Statisticsfor

BusinessandEconomics(14e)

MetricVersionAnderson,Sweeney,Williams,Camm,Cochran,Fry,Ohlmann?2020CengageLearning?2020Cengage.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart,exceptforuseaspermittedinalicensedistributedwithacertainproductorserviceorotherwiseonapassword-protectedwebsiteorschool-approvedlearningmanagementsystemforclassroomuse.1Chapter13-ExperimentalDesignandAnalysisofVariance13.1-AnIntroductiontoExperimentalDesignandAnalysisofVariance13.2-AnalysisofVarianceandtheCompletelyRandomizedDesign13.3-MultipleComparisonProcedures13.4-RandomizedBlockDesign13.5-FactorialExperiment2AnIntroductiontoExperimentalDesignandAnalysisofVariance(1of3)Statisticalstudiescanbeclassifiedasbeingeitherexperimentalorobservational.Inanexperimentalstudy,oneormorefactorsarecontrolledsothatdatacanbeobtainedabouthowthefactorsinfluencethevariablesofinterest.Inanobservationalstudy,noattemptismadetocontrolthefactors.Cause-and-effectrelationshipsareeasiertoestablishinexperimentalstudiesthaninobservationalstudies.Analysisofvariance(ANOVA)canbeusedtoanalyzethedataobtainedfromexperimentalorobservationalstudies.3AnIntroductiontoExperimentalDesignandAnalysisofVariance(2of3)Inthischapter,threetypesofexperimentaldesignsareintroduced:AcompletelyrandomizeddesignArandomizedblockdesignAfactorialexperiment4AnIntroductiontoExperimentalDesignandAnalysisofVariance(3of3)Afactorisavariablethattheexperimenterhasselectedforinvestigation.Atreatmentisalevelofafactor.Experimentalunits

aretheobjectsofinterestintheexperiment.Acompletelyrandomizeddesign

isanexperimentaldesigninwhichthetreatmentsarerandomlyassignedtotheexperimentalunits.5AnalysisofVariance:AConceptualOverview(1of4)AnalysisofVariance(ANOVA)canbeusedtotestfortheequalityofthreeormorepopulationmeans.Dataobtainedfromobservationalorexperimentalstudiescanbeusedfortheanalysis.Wewanttousethesampleresultstotestthefollowinghypotheses:6AnalysisofVariance:AConceptualOverview(2of4)

7AnalysisofVariance:AConceptualOverview(3of4)

8AnalysisofVariance:AConceptualOverview(4of4)

9AnalysisofVarianceandtheCompletelyRandomizedDesignBetween-TreatmentsEstimateofPopulationVarianceWithin-TreatmentsEstimateofPopulationVarianceComparingtheVarianceEstimates:TheFTestANOVATable10Between-TreatmentsEstimateofPopulationVarianceσ2Theestimateofσ2basedonthevariationofthesamplemeansiscalledthemeansquareduetotreatmentsandisdenotedbyMSTR.Numeratoriscalledthesumofsquaresduetotreatments

(SSTR).DenominatoristhedegreesoffreedomassociatedwithSSTR.11Within-TreatmentsEstimateofPopulationVarianceσ2Theestimateofσ2basedonthevariationofthesampleobservationswithineachsampleiscalledthemeansquareerrorandisdenotedbyMSE.Numeratoriscalledthesumofsquaresduetoerror

(SSE).DenominatoristhedegreesoffreedomassociatedwithSSE.12ComparingtheVarianceEstimates:TheFTest(1of2)

13ComparingtheVarianceEstimates:TheFTest(2of2)SamplingDistributionofMSTR/MSE14ANOVATableforaCompletelyRandomizedDesign(1of3)SSTispartitionedintoSSTRandSSE.SST’sdegreesoffreedom(df)arepartitionedintoSSTR’sdfandSSE’sdf.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFp-ValueTreatmentsSSTRKminus1Beginequation.MSTRequalsStartfraction,SSTRoverkminus1.Endfraction.Endequation.Beginfraction.MSTRoverMSE.Endfraction.

emptycellErrorSSENsubscriptTbaselineminuskBeginequation.MSEequalsstartfractionSSEovernsubscriptTbaselineminuskendfraction.Endequation.

emptycell

emptycellTotalSSTNsubscriptbaselineminus1

emptycell

emptycell

emptycell15ANOVATableforaCompletelyRandomizedDesign(2of3)

16ANOVATableforaCompletelyRandomizedDesign(3of3)

17

HypothesesTestStatistic 18

19

AutoShine,Inc.isconsideringmarketingalong-lastingcarwax.Threedifferentwaxes(Type1,Type2,andType3)havebeendeveloped.Inordertotestthedurabilityofthesewaxes,5newcarswerewaxedwithType1,5withType2,and5withType3.Eachcarwasthenrepeatedlyrunthroughanautomaticcarwashuntilthewaxcoatingshowedsignsofdeterioration.Thenumberoftimeseachcarwentthroughthecarwashbeforeitswaxdeterioratedisshownonthenextslide.AutoShine,Inc.mustdecidewhichwaxtomarket.Arethethreewaxesequallyeffective?Factor...CarwaxTreatments...Type1,Type2,Type3Experimentalunits...CarsResponsevariable...Numberofwashes20

ObservationWaxType1WaxType2WaxType312733292302828329313042830325313031SampleMean29.030.430.0SampleVariance2.53.32.521

22

MeanSquareBetweenTreatments:(Becausethesamplesizesareallequal)

MeanSquareError:23

RejectionRule:24

TestStatistic:Conclusion:Thereisinsufficientevidencetoconcludethatthemeannumberofwashesforthethreewaxtypesarenotallthesame.25

ANOVATableSourceofVariationSumofSquaresDegreesofFreedomMeanSquaresFp-ValueTreatments5.222.600.9390.42Error33.2122.77Total38.41426

27

ObservationPlant1BuffaloPlant2PittsburghPlant3Detroit14873512546363357666145464545627456SampleMean556857SampleVariance26.026.524.528

Developthehypotheses.29

Specifythelevelofsignificance.α

=0.05Computethevalueoftheteststatistic.30

Computethevalueoftheteststatistic.31

ANOVATableSourceofVariationSumofSquaresDegreesofFreedomMeanSquareFp-ValueTreatment49022459.55.0033Error3081225.667Total7981432

Wecanconcludethatthemeannumberofhoursworkedperweekbydepartmentmanagersisnotthesameatall3plants.33

CriticalValueApproachDeterminethecriticalvalueandrejectionrule.

5.34MultipleComparisonProceduresSupposethatanalysisofvariancehasprovidedstatisticalevidencetorejectthenullhypothesisofequalpopulationmeans.Fisher’sleastsignificantdifference(LSD)procedurecanbeusedtodeterminewherethedifferencesoccur.35Fisher’sLSDProcedure(1of2)Hypotheses:TestStatistic:36Fisher’sLSDProcedure(2of2)RejectionRule:37

38

Example:ReedManufacturingRecallthatJanetReedwantstoknowifthereisanysignificantdifferenceinthemeannumberofhoursworkedperweekforthedepartmentmanagersatherthreemanufacturingplants.Analysisofvariancehasprovidedstatisticalevidencetorejectthenullhypothesisofequalpopulationmeans.Fisher’sleastsignificantdifference(LSD)procedurecanbeusedtodeterminewherethedifferencesoccur.39

40

LSDforPlants1and2Conclusion:ThemeannumberofhoursworkedatPlant1isnotequaltothemeannumberworkedatPlant2.

41

LSDforPlants1and3Conclusion:ThereisnosignificantdifferencebetweenthemeannumberofhoursworkedatPlant1andthemeannumberofhoursworkedatPlant3.42

LSDforPlants2and3Conclusion:ThemeannumberofhoursworkedatPlant2isnotequaltothemeannumberworkedatPlant43TypeIErrorRatesThecomparisonwiseTypeIerrorrate

αindicatesthelevelofsignificanceassociatedwithasinglepairwisecomparison.

44RandomizedBlockDesign(1of9)Experimentalunits

aretheobjectsofinterestintheexperiment.Acompletelyrandomizeddesign

isanexperimentaldesigninwhichthetreatmentsarerandomlyassignedtotheexperimentalunits.Iftheexperimentalunitsareheterogeneous,blockingcanbeusedtoformhomogeneousgroups,resultinginarandomizedblockdesign.45RandomizedBlockDesign(2of9)ANOVAProcedureForarandomizedblockdesignthesumofsquarestotal(SST)ispartitionedintothreegroups:sumofsquaresduetotreatments,sumofsquaresduetoblocks,andsumofsquaresduetoerror.46RandomizedBlockDesign(3of9)ANOVATableSourceofvariationSumofsquaresDegreesoffreedomMeansquareFP-valueTreatmentSSTRkminus1MSTRequalsStartfraction.SSTRoverkminus1Endfraction.Startfraction.MSTRoverMSEEMPTYCELLBlocksSSBLbminus1MSBLequalsStartfraction.SSBLoverbminus1Endfraction.EMPTYCELLEMPTYCELLErrorSSELeftparenthesiskminus1rightparenthesisleftparenthesisbminus1rightparenthesisMSEequalsStartfraction.SSEoverleftparenthesiskminus1rightparenthesisleftparenthesisbminus1rightparenthesisEMPTYCELLEMPTYCELLTotalSSTnsubscripttbaselineminus1EMPTYCELLEMPTYCELLEMPTYCELL47RandomizedBlockDesign(4of9)Example:CrescentOilCo.CrescentOilhasdevelopedthreenewblendsofgasolineandmustdecidewhichblendorblendstoproduceanddistribute.Astudyofthemilespergallonratingsofthethreeblendsisbeingconductedtodetermineifthemeanratingsarethesameforthethreeblends.Fiveautomobileshavebeentestedusingeachofthethreegasolineblendsandthemilespergallonratingsareshownonthenextslide.Factor…GasolineblendTreatments…BlendX,BlendY,BlendZBlocks…AutomobilesResponsevariable…Milespergallon48RandomizedBlockDesign(5of9)Automobile(Block)TypeofGasoline(Treatment)BlendXTypeofGasoline(Treatment)BlendYTypeofGasoline(Treatment)BlendZBlockMeans131303030.333230292929.333329292828.667433312931.000526252625.667TreatmentMeans29.828.828.4EMPTYCELL49RandomizedBlockDesign(6of9)MeanSquareDuetoTreatmentsMeanSquareDuetoBlocksMeanSquareDuetoError50RandomizedBlockDesign(7of9)ANOVATableSourceofvariationSumofsquaresDegreesofFreedomMeanSquareFp-valueTreatment5.2022.603.82.07Blocks51.33412.80EMPTYCELLEMPTYCELLError5.478.68EMPTYCELLEMPTYCELLTotal62.0014EMPTYCELLEMPTYCELLEMPTYCELL51RandomizedBlockDesign(8of9)RejectionRule52RandomizedBlockDesign(9of9)TestStatistic

53FactorialExperimentInsomeexperimentswewanttodrawconclusionsaboutmorethanonevariableorfactor.FactorialexperimentsandtheircorrespondingANOVAcomputationsarevaluabledesignswhensimultaneousconclusionsabouttwoormorefactorsarerequired.Thetermfactorialisusedbecausetheexperimentalconditionsincludeallpossiblecombinationsofthefactors.Forexample,foralevelsoffactorAandblevelsoffactorB,theexperimentwillinvolvecollectingdataonabtreatmentcombinations.54Two-FactorFactorialExperiment(1of9)

55Two-FactorFactorialExperiment(2of9)SourceofvariationSumofsquares

DegreesoffreedomMeansquareFTreatmentSSTRKminus1MSTRequalsStartfraction.SSTRoverkminus1Endfraction.Startfraction.MSTRoverMSE.Endfraction.BlocksSSBLBminus1MSBLequalsStartfraction.SSBLoverbminus1Endfraction.Startfraction.MSBoverMSE.Endfraction.ErrorSSELeftparenthesiskminus1rightparenthesisleftparenthesisbminus1rightparenthesisMSEequalsStartfraction.SSEoverleftparenthesiskminus1rightparenthesisleftparenthesisbminus1rightparenthesisStartfraction.MSAB

overMSE.Endfraction.TotalSSTNsubscripttbaselineminus1EmptycellEmptycellSourceofvariationSumofsquares

DegreesoffreedomMeansquareF56Two-FactorFactorialExperiment(3of9)Step1:Computethetotalsumofsquares.Step2:ComputethesumofsquaresforfactorA.Step3:ComputethesumofsquaresforfactorB.57Tw