商務(wù)與經(jīng)濟(jì)統(tǒng)計(jì)第13章B課件

StatisticsforBusiness

andEconomicsAndersonSweeneyWilliamsSlidesbyJohnLoucksSt.Edward’sUniversityStatisticsforBusiness

andEcChapter13,PartB

ExperimentalDesignandAnalysisofVarianceFactorialExperimentRandomizedBlockDesignChapter13,PartB

ExperimentExperimentalunits

aretheobjectsofinterestintheexperiment.Acompletelyrandomizeddesign

isanexperimentaldesigninwhichthetreatmentsarerandomlyassignedtotheexperimentalunits.Iftheexperimentalunitsareheterogeneous,blockingcanbeusedtoformhomogeneousgroups,resultinginarandomizedblockdesign.RandomizedBlockDesignExperimentalunitsaretheobjForarandomizedblockdesignthesumofsquarestotal(SST)ispartitionedintothreegroups:sumofsquaresduetotreatments,sumofsquaresduetoblocks,andsumofsquaresduetoerror.ANOVAProcedureRandomizedBlockDesignSST=SSTR+SSBL+SSEThetotaldegreesoffreedom,nT-1,arepartitionedsuchthatk-1degreesoffreedomgototreatments,b-1gotoblocks,and(k-1)(b-1)gototheerrorterm.ForarandomizedblockdesignSourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatmentsErrorTotalk-1nT-1SSTRSSESSTRandomizedBlockDesignANOVATableBlocksSSBLb-1(k–1)(b–1)p-ValueSourceofSumofDegreesofMeanFRandomizedBlockDesignExample:CrescentOilCo.

CrescentOilhasdevelopedthreenewblendsofgasolineandmustdecidewhichblendorblendstoproduceanddistribute.Astudyofthemilespergallonratingsofthethreeblendsisbeingconductedtodetermineifthemeanratingsarethesameforthethreeblends.RandomizedBlockDesignExampleRandomizedBlockDesignExample:CrescentOilCo.

Fiveautomobileshavebeentestedusingeachofthethreegasolineblendsandthemilespergallonratingsareshownonthenextslide.Factor...GasolineblendTreatments...BlendX,BlendY,BlendZBlocks...AutomobilesResponsevariable...MilespergallonRandomizedBlockDesignExampleRandomizedBlockDesign29.8 28.8 28.4TreatmentMeans1234531302933263029293125302928292630.33329.33328.66731.00025.667TypeofGasoline(Treatment)BlockMeansBlendXBlendYBlendZAutomobile(Block)RandomizedBlockDesign29.8 MeanSquareDuetoErrorRandomizedBlockDesignMSE=5.47/[(3-1)(5-1)]=.68SSE=62-5.2-51.33=5.47MSBL=51.33/(5-1)=12.8SSBL=3[(30.333-29)2+...+(25.667-29)2]=51.33MSTR=5.2/(3-1)=2.6SSTR=5[(29.8-29)2+(28.8-29)2+(28.4-29)2]=5.2Theoverallsamplemeanis29.Thus,MeanSquareDuetoTreatmentsMeanSquareDuetoBlocksMeanSquareDuetoErrorRandomSourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatmentsErrorTotal2145.205.4762.0082.60.683.82ANOVATableRandomizedBlockDesignBlocks51.3312.804p-Value.07SourceofSumofDegreesofMeanFRejectionRuleRandomizedBlockDesignFor=.05,F.05=4.46(2d.f.numeratorand8d.f.denominator)p-ValueApproach:

RejectH0ifp-value<.05CriticalValueApproach:

RejectH0ifF

>4.46RejectionRuleRandomizedBlockConclusionRandomizedBlockDesignThereisinsufficientevidencetoconcludethatthemilespergallonratingsdifferforthethreegasolineblends.Thep-valueisgreaterthan.05(whereF=4.46)andlessthan.10(whereF=3.11).(Excelprovidesap-valueof.07).Therefore,wecannotrejectH0.F=MSTR/MSE=2.6/.68=3.82TestStatisticConclusionRandomizedBlockDesFactorialExperimentInsomeexperimentswewanttodrawconclusionsaboutmorethanonevariableorfactor.FactorialexperimentsandtheircorrespondingANOVAcomputationsarevaluabledesignswhensimultaneousconclusionsabouttwoormorefactorsarerequired.Forexample,foralevelsoffactorAandblevelsoffactorB,theexperimentwillinvolvecollectingdataonabtreatmentcombinations.Thetermfactorialisusedbecausetheexperimentalconditionsincludeallpossiblecombinationsofthefactors.FactorialExperimentInsomeexTheANOVAprocedureforthetwo-factorfactorialexperimentissimilartothecompletelyrandomizedexperimentandtherandomizedblockexperiment.ANOVAProcedureSST=SSA+SSB+SSAB+SSEThetotaldegreesoffreedom,nT-1,arepartitionedsuchthat(a–1)d.fgotoFactorA,(b–1)d.fgotoFactorB,(a–1)(b–1)d.f.gotoInteraction,andab(r–1)gotoError.Two-FactorFactorialExperimentWeagainpartitionthesumofsquarestotal(SST)intoitssources.TheANOVAprocedureforthetwSourceofVariationSumofSquaresDegreesofFreedomMeanSquareFFactorAErrorTotala-1nT-1SSASSESSTFactorBSSBb-1ab(r–1)Two-FactorFactorialExperimentInteractionSSAB(a–1)(b–1)p-ValueSourceofSumofDegreesofMeanFStep3ComputethesumofsquaresforfactorBTwo-FactorFactorialExperimentStep1ComputethetotalsumofsquaresStep2ComputethesumofsquaresforfactorAStep3ComputethesumofsqStep4ComputethesumofsquaresforinteractionTwo-FactorFactorialExperimentSSE=SST–SSA–SSB-SSABStep5ComputethesumofsquaresduetoerrorStep4ComputethesumofsqAsurveywasconductedofhourlywagesforasampleofworkersintwoindustriesatthreelocationsinOhio.Partofthepurposeofthesurveywastodetermineifdifferencesexistinbothindustrytypeandlocation.Thesampledataareshownonthenextslide.Example:StateofOhioWageSurveyTwo-FactorFactorialExperimentAsurveywasconductedofExample:StateofOhioWageSurveyTwo-FactorFactorialExperimentIndustryCincinnatiClevelandColumbusI$12.10$11.80$12.90I11.8011.2012.70I12.1012.0012.20II12.4012.6013.00II12.5012.0012.10II12.0012.5012.70Example:StateofOhioWageSFactorsTwo-FactorFactorialExperimentEachexperimentalconditionisrepeated3timesFactorB:Location(3levels)FactorA:IndustryType(2levels)ReplicationsFactorsTwo-FactorFactorialExSourceofVariationSumofSquaresDegreesofFreedomMeanSquareFFactorAErrorTotal117.501.433.49ANOVATableFactorB1.12.562Two-FactorFactorialExperimentInteraction.37.1924.691.55p-Value.06.03.25SourceofSumofDegreesofMeanFConclusionsUsingtheCriticalValueApproachTwo-FactorFactorialExperimentInteractionisnotsignificant.Interaction:F=1.55<

Fa=3.89Meanwagesdifferbylocation.Locations:F=4.69>Fa=3.89Meanwagesdonotdifferbyindustrytype.Industries:F=4.19<

Fa=4.75ConclusionsUsingtheCriticalConclusionsUsingthep-ValueApproach

Two-FactorFactorialExperimentInteractionisnotsignificant.Interaction:p-value=.25>a=.05Meanwagesdifferbylocation.Locations:p-value=.03<

a=.05Meanwagesdonotdifferbyindustrytype.Industries:p-value=.06>a=.05ConclusionsUsingthep-ValueEndofChapter13,PartBEndofChapter13,PartBStatisticsforBusiness

andEconomicsAndersonSweeneyWilliamsSlidesbyJohnLoucksSt.Edward’sUniversityStatisticsforBusiness

andEcChapter13,PartB

ExperimentalDesignandAnalysisofVarianceFactorialExperimentRandomizedBlockDesignChapter13,PartB

ExperimentExperimentalunits

aretheobjectsofinterestintheexperiment.Acompletelyrandomizeddesign

isanexperimentaldesigninwhichthetreatmentsarerandomlyassignedtotheexperimentalunits.Iftheexperimentalunitsareheterogeneous,blockingcanbeusedtoformhomogeneousgroups,resultinginarandomizedblockdesign.RandomizedBlockDesignExperimentalunitsaretheobjForarandomizedblockdesignthesumofsquarestotal(SST)ispartitionedintothreegroups:sumofsquaresduetotreatments,sumofsquaresduetoblocks,andsumofsquaresduetoerror.ANOVAProcedureRandomizedBlockDesignSST=SSTR+SSBL+SSEThetotaldegreesoffreedom,nT-1,arepartitionedsuchthatk-1degreesoffreedomgototreatments,b-1gotoblocks,and(k-1)(b-1)gototheerrorterm.ForarandomizedblockdesignSourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatmentsErrorTotalk-1nT-1SSTRSSESSTRandomizedBlockDesignANOVATableBlocksSSBLb-1(k–1)(b–1)p-ValueSourceofSumofDegreesofMeanFRandomizedBlockDesignExample:CrescentOilCo.

CrescentOilhasdevelopedthreenewblendsofgasolineandmustdecidewhichblendorblendstoproduceanddistribute.Astudyofthemilespergallonratingsofthethreeblendsisbeingconductedtodetermineifthemeanratingsarethesameforthethreeblends.RandomizedBlockDesignExampleRandomizedBlockDesignExample:CrescentOilCo.

Fiveautomobileshavebeentestedusingeachofthethreegasolineblendsandthemilespergallonratingsareshownonthenextslide.Factor...GasolineblendTreatments...BlendX,BlendY,BlendZBlocks...AutomobilesResponsevariable...MilespergallonRandomizedBlockDesignExampleRandomizedBlockDesign29.8 28.8 28.4TreatmentMeans1234531302933263029293125302928292630.33329.33328.66731.00025.667TypeofGasoline(Treatment)BlockMeansBlendXBlendYBlendZAutomobile(Block)RandomizedBlockDesign29.8 MeanSquareDuetoErrorRandomizedBlockDesignMSE=5.47/[(3-1)(5-1)]=.68SSE=62-5.2-51.33=5.47MSBL=51.33/(5-1)=12.8SSBL=3[(30.333-29)2+...+(25.667-29)2]=51.33MSTR=5.2/(3-1)=2.6SSTR=5[(29.8-29)2+(28.8-29)2+(28.4-29)2]=5.2Theoverallsamplemeanis29.Thus,MeanSquareDuetoTreatmentsMeanSquareDuetoBlocksMeanSquareDuetoErrorRandomSourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatmentsErrorTotal2145.205.4762.0082.60.683.82ANOVATableRandomizedBlockDesignBlocks51.3312.804p-Value.07SourceofSumofDegreesofMeanFRejectionRuleRandomizedBlockDesignFor=.05,F.05=4.46(2d.f.numeratorand8d.f.denominator)p-ValueApproach:

RejectH0ifp-value<.05CriticalValueApproach:

RejectH0ifF

>4.46RejectionRuleRandomizedBlockConclusionRandomizedBlockDesignThereisinsufficientevidencetoconcludethatthemilespergallonratingsdifferforthethreegasolineblends.Thep-valueisgreaterthan.05(whereF=4.46)andlessthan.10(whereF=3.11).(Excelprovidesap-valueof.07).Therefore,wecannotrejectH0.F=MSTR/MSE=2.6/.68=3.82TestStatisticConclusionRandomizedBlockDesFactorialExperimentInsomeexperimentswewanttodrawconclusionsaboutmorethanonevariableorfactor.FactorialexperimentsandtheircorrespondingANOVAcomputationsarevaluabledesignswhensimultaneousconclusionsabouttwoormorefactorsarerequired.Forexample,foralevelsoffactorAandblevelsoffactorB,theexperimentwillinvolvecollectingdataonabtreatmentcombinations.Thetermfactorialisusedbecausetheexperimentalconditionsincludeallpossiblecombinationsofthefactors.FactorialExperimentInsomeexTheANOVAprocedureforthetwo-factorfactorialexperimentissimilartothecompletelyrandomizedexperimentandtherandomizedblockexperiment.ANOVAProcedureSST=SSA+SSB+SSAB+SSEThetotaldegreesoffreedom,nT-1,arepartitionedsuchthat(a–1)d.fgotoFactorA,(b–1)d.fgotoFactorB,(a–1)(b–1)d.f.gotoInteraction,andab(r–1)gotoError.Two-FactorFactorialExperimentWeagainpartitionthesumofsquarestotal(SST)intoitssources.TheANOVAprocedureforthetwSourceofVariationSumofSquaresDegreesofFreedomMeanSquareFFactorAErrorTotala-1nT-1SSASSESSTFactorBSSBb-1ab(r–1)Two-FactorFactorialExperimentInteractionSSAB(a–1)(b–1)p-ValueSourceofSumofDegreesofMeanFStep3ComputethesumofsquaresforfactorBTwo-FactorFactorialExperimentStep1ComputethetotalsumofsquaresStep2ComputethesumofsquaresforfactorAStep3ComputethesumofsqStep4ComputethesumofsquaresforinteractionTwo-FactorFactorialExperimentSSE=SST–SSA–SSB-SSABStep5ComputethesumofsquaresduetoerrorStep4ComputethesumofsqAsurveywasconductedofhourlywagesforasampleofworkersintwoindustriesatthreelocationsinOhio.Partofthepurposeofthesurveywastodetermineifdifferencesexistinbothindustrytypeandlocation.Thesampledataareshownonthenextslide.Example:StateofOhioWageSurveyTwo-FactorFactorialExperimentAsurveywasconductedofExample:StateofOhioWageSurveyTwo-Fact