jf Sf Mis/ jelli HyporlJ(~s 131
8 .11 Run Test a nd Test for Ran dom ness
8. L L- 2 The cOllloineU ordered ~lIlLp lf' is:
13.IXI 15.50 IG.75 17.25 17.50 HLOO
Y ,. r x y y
19.25 19.75 20.50 20.75 21.50
x y .,. x y
22 .00 22.50 22.75 23.1".0 24.75
x x y y y
For t.hc:;e data, ,. = 9. Abo,
E(R) ~ 2(8)(") R+8
j- I = !l
so we d early !In·cpt. 110 .
8. LL- "J " x x x " x. r x r " E r.
x x x x x x , " x r x r .", x x L X X x, " " x x " x ,
" r x x x :1'. " x x x x L ,
,r x x x ,. x, " r x r x x.
8.11 - 6 Thl' cOInbiucd ordl'rc'd 1>lUll plt· is:
- 2.0 182 - 1.57,18 ~ 1.2311 - 1.0228 - 0.8830 - 0 ,8797 - 0.7170
r , y y y " x
- 0.6684 - 0 .6 157 -O.57~[, - 0.49U7 - 0.205 1 - 0.1019 - 0.0297
II Y Y ,. " y II
0. 165 1 0.2893 0.3186 O. ~~550 0.378J
x x x :r y
0.'1056 O.6!J75 0.7 113 0.7:t77
x x x "
0.7400 O .~47!J 1.0901 1.1 :\97 1.1718 1.2921 1.7:15/.i
y y y y y y " For thCSt' dat.a , 1 ill' HlIlIl tx' r of fUllS b 1" \1 . T Ill' /H'nlw\ or this test i ~
TJ-\'ulue = P(R S II ) ~ p( Z $ 1l.5 - IG.O) ,/1',(14)/29
= O.().l7:i.
Thus w{' wou ld rej~'t I/o Ill. an (10 ;: 0.0·173 ~ (J.D;) il iguifi('il/we 11'\'('1.
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1:12 Clwptf'r 8
8. L 1- 8 The median is 2:J.45. Replaci ng observu tions lielow Lhe mediulJ with L aIJd uoove the median with U , we bave
or ,. = 12 rHUS. Si llce
P(R " 12) (2 + 14+98+294+882)/ 12, 870
1290/ 12, 870 = 0.10 a nd
P(R 2: 1:.\) = 408/ 12, 87U = O.O;.J7J,
WI' WOl iid rejecL ti ll' iIYIXlt.ht!s is of ralldo l [lll (~ if Ct = 0. 10 but wou ld nut reject i f 0 = O.O~117.
8. U - 10 For these dnta, the 1I1f'dil111 is 21.55. R.cplncing lower nnd Il pper values with [oJ Bud U , respcct,ively, gives I,he followiug displi.lys:
"'- lL L /, U U If U "'- lL "'- lL L I. lL "'- U U "'- U U U U
' oJ L L U U it. !l... it. Y.. [. LIs!!.. L L
We S(:!t! t,llIlt tlwre Mf' ,. = 23 rullli. The value o f tlw sLrul(ll;Ird Ilonna.! test statistic is
23 - 20 ,~ ~~"'" / (19)( 18)/37
D.US7.
Thus we would 110t. rejP.Ct the hy p ot,hcsis of mndOllllll.:SS lit allY I"CIl.>;OIIIlbip s ignificauce level.
8 .11- 12 (It) TIll' IlIlmher of 1"1 111'" is r = :18. The I~valll(: of the test is
p(z > 37.5 - 28.964 ) - / (27.964)(26.96" )/ 5<928
P (Z :::: 2.30) = 0.0107,
sc) we would 1I0t rejl."1' t the hYIJotlll.:sis of ["(lIldo lllJl(.'s:,; iu fa vor o f Il cyclic effect. at
0 = 0.01. but the cvidc llce is strong tilat Llw laLt.er might exist. This , howel'cr, is !lOt bud.
(L) The d ifferent vCOIious of Lbc tl.,'lit were 1I0t writtCil ill such a way that Rilowed students to fin ish ellrlit' 1" 011 OU(' LImn 0 11 t he otlu~r.
8 . Li - L'l T he number of ruus is r = 30. The p- va ille of the tL's l, is
II-value = P( n 2:. 30) ~ (
095 - 35.886 ) P Z > --,""~-;;:,,;;."""""';;,,"""" - /(34.886)(33.886)/ 69.772
P(Z 2: 1.55) = 0.9:19.1,
~ we wou ld nm reject t he hYP011\1~ i s of ralldoUlncRs, lIlt.hOllgh there sccms to Ix: a tcmlcllcy of too few runs. A display of I he data shows that there is a cyclic effCt't with 101lg cycles.
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Chapter 9
Theory of Statistical Tests
9.1 Powe r of a Statis t ical Test
9 . 1- 2 (It) /{ (/I ) P( X ::5 :l 54.0.'); II)
(z :l[).I.0 5 - I I ) =: p < . I' - 2/ .;T'i ,
'I) . ( :i !i .. L!J5 - II)
2/ .;T'i ,
• ( :1.')4 .0.') - 355) (b) u :=: I, (:\55) = (II Ii<> =- !P( - 1.6·lr}) = D.Oa; 2/ v 12
(e) K (:l5.J .05) <1>(0) = 0.5;
(1)( 1.645) = 0 .95. /( (:15:1.1)
K(~)
1 .00 t-_~
0.75
0.50
0.25
.- 353 ~353.5 · ' 354 -·354.5~5 ··
Fig ure !.I. I 2: 1«11) = IlJ ([:1"i 1.0,'l - ltIl12/ fi2 ])
(d ) :r - :~;.L8J < :\5 .. 1.05, rejt'ft 1-10:
(e) p-vnluc = P ( X ::5 ;\!):Ut\ ; 11 = :j!)5)
P(Z::5 - 2.0:1) = tl 0212.
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9. 1- 4 (ll) J((J, ) P( X ;?:&1: /.1 )
= p(z > 8:J - 11) = I '11(8:1- 1/) . - 10/5 2 '
(b) 0 ~ 1( 80) ~ I - <1>(1.5) ~ 0.0668;
(e) K(80) a = 0.0668,
(d)
1« 8")
(( SG)
K(~)
1.00
0.75
0.50
0.25
78
1 - '~ (O) = 0.:)000,
1 - .1>( - I .u) = O.!l:tU:
8-0~'~ 82~'-' 84 ' . . 86 .
Figure 9. 1 4: K(/, ) = I - /1) (183 - 111/2)
(e) Ir vuJllc P( X 2: 8:1.4 I; P = 8U)
P(Z:;::" 1.70r)) = 0.0·1111.
0 . 1- 0 (n)
(b)
(e)
I( (11) P( X 5 G68.!H j II )
,,(6G8,94 - 11). 1 140/G '
= P (z < (i08.9·1 - II , t) - 140/5 , I
(} = J( (7IS ) 'I' (668.9-1- 71 5) 140/ 5
<1>(- 1.645) = 0.05:
A·(GGS. 94 ) " (0) = 0.5;
1< (G22.88) <I>( 1.G45) = 0.95;
G'JlIlJ)I,( ' /' (J
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K(p)
1.00
Theory of Statistical Inference
(d) K(Il)
1.00
0.75
0.50
0.25
600 620 640 660 680 700 720 746
Figure 9.1-6: K(/l) = <1)([668.94 - /lJ/[140/5])
(e) x = 667.992 < 668.94, reject Ho;
(f) p-value P( X :::; 667.92; /l = 715)
P(Z:::; -1.68) = 0.0465.
9.1-8 (a) and (b)
0.5 0.6 P
0.7
K(p)
1.00
0.75
0.50
0.25
Figure 9.1-8: Power functions corresponding to different critical regions and different sample sizes
8
9.1-10 Let Y = L Xi. Then Y has a Poisson distribution with mean /l = 8A. i=l
(a) Q: P(Y 2: 8; A = 0.5) = 1 - P(Y :::; 7; A) = 0.5
1 - 0.949 = 0.051.
(b) K(A) = 1 - t (8A)Ye- 8,\
y=o y!
(c) K(0.75)
K(1.00)
K(1.25)
1 - 0.744 = 0.256,
1 - 0.453 = 0.547,
1 - 0.220 = 0.780.
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135
136
,
KOJ 1.00
0 .75
0.50
0.25
j
'---"-<;;0.·5-~71 .0-- 1 ·".5-~"2.0
Figure 9.1 10: l\' (..\) = I - PO~:::; 7;,,\)
. A 2.5
9.1 - 12 (a) L X. has gamma dist ribution with parameters 0 '- :ll,\ud O. Thus _.,
(b)
K(a)
1.00
0.75
0 .50
0.25
(0) 1«2)
/((1 )
1(1 / 2)
1«1 / 4)
2~3 [_ 0.r.2e-:<"/0 - 202:ce -:rIO - 20·l e-:r/8J:
.:f. (2/ 9)' _'i'. L. --,-e , 1''"'0 y.
~:;;:;=:=;. e 1.5 2.0 2.5
Figure 9.1 12; 1\"(0) = PO=~,",l X, :5 2)
2 l"e- 1 I - L - ,- = I 0.920 = 0.080;
,,=0 y.
I - 0.677 = O.:t2:1;
I - 0.238 ~ 0.762,
I - 0.0 14 = 0.986.
Cllllpff'r 9
C2005 Poarson EdUGatioo. Inc., Upper SarkIe River , NJ J\II rig!~M 11,186/_ 1. l'tlIs malerlalls proteclOO I,lIl(lQr eu oopyriCJ/IIlaws as lOOy currently axial.. NIl DO"N'In oI1hi11 malM1111 mav bII roomducBd In anY Iorm Of IW 8!'IV rnBal'l5 wiIhou1 OEIrrnIMIoo In writIrla 110m Ihe [)ublisnOf
Theory of Stl1tisti<;ul IllfcrclI (,c
9.2 Best Critical Regions
9.2- 2 (lI) L(4) L(16)
(I / 2.)2i )" cx p[ - I: J': / 8 J
(1 / 4v"2i )1l cxp[- E x? /:12 1
2" l'xp[-3E x; / :l2 ! :5 k
9.2- 4
, " _ ":"- ~ L2 <_
J2 L ' .. , " '" LL, ~ ,.,
Inl..·- 11I2"
(32) - "3 {lul..·- ln2") = c;
=" P L..,., . I · . >~ '(f2 = .J ( '~" y' ) 4 - .1'
Thus:' = \'5 05( 15) = 25 uud {' = 100. 4
(e) ,J ~ p(f: X; < l()(l,u' ~ IG) ,., p " .. I X i < 100 (l:" ,
16 16 6.2,,) ~ 0.025.
(a) 1,(0.9) (O.!))1:"" (0.1 ),, - 1: .£;
S I ' L(O.S) ~ (0.8)1: .... (0.2),, - 1:..,·
[mmll:;r.[~r s k
(t ".)t,,(o/. ) s lnk + llln2 ,. , " III 1..' + ,,1 11 2
Y ~ Lr, S 1,, (9/·1) = (" .
, , Ill'(·/dl that tilt' distribu tion of l llc sum of Oe ruo uUi trinh •. Y . is b(fl. I' )·
(b) 0.10 ~ P[ I'S " (U.S5j;,) ~ 0.9[
P [_Y:,:",;;" ;;:(O';;;'Of,,) < ',,(0.85) - ,,(0.9) 'I' ~ 0.9] . ) ,, (0 .0)(0.1) - ) ,, (0.9)(0. 1)
" is true, lLpprox.iuJaI.('ly, t hat
(c) P[ Y > ,,(O.~~) = fJ L p = 0.81
(d ) Yes.
n(- O.Of,) fo(O.:I)
1.21:S2
11 5!). 17 vr n = (;O.
[ )' - 60(0.8) 51 - . 8 . ]
p )60(0.8)(0.2) > J91i' p ~ 0 .•
~ P(Z ~ 0.07) ~ 0.166.
1:17
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(:is
(b)
Thus
c, - 80 ----y;-
c ,
Similurly.
C'J - 8U
----:iT4 ",
(';1
3/4
CJ
/( , ( / 1)
[(1 (/1 )
1('(/ ' )
=
I - II) --- . (
C. 80) 3/ 4
1.6<1 5
81.234.
- 1.6.Jn
78.766;
1.9(;
1.<17.
1 - '1>([8 1.2:1'1- 1.1/[3/>1[);
'~ ([78 . 766 1.1/[3/ 4[;
1 - '1>([8 1.47 1.1/[:I/ 4[) + ">([7S.S3 - 1.1/[3/ 4[).
K(~)
1.00
0.75
0.50
0.25
~~' .. "
""
\, K2
,
\ K, ,
,/ "
/ ,/
.
/.,/'
K I ./ , .
Figure 9.2 6: T hree power fuuClions
Chllpter 9
C2005 Paaf$Qfl EdlJC8i1Orl. 1rIc.. Upper Sa!kIe RIVOI . NJ All nghl!i roservocl This malOlial is proledod undor all OOf'yrl£lll/ laWS as IhcIy cu rronlly allisl . No 0011100 til ItlIR mil llIfIiIl miI~ hIIl'I'IOI'tldtlCfl(! In anY form nr bv anY ITIMnII. wiIhout DfIfITIIuInn In WrilII'lD hom rtIe nllbl~l8r
9.3 Likelihood Ratio Tests
9.a- 2 en) If I' ( w (that is, p 2: 10.:'1;1). t lll' lI ;; =- 7 if r 2: 10.:1:), bil l. ii = 1O.3G if l' < 10 :'15.
Thus). = I if J' ?: 1O.:lfl; but . if r < 1O . :!~. t )II: 1I
• II 1(0.3)(2')1"1' <xp1 L:;' (x, 10·3" )' /(0.06)1 (l / (0.3)(2')I "I' <x ,,[- L;' v, - x)' /(O.IKi )1
<
[ " exp - 0.06 (1" IO.3S )J] " " - 0.00 (:r 1 0 .~")' "
:i. - lO.3!"1
/0.0:'/" " ( b) .1 0 .",.::;:"=-~llJ;;;.3o:.5 o· . - .. - 1. :.i.I >
/ U.03/ 50 1.6Hi ; do not rcjl't:L Ho.
(c ) IH'R lm.' = P(Z :5 - I.G:-I:i) O.Of, I :.t
9 .3 <I (n) 1.<:1 = !:r. - !in [ ?: l.UG; IS/ .J/i
[56. 1:1 - 59 1 (b) 1=1 = 15/ 10 = 1- I.!JI ;SI < 1.96, do nOL r( ·jl.'~ .. t flo:
(e) IHl} lu(' = P(]ZI?: l.OU) O.fl55S.
32-1 .8 a35 9.3- 6 t = ==--~:::
10/ y'i7 1.05 1 > - 1.:1:37 , do not rcjf'<:L Ho.
! .
k
III k
J-2Iu~' -':0 00
= - 1.li45.
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140
9.3- 8 1110, ;; = :E. T hw;,
(1/ 00 )" expI- L:: ,./00 1
(I tt)" e..xp! ~;' .c,/x] :5 k
(~)" e.xp[- n(x/ Oo - I )l :5 k.
C/lIlpt,e.r 9
P lottillg A as a function of w = £/00 , we see lImt ,\ = 0 whell :E/ Oo = 0, iL hus tL ll'ln..ximulll
when ?f/ Oo - 1, and it upproacllt..'S 0 as x/Oo becoHlt..'s large. T hlls ,\ .S: k wheu x :5 CI or J; ~ C2'
.) " Siuee the di~lr ibution of i L X, is x2(2n) when 110 is Lru~, we conkl lei the criticlll
o 1 ... 1
region be such that we reject No if
0.25
? " o LX, :5 XLn/2(2n.) or (j i= t
,
,
, i '
, , ,
, , ,
.... t' \
-- ------- ---'~ --- -------, '.
LL.-"o/ C;c-'---;--;:- -~~',,;.--'=;--.,::::;~ 0.5 1.0 1.5 2.0 2.5 3.0
Figure 9.:\- 8: Likchood fUIiCtiOIlS: solid , 11 = 5; dot.t(.'(I, 11 _ 10
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Chapter 10
Quality Improvement Through Statistical Methods
10.1 Tirne Sequences
10: ' 2
3.6
3.5
3.4
3.3
3.2
3" 1 3.0
Fig llrl' 10. 1 2: Apple' Wl'ights fl'OIll S('ll!'=S !i (Iud G
III
02006 PearSQII Educa.lon. Inc , Upper Saddle R,ver. N.J. All r1gt11S r868fVBd. ThIS malllna. is pIOl8CIecl undef a. oopyr~ IiIws illS they currendy tllOSf. No DQf.oon oIlhIII mlnena' mav he feOfoduCfid. k> a"", Inrm Of DII a"", means. WI1hau' 08ml1!11101)tl in wriIina .rom IIlQ l'Iul)lo!ihllf
160
140
120
100
80
142
10.1- 4 (a) 24
23
22 2 1
20
19
18
17
16
15
14
\ , \ 1\
' ,' ' " , ,
" '
Pigurc 10.1 4: US birth wciglllS. 1!)(j()..l!}!)i
( b ) From l!)GO to the mid 1970's there is l1 d OWllwn.rd trcuci llud theu /I faiJ·Jy s teady rat.c followed by a short upward I.rend ami then nJlOLhcr downward t rend.
10. L 6 (n) /Iud (b)
It(x)
0.14
0.12
0.10
0.08
.-
0.08
0.04
,.., 0.02
,
!-
~. .-j ! - !
I I r-
I , I
. 0 ---'0 -- "20"' '~3(i " 4C} .-. '50'- " sO~ '-70' 78 87 96 105114 123 132 141 150 159 x
Figure 10. J 6: FOI'(,t! rt·quired to pull Ollt swds
(e) 'fhe data arc cyclic. leading to a bimodal distri bution .
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