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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: