Chapter10,PartB

ComparisonsInvolvingMeans,

ExperimentalDesign,andAnalysisofVarianceAnIntroductiontoExperimentalDesignandAnalysisofVarianceAnalysisofVarianceandtheCompletelyRandomizedDesignStatisticalstudiescanbeclassifiedasbeingeitherexperimentalorobservational.Inanexperimentalstudy,oneormorefactorsarecontrolledsothatdatacanbeobtainedabouthowthefactorsinfluencethevariablesofinterest.Inanobservationalstudy,noattemptismadetocontrolthefactors.Cause-and-effectrelationshipsareeasiertoestablishinexperimentalstudiesthaninobservationalstudies.AnIntroductiontoExperimentalDesignandAnalysisofVarianceAnalysisofvariance(ANOVA)canbeusedtoanalyzethedataobtainedfromexperimentalorobservationalstudies.AnIntroductiontoExperimentalDesignandAnalysisofVarianceThreetypesofexperimentaldesignsareintroduced.acompletelyrandomizeddesignarandomizedblockdesignafactorialexperimentAnIntroductiontoExperimentalDesignandAnalysisofVarianceAfactorisavariablethattheexperimenterhasselectedforinvestigation.Atreatmentisalevelofafactor.Experimentalunits

aretheobjectsofinterestintheexperiment.Acompletelyrandomizeddesign

isanexperimentaldesigninwhichthetreatmentsarerandomlyassignedtotheexperimentalunits.AnalysisofVariance:AConceptualOverview

AnalysisofVariance(ANOVA)canbeusedtotestfortheequalityofthreeormorepopulationmeans.Dataobtainedfromobservationalorexperimentalstudiescanbeusedfortheanalysis.Wewanttousethesampleresultstotestthefollowinghypotheses:H0:

1

=

2

=

3

=

...=

kHa:NotallpopulationmeansareequalH0:

1

=

2

=

3

=

...=

kHa:NotallpopulationmeansareequalIfH0isrejected,wecannotconcludethatallpopulationmeansaredifferent.RejectingH0meansthatatleasttwopopulationmeanshavedifferentvalues.AnalysisofVariance:AConceptualOverviewForeachpopulation,theresponse(dependent)variableisnormallydistributed.Thevarianceoftheresponsevariable,denoted

2,isthesameforallofthepopulations.Theobservationsmustbeindependent.AssumptionsforAnalysisofVarianceAnalysisofVariance:AConceptualOverviewSamplingDistributionofGivenH0isTrue

SamplemeansareclosetogetherbecausethereisonlyonesamplingdistributionwhenH0istrue.AnalysisofVariance:AConceptualOverviewSamplingDistributionofGivenH0isFalse

3

1

2SamplemeanscomefromdifferentsamplingdistributionsandarenotasclosetogetherwhenH0isfalse.AnalysisofVariance:AConceptualOverviewAnalysisofVarianceBetween-TreatmentsEstimateofPopulationVarianceWithin-TreatmentsEstimateofPopulationVarianceComparingtheVarianceEstimates:TheFTestANOVATableBetween-TreatmentsEstimateofPopulationVariances2DenominatoristhedegreesoffreedomassociatedwithSSTRNumeratoriscalledthesumofsquaresduetotreatments

(SSTR)Theestimateof

2basedonthevariationofthesamplemeansiscalledthemeansquaredueto

treatmentsandisdenotedbyMSTR.Theestimateof

2basedonthevariationofthesampleobservationswithineachsampleiscalledthemeansquareerrorandisdenotedbyMSE.Within-TreatmentsEstimate

ofPopulationVariances2DenominatoristhedegreesoffreedomassociatedwithSSENumeratoriscalledthesumofsquaresduetoerror

(SSE)ComparingtheVarianceEstimates:TheFTestIfthenullhypothesisistrueandtheANOVAassumptionsarevalid,thesamplingdistributionofMSTR/MSEisanFdistributionwithMSTRd.f.equaltok-1andMSEd.f.equaltonT-k.Ifthemeansofthekpopulationsarenotequal,thevalueofMSTR/MSEwillbeinflatedbecauseMSTRoverestimates

2.Hence,wewillrejectH0iftheresultingvalueofMSTR/MSEappearstobetoolargetohavebeenselectedatrandomfromtheappropriateF

distribution.SamplingDistributionofMSTR/MSEDoNotRejectH0RejectH0MSTR/MSECriticalValueF

SamplingDistributionofMSTR/MSEaComparingtheVarianceEstimates:TheFTestSourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatmentsErrorTotalk-1nT-1SSTRSSESSTnT-kSSTispartitionedintoSSTRandSSE.SST’sdegreesoffreedom(d.f.)arepartitionedintoSSTR’sd.f.andSSE’sd.f.ANOVATablep-ValueANOVATableSSTdividedbyitsdegreesoffreedomnT–1istheoverallsamplevariancethatwouldbeobtainedifwetreatedtheentiresetofobservationsasonedataset.Withtheentiredatasetasonesample,theformulaforcomputingthetotalsumofsquares,SST,is:ANOVATableANOVAcanbeviewedastheprocessofpartitioningthetotalsumofsquaresandthedegreesoffreedomintotheircorrespondingsources:treatmentsanderror.DividingthesumofsquaresbytheappropriatedegreesoffreedomprovidesthevarianceestimatesandtheFvalueusedtotestthehypothesisofequalpopulationmeans.TestfortheEqualityofkPopulationMeansF=MSTR/MSEH0:

1

=

2

=

3

=

...=

k

Ha:NotallpopulationmeansareequalHypotheses

TestStatistic

TestfortheEqualityofkPopulationMeansRejectionRulewherethevalueofF

isbasedonanFdistributionwithk-1numeratord.f.andnT-kdenominatord.f.RejectH0ifp-value<

ap-valueApproach:CriticalValueApproach:RejectH0ifF

>

FaAutoShine,Inc.isconsideringmarketingalong-lastingcarwax.Threedifferentwaxes(Type1,Type2,andType3)havebeendeveloped.Example:AutoShine,Inc.Inordertotestthedurabilityofthesewaxes,5newcarswerewaxedwithType1,5withType2,and5withType3.Eachcarwasthenrepeatedlyrunthroughanautomaticcarwashuntilthewaxcoatingshowedsignsofdeterioration.TestingfortheEqualityofkPopulationMeans:ACompletelyRandomizedExperimentalDesignThenumberoftimeseachcarwentthroughthecarwashbeforeitswaxdeterioratedisshownonthenextslide.AutoShine,Inc.mustdecidewhichwaxtomarket.Arethethreewaxesequallyeffective? Example:AutoShine,Inc.TestingfortheEqualityofkPopulationMeans:ACompletelyRandomizedExperimentalDesignFactor...CarwaxTreatments...TypeI,Type2,Type3Experimentalunits...CarsResponsevariable...Numberofwashes12345273029283133283130302928303231SampleMeanSampleVarianceObservationWaxType1WaxType2WaxType3

2.5 3.3 2.529.030.4 30.0TestingfortheEqualityofkPopulationMeans:ACompletelyRandomizedExperimentalDesignHypotheseswhere:

1=meannumberofwashesusingType1wax

2=meannumberofwashesusingType2wax

3=meannumberofwashesusingType3waxH0:

1

=

2

=

3

Ha:NotallthemeansareequalTestingfortheEqualityofkPopulationMeans:ACompletelyRandomizedExperimentalDesignBecausethesamplesizesareallequal:MSE=33.2/(15-3)=2.77MSTR=5.2/(3-1)=2.6SSE=4(2.5)+4(3.3)+4(2.5)=33.2SSTR=5(29–29.8)2+5(30.4–29.8)2+5(30–29.8)2=5.2MeanSquareErrorMeanSquareBetweenTreatments=(29+30.4+30)/3=29.8TestingfortheEqualityofkPopulationMeans:ACompletelyRandomizedExperimentalDesignRejectionRulewhereF.05=3.89isbasedonanFdistributionwith2numeratordegreesoffreedomand12denominatordegreesoffreedomp-ValueApproach:RejectH0ifp-value<.05CriticalValueApproach:RejectH0ifF

>3.89TestingfortheEqualityofkPopulationMeans:ACompletelyRandomizedExperimentalDesignTestStatisticThereisinsufficientevidencetoconcludethatthemeannumberofwashesforthethreewaxtypesarenotallthesame.ConclusionF=MSTR/MSE=2.60/2.77=.939Thep-valueisgreaterthan.10,whereF=2.81.(Excelprovidesap-valueof.42.)Therefore,wecannotrejectH0.TestingfortheEqualityofkPopulationMeans:ACompletelyRandomizedExperimentalDesignSourceofVariationSumofSquaresDegreesofFreedomMeanSquaresFTreatmentsErrorTotal2145.233.238.4122.602.77.939ANOVATableTestingfortheEqualityofkPopulationMeans:ACompletelyRandomizedExperimentalDesignp-Value.42Example:ReedManufacturingJanetReedwouldliketoknowifthereisanysignificantdifferenceinthemeannumberofhoursworkedperweekforthedepartmentmanagersatherthreemanufacturingplants(inBuffalo,Pittsburgh,andDetroit).AnFtestwillbeconductedusinga=.05.TestingfortheEqualityofkPopulationMeans:AnObservationalStudyExample:ReedManufacturingAsimplerandomsampleoffivemanagersfromeachofthethreeplantswastakenandthenumberofhoursworkedbyeachmanagerinthepreviousweekisshownonthenextslide.TestingfortheEqualityofkPopulationMeans:AnObservationalStudyFactor...ManufacturingplantTreatments...Buffalo,Pittsburgh,DetroitExperimentalunits...ManagersResponsevariable...Numberofhoursworked12345485457546273636664745163615456Plant1BuffaloPlant2PittsburghPlant3DetroitObservationSampleMeanSampleVariance55 68 5726.0 26.5 24.5TestingfortheEqualityofkPopulationMeans:AnObservationalStudyH0:

1

=

2

=

3

Ha:Notallthemeansareequalwhere:

1=meannumberofhoursworkedper weekbythemanagersatPlant1

2=meannumberofhoursworkedperweekbythemanagersatPlant2

3=meannumberofhoursworkedperweekbythemanagersatPlant31.Developthehypotheses.

p-ValueandCriticalValueApproachesTestingfortheEqualityofkPopulationMeans:AnObservationalStudy2.Specifythelevelofsignificance.a=.05

p-ValueandCriticalValueApproaches3.Computethevalueoftheteststatistic.MSTR=490/(3-1)=245SSTR=5(55-60)2+5(68-60)2+5(57-60)2=490=(55+68+57)/3=60(Samplesizesareallequal.)MeanSquareDuetoTreatmentsTestingfortheEqualityofkPopulationMeans:AnObservationalStudy3.Computethevalueoftheteststatistic.MSE=308/(15-3)=25.667SSE=4(26.0)+4(26.5)+4(24.5)=308MeanSquareDuetoError(con’t.)F=MSTR/MSE=245/25.667=9.55

p-ValueandCriticalValueApproachesTestingfortheEqualityofkPopulationMeans:AnObservationalStudyTreatmentErrorTota