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7/21/2019 ANOVA Dasar

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Analysis of VarianceAnalysis of Variance

Introduction to Analysis of VarianceIntroduction to Analysis of Variance

Analysis of Variance: Testing for the Equality ofAnalysis of Variance: Testing for the Equality of

kk Population MeansPopulation Means

Multiple Comparison ProceduresMultiple Comparison Procedures


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A!A"ISIS VA#IA!S $A!%VA&A!A"ISIS VA#IA!S $A!%VA&

ISI:ISI:

' Men(elas)an )onsep dasar A!%VAMen(elas)an )onsep dasar A!%VA

$*(i +&$*(i +&

' #umus dan cara penghitungan ,er,agai#umus dan cara penghitungan ,er,agai

item dalam A!%VAitem dalam A!%VA' Contoh penggunaan A!%VAContoh penggunaan A!%VA

' +isher test untu) u(i per,edaan diantara+isher test untu) u(i per,edaan diantara

rata-rata sampel yang adarata-rata sampel yang ada


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Analysis of VarianceAnal

ysis of Variance$A!%VA& can ,e used to test$A!%VA& can ,e used to testfor the equality of three or more population means/for the equality of three or more population means/

Analysis of VarianceAnal

ysis of Variance$A!%VA& can ,e used to test$A!%VA& can ,e used to testfor the equality of three or more population means/for the equality of three or more population means/

0ata o,tained from o,serational or e2perimental0ata o,tained from o,serational or e2perimental

studies can ,e used for the analysis/studies can ,e used for the analysis/

0ata o,tained from o,serational or e2perimental0ata o,tained from o,serational or e2perimentalstudies can ,e used for the analysis/studies can ,e used for the analysis/

3e 4ant to use the sample results to test the3e 4ant to use the sample results to test thefollo4ing hypotheses:follo4ing hypotheses:

3e 4ant to use the sample results to test the3e 4ant to use the sample results to test thefollo4ing hypotheses:follo4ing hypotheses:

HH55:: 116666..66/ / // / / 66 ))

HHaa: !ot all population means are equal: !ot all population means are equal


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7


HH55:: 116666..66/ / // / / 66 ))


IfIf HH55is re(ected8 4e cannot conclude thatis re(ected8 4e cannot conclude that allallpopulation means are di9erent/population means are di9erent/

IfIf HH55is re(ected8 4e cannot conclude thatis re(ected8 4e cannot conclude that allallpopulation means are di9erent/population means are di9erent/

#e(ecting#e(ecting HH55means that at least t4o populationmeans that at least t4o population

means hae di9erent alues/means hae di9erent alues/

#e(ecting#e(ecting HH55means that at least t4o populationmeans that at least t4o population

means hae di9erent alues/means hae di9erent alues/


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Sampling 0istri,ution of ;ienSampling 0istri,ution of ;ien HH55is Trueis Truexx


1x1x .x.xxx

Sample means are close togetherSample means are close together,ecause there is only,ecause there is only

one sampling distri,utionone sampling distri,ution4hen4hen HH55is true/is true/

xn

=

xn

=


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+or each population8 the response aria,le is+or each population8 the response aria,le isnormally distri,uted/normally distri,uted/+or each population8 the response aria,le is+or each population8 the response aria,le isnormally distri,uted/normally distri,uted/

Assumptions for Analysis of VarianceAssumptions for Analysis of Variance

The ariance of the response aria,le8 denotedThe ariance of the response aria,le8 denoted 88

is the same for all of the populations/is the same for all of the populations/

The ariance of the response aria,le8 denotedThe ariance of the response aria,le8 denoted 88is the same for all of the populations/is the same for all of the populations/

The o,serations must ,e independent/The o,serations must ,e independent/The o,serations must ,e independent/The o,serations must ,e independent/


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:Testing for the Equality ofTesting for the Equality of kkPopulationPopulation

MeansMeans ?et4een-Treatments Estimate of Population Variance?et4een-Treatments Estimate of Population Variance

3ithin-Treatments Estimate of Population Variance3ithin-Treatments Estimate of Population Variance

Comparing the Variance Estimates: TheComparing the Variance Estimates: The FF TestTest

A!%VA Ta,leA!%VA Ta,le


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?et4een-Treatments Estimate?et4een-Treatments Estimateof Population Varianceof Population Variance

A ,et4een-treatment estimate ofA ,et4een-treatment estimate of is called theis called the

mean square treatmentmeansquare treatmentand is denoted MST#/and is denoted MST#/

1

$ &

MST# 1

k

j jj

n x x

k

=

=

1

$ &

MST# 1

k

j j

j

n x x

k

=

=

0enominator represents0enominator representsthethe degrees of freedomdegrees of freedomassociated 4ith SST#associated 4ith SST#

!umerator is the!umerator is the

sum of squaressum ofsquaresdue to treatmentsdue to treatmentsand is denoted SST#and is denoted SST#


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The estimate ofThe estimate of ,ased on the ariation of the,ased on the ariation of the

sample o,serations 4ithin each sample is calledsample o,serations 4ithin each sample is calledthethe mean square errormean square errorand is denoted ,y MSE/and is denoted ,y MSE/

3ithin-Samples Estimate3ithin-Samples Estimateof Population Varianceof Population Variance

kn

sn

T

k

j

jj

=

=1

2)1(

MSEkn

sn

T

k

j

jj

=

=1

2)1(

MSE

0enominator represents0enominator representsthethe degrees of freedomdegrees of freedom

associated 4ith SSEassociated 4ith SSE

!umerator is the!umerator is the

sum of squaressum of squaresdue to errordue to error

and is denoted SSEand is denoted SSE


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Comparing the Variance Estimates: TheComparing the Variance Estimates: The FFTestTest

If the null hypothesis is true and the A!%VAIf the null hypothesis is true and the A!%VA

assumptions are alid8 the sampling distri,ution oassumptions are alid8 the sampling distri,ution o MST#MSE is anMST#MSE is an FFdistri,ution 4ith MST# d/f/distri,ution 4ith MST# d/f/ equal toequal to kk- 1 and MSE d/f/ equal to- 1 and MSE d/f/ equal to nnTT-- kk//

If the means of theIf the means of the kkpopulations are not equal8 thpopulations are not equal8 th

alue of MST#MSE 4ill ,e inBated ,ecause MST#alue of MST#MSE 4ill ,e inBated ,ecause MST# oerestimatesoerestimates //

ence8 4e 4ill re(ectence8 4e 4ill re(ect HH55if the resulting alue ofif the resulting alue of

MST#MSE appears to ,e too large to hae ,eenMST#MSE appears to ,e too large to hae ,een

selected at random from the appropriateselected at random from the appropriate FF distri,ution/distri,ution/


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Test for the Equality ofTest for the Equality of kk PopulationPopulationMeansMeans

FF6 MST#MSE6 MST#MSE

HH55:: 116666..66/ / // / / 66 ))


ypothesesypotheses

Test StatisticTest Statistic


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#e(ection #ule#e(ection #ule

4here the alue of4here the alue of FF

is ,ased on anis ,ased on an

FFdistri,ution 4ithdistri,ution 4ith kk- 1 numerator d/f/- 1 numerator d/f/andand nnTT-- kkdenominator d/f/denominator d/f/

#e(ect#e(ect HH55ififpp-alue-alue DDpp-alue Approach:-alue Approach:

Critical Value Approach:Critical Value Approach: #e(ect#e(ect HH55ifif FFFF


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Sampling 0istri,ution of MST#MSESampling 0istri,ution of MST#MSE

#e(ection #egion#e(ection #egion

0o !ot #e(ect H5

#e(ect H5

MST#MSE

Critical ValueF

Sampling 0istri,utionSampling 0istri,utionof MST#MSEof MST#MSE


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SST isSST is

partitionedpartitionedinto SST# andinto SST# and

SSE/SSE/

SSTFs degrees ofSSTFs degrees offreedomfreedom

$d/f/& are partitioned$d/f/& are partitionedintointo

SST#Fs d/f/ and SSEFsSST#Fs d/f/ and SSEFsd/f/d/f/

TreatmentTreatment

ErrorError

TotalTotal

SST#SST#

SSESSE

SSTSST

kkG 1G 1

nnTT GG kk

nnTT- 1- 1

MST#MST#

MSEMSE

Source ofSource ofVariationVariation Sum ofSum ofSquaresSquares 0egrees of0egrees of+reedom+reedom MeanMeanSquaresSquares

MST#MSEMST#MSE

FF


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A!%VA can ,e ie4ed as the process of partitioningA!%VA can ,e ie4ed as the process of partitioningthe total sum of squares and the degrees of freedomthe total sum of squares and the degrees of freedominto their corresponding sources: treatments and errinto their corresponding sources: treatments and err

A!%VA can ,e ie4ed as the process of partitioningA!%VA can ,e ie4ed as the process of partitioningthe total sum of squares and the degrees of freedomthe total sum of squares and the degrees of freedominto their corresponding sources: treatments and errointo their corresponding sources: treatments and erro

0iiding the sum of squares ,y the appropriate0iiding the sum of squares ,y the appropriatedegrees of freedom proides the ariance estimatesdegrees of freedom proides the ariance estimatesand theand the FFalue used to test the hypothesis of equalalue used to test the hypothesis of equalpopulation means/population means/

0iiding the sum of squares ,y the appropriate0iiding the sum of squares ,y the appropriatedegrees of freedom proides the ariance estimatesdegrees of freedom proides the ariance estimatesand theand the FFalue used to test the hypothesis of equalalue used to test the hypothesis of equalpopulation means/population means/

f h li fT t f th E lit f kk l iP l ti


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E2ample: #eed ManufacturingE2ample: #eed Manufacturing


Hanet #eed 4ould li)e to )no4 ifHanet #eed 4ould li)e to )no4 if

there is any signicant di9erence inthere is any signicant di9erence in

the mean num,er of hours 4or)ed perthe mean num,er of hours 4or)ed per

4ee) for the department managers4ee) for the department managers

at her three manufacturing plantsat her three manufacturing plants

$in ?u9alo8 Pitts,urgh8 and 0etroit&/$in ?u9alo8 Pitts,urgh8 and 0etroit&/

T f h E li fT t f th E lit f kk P l iP l ti


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A simple random sample of eA simple random sample of e

managers from each of the three plantsmanagers from each of the three plants

4as ta)en and the num,er of hours4as ta)en and the num,er of hours

4or)ed ,y each manager for the4or)ed ,y each manager for the

preious 4ee) is sho4n on the ne2tpreious 4ee) is sho4n on the ne2t

slide/slide/

Conduct anConduct an FFtest usingtest using 6 /5/6 /5/

T t f th E lit fT t f th E lit f kk P l tiP l ti


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5

11

..

77

7>7>

77

==

77


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HH55:: 1166 66 ..

HHaa: !ot all the means are equal: !ot all the means are equal

4here:4here: 11 6 mean num,er of hours 4or)ed per6 mean num,er of hours 4or)ed per

4ee) ,y the managers at Plant 14ee) ,y the managers at Plant 1 6 mean num,er of hours 4or)ed per6 mean num,er of hours 4or)ed per

4ee) ,y the managers at Plant 4ee) ,y the managers at Plant .. 6 mean num,er of hours 4or)ed per6 mean num,er of hours 4or)ed per

4ee) ,y the managers at Plant .4ee) ,y the managers at Plant .

1/ 0eelop the hypotheses/1/ 0eelop the hypotheses/

pp-Value and Critical Value Approaches-Value and Critical Value Approaches

T t f th E lit fT t f th E lit f kk P l tiP l ti


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/ Specify the leel of signicance// Specify the leel of signicance/ 6 /56 /5


pp-Value and Critical Value Approaches-Value and Critical Value Approaches

./ Compute the alue of the test statistic/./ Compute the alue of the test statistic/

MST# 6 7@5$. - 1& 6 7MST# 6 7@5$. - 1& 6 7SST# 6 $ -


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./ Compute the alue of the test statistic/./ Compute the alue of the test statistic/


MSE 6 .5>$1 - .& 6 /


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TreatmentTreatment

ErrorError

TotalTotal

7@57@5

.5>.5>

=@>=@>

11

1717

77

/


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/ 0etermine 4hether to re(ect/ 0etermine 4hether to re(ect HH55//

3e hae suLcient eidence to conclude that3e hae suLcient eidence to conclude thatthe mean num,er of hours 4or)ed per 4ee)the mean num,er of hours 4or)ed per 4ee),y department managers is not the same at,y department managers is not the same atall . plant/all . plant/

TheThepp-alue-alue DD/58/58so 4e re(ectso 4e re(ect HH55//

3ith numerator d/f/ and 13ith numerator d/f/ and 1denominator d/f/8denominator d/f/8

thethepp-alue is /51 for-alue is /51 for FF6


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@8 4e re(ect./>@8 4e re(ect HH55//

Critical Value ApproachCritical Value Approach

7/ 0etermine the critical alue and re(ection rule/7/ 0etermine the critical alue and re(ection rule/

#e(ect#e(ect HH55ifif FF./>@./>@


3e hae suLcient eidence to conclude that3e hae suLcient eidence to conclude thatthe mean num,er of hours 4or)ed per 4ee)the mean num,er of hours 4or)ed per 4ee),y department managers is not the same at,y department managers is not the same atall . plant/all . plant/

?ased on an?ased on an FFdistri,ution 4ith numeratordistri,ution 4ith numeratord/f/ and 1 denominator d/f/8d/f/ and 1 denominator d/f/8 FF/5/56 ./>@/6 ./>@/


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=

Multiple Comparison ProceduresMultiple Comparison Procedures

Suppose that analysis of ariance hasSuppose that analysis of ariance has

proided statistical eidence to re(ect the nullproided statistical eidence to re(ect the nullhypothesis of equal population means/hypothesis of equal population means/

+isherFs least signicant di9erence $"S0& procedure can+isherFs least signicant di9erence $"S0& procedure can,e used to determine 4here the di9erences occur/,e used to determine 4here the di9erences occur/


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+isherFs "S0 Procedure+isherFs "S0 Procedure

1 1MSE$ &

i j

i j

x xt

n n

=

+1 1MSE$ &

i j

i j

x xt

n n

=

+


ypothesesypotheses

5 : i jH 5 : i jH

:a i jH :a i jH


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4here the alue of4here the alue of ttaa is ,ased on ais ,ased on a

ttdistri,ution 4ithdistri,ution 4ith nnTT-- kkdegrees of freedom/degrees of freedom/


#e(ect#e(ect HH55ififpp-alue-alue DD

pp-alue Approach:-alue Approach:

Critical Value Approach:Critical Value Approach:

#e(ect#e(ect HH55ifif ttD -D -ttaa oror tt ttaa



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.5


+isher s "S0 Procedure+isher s "S0 Procedure?ased on the Test Statistic?ased on the Test Statisticxxii--xxjj

A 1 1"S0 MSE$ &i j

t n n= +A 1 1"S0 MSE$ &i jt n n= +

4here4here

i jx xi jx x

#e(ect#e(ect HH55if "S0if "S0i jx xi jx x

ypothesesypotheses


5 : i jH 5 : i jH

:a i jH :a i jH



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#ecall that Hanet #eed 4ants to )no4#ecall that Hanet #eed 4ants to )no4

if there is any signicant di9erence inif there is any signicant di9erence in

the mean num,er of hours 4or)ed perthe mean num,er of hours 4or)ed per

4ee) for the department managers4ee) for the department managers

at her three manufacturing plants/at her three manufacturing plants/

Analysis of ariance has proidedAnalysis of ariance has proided

statistical eidence to re(ect the nullstatistical eidence to re(ect the null

hypothesis of equal population means/hypothesis of equal population means/

+isherFs least signicant di9erence $"S0&+isherFs least signicant di9erence $"S0&

procedureprocedure

can ,e used to determine 4here the di9erencescan ,e used to determine 4here the di9erences

occur/occur/



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.

+or+or 6 /5 and6 /5 and nnTT-- kk6 1 G . 6 16 1 G . 6 1

degrees of freedom8degrees of freedom8 tt..55 6 /1=@6 /1=@

"S0 = + = 1=@ :


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..

"S0 for Plants 1 and "S0 for Plants 1 and


' ConclusionConclusion

' Test StatisticTest Statistic

1 x x1 x x 6 N6 N N 6 1.N 6 1.

#e(ect#e(ect HH55ifif 1 x x

1 x x

' #e(ection #ule#e(ection #ule

5 1 :H 5 1 :H

1 :aH 1 :aH

' ypotheses $A&ypotheses $A&

The mean num,er of hours 4or)ed at Plant 1 isThe mean num,er of hours 4or)ed at Plant 1 is

not equalnot equalto the mean num,er 4or)ed at Plant /to the mean num,er 4or)ed at Plant /



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.7

"S0 for Plants 1 and ."S0 for Plants 1 and .




1 .x x1 .x x 6 N6 N =N 6 =N 6

#e(ect#e(ect HH55ifif 1 .x x

1 .x x


5 1 .:H 5 1 .:H

1 .:aH 1 .:aH

' ypotheses $?&ypotheses $?&

There isThere is no signicant di9erenceno signicant di9erence,et4een the me,et4een the me

num,er of hours 4or)ed at Plant 1 andnum,er of hours 4or)ed at Plant 1 and the meathe mea

num,er of hours 4or)ed at Plant ./num,er of hours 4or)ed at Plant ./



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.

"S0 for Plants and ."S0 for Plants and .




.x x .x x 6 N6 N =N 6 11=N 6 11

#e(ect#e(ect HH55ifif

.x x

.x x


5 .:H 5 .:H

.:aH .:aH

' ypotheses $C&ypotheses $C&

The mean num,er of hours 4or)ed at Plant isThe mean num,er of hours 4or)ed at Plant is

not equalnot equalto the mean num,er 4or)ed at Plant ./to the mean num,er 4or)ed at Plant ./


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End of ChapterEnd of Chapter

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