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7/24/2019 Statistics for Business and Economics: bab 13
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Slides Prepared by
JOHN S. LOUCKSSt. Edwards University
2002 South-Western/Thomson Learning
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Chapter 13Analysis of arian!e and "#perimental
$esign An %ntrodu!tion to Analysis of arian!e Analysis of arian!e& Testing for the "'uality
of
k (opulation )eans )ultiple Comparison (ro!edures An %ntrodu!tion to "#perimental $esign Completely *andomi+ed $esigns *andomi+ed ,lo! $esign
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Analysis of arian!e.AA !an e used to
test for the e'uality of three or morepopulation means using data otained fromoserational or e#perimental studies4
We 5ant to use the sample results to test the
follo5ing hypotheses4
H0& 1626364 4 4 6
Ha& ot all population means are e'ual
%f H0is re7e!ted8 5e !annot !on!lude that all
population means are di9erent4
*e7e!ting H0
means that at least t5o
population means hae di9erent alues4
An %ntrodu!tion to Analysis of arian!e
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Assumptions for Analysis of arian!e
;or ea!h population8 the response ariale is
normally distriuted4 The arian!e of the response ariale8 denoted
28 is the same for all of the populations4 The oserations must e independent4
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,et5een-Samples "stimateof (opulation arian!e
A et5een-samples estimate of 2 is !alled
the mean s'uare et5een.)S,4
The numerator of )S, is !alled the sum ofs'uares et5een.SS,4
The denominator of )S, represents the
degrees of freedom asso!iated 5ith SS,4
)S, =
=
n x x
k
j jj
k
. 2
1
1
)S, =
=
n x x
k
j jj
k
. 2
1
1
66??
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Within-Samples "stimateof (opulation arian!e
The estimate of 2ased on the ariation of
the sample oserations 5ithin ea!h sample is!alled the mean s'uare 5ithin.)SW4
The numerator of )SW is !alled the sum ofs'uares 5ithin.SSW4
The denominator of )SW represents thedegrees of freedom asso!iated 5ith SSW4
)SW =
=
. n s
n k
j jj
k
T
1 2
1
)SW =
=
. n s
n k
j jj
k
T
1 2
1
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Comparing the arian!e "stimates& The FTest
%f the null hypothesis is true and the AA
assumptions are alid8 the samplingdistriution of )S,/)SW is an F distriution5ith )S, d4f4 e'ual to k- 1 and )SW d4f4 e'ualto nT- k4
%f the means of the k populations are note'ual8 the alue of )S,/)SW 5ill e inBatede!ause )S, oerestimates 24
en!e8 5e 5ill re7e!t H0if the resulting alue of
)S,/)SW appears to e too large to haeeen sele!ted at random from the appropriateFdistriution4
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Test for the "'uality of k (opulation)eans
ypotheses
H0& 1626364 4 4 6
Ha& ot all population means are e'ual
Test Statisti!
; 6 )S,/)SW *e7e!tion *ule
*e7e!t H0if FE F
5here the alue of F is ased on an Fdistriution 5ith k- 1 numerator degrees offreedom and nT- 1 denominator degrees offreedom4
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The Fgure elo5 sho5s the re7e!tion region
asso!iated 5ith a leel of signiF!an!e e'ual to5here F
denotes the !riti!al alue4
Sampling $istriution of )ST*/)S"
$o ot *e7e!t H0$o ot *e7e!t H0 *e7e!t H0*e7e!t H0
)ST*/)S")ST*/)S"
Criti!al alueCriti!al alue
F
F
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The AA Tale
Sour!e of Sum of $egrees of )ean
ariation S'uares ;reedom S'uares;
Treatment SST* k- 1 )ST* )ST*/)S"
"rror SS" nT - k )S"
Total SST nT- 1
SST diided y its degrees of freedom nT- 1 is
simply the oerall sample arian!e that 5ould e
otained if 5e treated the entire nToserations
as one data set4= =
+==k
j
n
i
ij
j
xx1 1
2 SSESST!"SST
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"#ample& *eed )anufa!turing
Analysis of arian!e
G4 *4 *eed 5ould lie to no5 if the meannumer of
hours 5ored per 5ee is the same for thedepartment
managers at her three manufa!turing plants.,u9alo8
(ittsurgh8 and $etroit4
A simple random sample of < managers from
ea!h of
the three plants 5as taen and the numer ofhours
5ored y ea!h manager for the preious 5ee
is
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Analysis of arian!e
(lant 1 (lant 2 (lant 3
seration ,u9alo (ittsurgh$etroit
1 : @3 3
3 > >1
: : 2 @: Sample )ean 40 2>41>
Analysis of arian!e F- Test
%f H0is true8 the ratio )S,/)SW should e
near 1
sin!e oth )S, and )SW are estimating 24
%f Ha
is true8 the ratio should e signiF!antlylarger than
1 sin!e )S, tends to oerestimate 2
4 *e7e!tion *ule
Assuming 6 40
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"#ample& *eed )anufa!turing
Analysis of arian!e
Test Statisti!
F6 )S,/)SW 6 2:@ 6 D4
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Analysis of arian!e
AA Tale
Sour!e of Sum of $egrees of )ean
ariation S'uares ;reedom S'uare
; Treatments :D0 2 2:3>
Completely *andomi+ed $esign AA Tale
Sour!e of Sum of $egrees of)ean
ariation S'uares ;reedomS'uares ;
Treatments
:D0 2 2: