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StatisticsforBusiness
andEconomicsAndersonSweeneyWilliamsSlidesbyJohnLoucksSt.Edward’sUniversityStatisticsforBusiness
andEcChapter12
TestsofGoodnessofFitandIndependence
GoodnessofFitTest:AMultinomialPopulationGoodnessofFitTest: PoissonandNormalDistributionsTestofIndependenceChapter12
TestsofGoodnessHypothesis(GoodnessofFit)Test
forProportionsofaMultinomialPopulation1.Statethenullandalternativehypotheses.H0:ThepopulationfollowsamultinomialdistributionwithspecifiedprobabilitiesforeachofthekcategoriesHa:ThepopulationdoesnotfollowamultinomialdistributionwithspecifiedprobabilitiesforeachofthekcategoriesHypothesis(GoodnessofFit)THypothesis(GoodnessofFit)Test
forProportionsofaMultinomialPopulation2.Selectarandomsampleandrecordtheobservedfrequency,fi,foreachofthekcategories.3.AssumingH0istrue,computetheexpectedfrequency,ei,ineachcategorybymultiplyingthecategoryprobabilitybythesamplesize.Hypothesis(GoodnessofFit)THypothesis(GoodnessofFit)Test
forProportionsofaMultinomialPopulation4.Computethevalueoftheteststatistic.Note:Theteststatistichasachi-squaredistributionwithk–1dfprovidedthattheexpectedfrequenciesare5ormoreforallcategories.fi=observedfrequencyforcategoryiei=expectedfrequencyforcategoryik=numberofcategorieswhere:Hypothesis(GoodnessofFit)THypothesis(GoodnessofFit)Test
forProportionsofaMultinomialPopulationwhereisthesignificancelevelandtherearek-1degreesoffreedomp-valueapproach:Criticalvalueapproach:RejectH0ifp-value<
a5.Rejectionrule:RejectH0ifHypothesis(GoodnessofFit)TMultinomialDistributionGoodnessofFitTestExample:FingerLakesHomes(A)FingerLakesHomesmanufacturesfourmodelsofprefabricatedhomes,atwo-storycolonial,alogcabin,asplit-level,andanA-frame.Tohelpinproductionplanning,managementwouldliketodetermineifpreviouscustomerpurchasesindicatethatthereisapreferenceinthestyleselected.MultinomialDistributionGoodnSplit-A-ModelColonialLogLevelFrame#Sold
30203515Thenumberofhomessoldofeachmodelfor100salesoverthepasttwoyearsisshownbelow.MultinomialDistributionGoodnessofFitTestExample:FingerLakesHomes(A)
HypothesesMultinomialDistributionGoodnessofFitTestwhere:pC=populationproportionthatpurchaseacolonialpL=populationproportionthatpurchasealogcabinpS=populationproportionthatpurchaseasplit-levelpA=populationproportionthatpurchaseanA-frameH0:pC=pL=pS=pA=.25Ha:Thepopulationproportionsarenot
pC=.25,pL=.25,pS=.25,andpA=.25HypothesesMultinomialDistribuRejectionRule27.815DoNotRejectH0RejectH0MultinomialDistributionGoodnessofFitTestWith=.05andk-1=4-1=3degreesoffreedom
RejectH0ifp-value<.05orc2>7.815.RejectionRule27.815DoNotExpectedFrequencies
TestStatisticMultinomialDistributionGoodnessofFitTeste1=.25(100)=25e2=.25(100)=25e3=.25(100)=25e4=.25(100)=25=1+1+4+4=10ExpectedFrequenciesMultinomiaMultinomialDistributionGoodnessofFitTestConclusionUsingthep-ValueApproachThep-value<
a.Wecanrejectthenullhypothesis.Becausec2
=10isbetween9.348and11.345,theareaintheuppertailofthedistributionisbetween.025and.01.AreainUpperTail.10.05.025.01.005c2Value(df=3)6.2517.8159.34811.34512.838Actualp-valueis.0186MultinomialDistributionGoodnConclusionUsingtheCriticalValueApproachMultinomialDistributionGoodnessofFitTestWereject,atthe.05levelofsignificance,theassumptionthatthereisnohomestylepreference.c2=10>7.815ConclusionUsingtheCriticalTestofIndependence:ContingencyTables1.Setupthenullandalternativehypotheses.2.Selectarandomsampleandrecordtheobservedfrequency,fij,foreachcellofthecontingencytable.3.Computetheexpectedfrequency,eij,foreachcell.H0:ThecolumnvariableisindependentoftherowvariableHa:ThecolumnvariableisnotindependentoftherowvariableTestofIndependence:ContingTestofIndependence:ContingencyTables5.Determinetherejectionrule.RejectH0ifp-value<
aor.4.Computetheteststatistic.whereisthesignificanceleveland,withnrowsandmcolumns,thereare(n-1)(m-1)degreesoffreedom.TestofIndependence:ContingEachhomesoldbyFingerLakesHomescanbeclassifiedaccordingtopriceandtostyle.FingerLakes’managerwouldliketodetermineifthepriceofthehomeandthestyleofthehomeareindependentvariables.ContingencyTable(Independence)TestExample:FingerLakesHomes(B)EachhomesoldbyFingerPriceColonialLogSplit-LevelA-FrameThenumberofhomessoldforeachmodelandpriceforthepasttwoyearsisshownbelow.Forconvenience,thepriceofthehomeislistedaseither$99,000orlessormorethan$99,000.>$99,0001214 163<$99,00018 61912ContingencyTable(Independence)TestExample:FingerLakesHomes(B)PriceColonialLHypothesesContingencyTable(Independence)TestH0:PriceofthehomeisindependentofthestyleofthehomethatispurchasedHa:PriceofthehomeisnotindependentofthestyleofthehomethatispurchasedHypothesesContingencyTable(IExpectedFrequenciesContingencyTable(Independence)TestPrice
ColonialLogSplit-LevelA-FrameTotal<$99K>$99KTotal3020351510012 1416345186191255ExpectedFrequenciesContingencRejectionRuleContingencyTable(Independence)TestWith=.05and(2-1)(4-1)=3d.f.,RejectH0ifp-value<.05or2
>7.815=.1364+2.2727+...+2.0833=9.149TestStatisticRejectionRuleContingencyTablConclusionUsingthep-ValueApproachThep-value<
a.Wecanrejectthenullhypothesis.Becausec2
=9.145isbetween7.815and9.348,theareaintheuppertailofthedistributionisbetween.05and.025.AreainUpperTail.10.05.025.01.005c2Value(df=3)6.2517.8159.34811.34512.838ContingencyTable(Independence)TestActualp-valueis.0274ConclusionUsingthep-ValueAConclusionUsingtheCriticalValueApproachContingencyTable(Independence)TestWereject,atthe.05levelofsignificance,theassumptionthatthepriceofthehomeisindependentofthestyleofhomethatispurchased.c2=9.145>7.815ConclusionUsingtheCriticalGoodnessofFitTest:PoissonDistribution1.Statethenullandalternativehypotheses.2.Selectarandomsampleand
a.RecordtheobservedfrequencyfiforeachvalueofthePoissonrandomvariable.
b.Computethemeannumberofoccurrences.H0:ThepopulationhasaPoissondistributionHa:ThepopulationdoesnothaveaPoissondistributionGoodnessofFitTest:PoissonGoodnessofFitTest:PoissonDistribution3.Computetheexpectedfrequencyofoccurrencesei
foreachvalueofthePoissonrandomvariable.MultiplythesamplesizebythePoissonprobabilityofoccurrenceforeachvalueofthePoissonrandom
variable.Iftherearefewerthanfiveexpectedoccurrences
forsomevalues,combineadjacentvaluesand
reducethenumberofcategoriesasnecessary.GoodnessofFitTest:PoissonGoodnessofFitTest:PoissonDistribution4.Computethevalueoftheteststatistic.fi=observedfrequencyforcategoryiei=expectedfrequencyforcategoryik=numberofcategorieswhere:GoodnessofFitTest:Poissonwhereisthesignificancelevelandtherearek-2degreesoffreedomp-valueapproach:Criticalvalueapproach:RejectH0ifp-value<
a5.Rejectionrule:RejectH0ifGoodnessofFitTest:PoissonDistributionwhereisthesignificancelExample:TroyParkingGarageInstudyingtheneedforanadditionalentrancetoacityparkinggarage,aconsultanthasrecommendedananalysisapproachthatisapplicableonlyinsituationswherethenumberofcarsenteringduringaspecifiedtimeperiodfollowsaPoissondistribution.GoodnessofFitTest:PoissonDistributionExample:TroyParkingGarage Arandomsampleof100one-minutetimeintervalsresultedinthecustomerarrivalslistedbelow.AstatisticaltestmustbeconductedtoseeiftheassumptionofaPoissondistributionisreasonable.GoodnessofFitTest:PoissonDistributionExample:TroyParkingGarage#Arrivals
0123456789101112Frequency014101420121298631 Arandomsampleof100one-mHypothesesHa:Numberofcarsenteringthegarageduringaone-minuteintervalisnotPoissondistributedH0:Numberofcarsenteringthegarageduringaone-minuteintervalisPoissondistributedGoodnessofFitTest:PoissonDistributionHypothesesHa:NumberofcarsEstimateofPoissonProbabilityFunctionotalArrivals=0(0)+1(1)+2(4)+...+12(1)=600Hence,Estimateof=600/100=6TotalTimePeriods=100GoodnessofFitTest:PoissonDistributionEstimateofPoissonProbabilit
ExpectedFrequenciesx
f(x)nf(x)012345613.7710.336.81100.00.1377.1033.0688.0413.0225.02011.0000789101112+Total.0025.0149.0446.0892.1339.1606.1606.251.494.468.9213.3916.0616.06x
f(x) nf(x)GoodnessofFitTest:PoissonDistribution ExpectedFrequenciesxf
ObservedandExpectedFrequencies
i
fi
ei
fi-ei-1.201.080.613.94-4.06-1.77-1.331.121.616.208.9213.3916.0616.0613.7710.336.888.395101420121298100or1or2345678910ormoreGoodnessofFitTest:PoissonDistribution ObservedandExpectedFrequenTestStatisticWith=.05andk-p-1=9-1-1=7d.f.(wherek=numberofcategoriesandp=numberofpopulationparametersestimated),RejectH0ifp-value<.05or2
>14.067.RejectionRuleGoodnessofFitTest:PoissonDistributionTestStatisticWith=.0ConclusionUsingthep-ValueApproachThep-value>a.Wecannotrejectthenullhypothesis.ThereisnoreasontodoubttheassumptionofaPoissondistribution.Becausec2
=3.268isbetween2.833and12.017intheChi-SquareDistributionTable,theareaintheuppertailofthedistributionisbetween.90and.10.AreainUpperTail.90.10.05.025.01c2Value(df=7)2.83312.01714.06716.01318.475GoodnessofFitTest:PoissonDistributionActualp-valueis.8591ConclusionUsingthep-ValueAGoodnessofFitTest:NormalDistribution1.Statethenullandalternativehypotheses.Computetheexpectedfrequency,ei,foreachinterval.(Multiplythesamplesizebytheprobabilityofanormalrandomvariablebeingintheinterval.2.Selectarandomsampleand
a.Computethemeanandstandarddeviation.
b.Defineintervalsofvaluessothattheexpected frequencyisatleast5foreachinterval.
c.Foreachinterval,recordtheobservedfrequenciesH0:ThepopulationhasanormaldistributionHa:ThepopulationdoesnothaveanormaldistributionGoodnessofFitTest:Normal4.Computethevalueoftheteststatistic.GoodnessofFitTest:NormalDistribution5.RejectH0if (whereisthesignificancelevelandtherearek-3degreesoffreedom).4.ComputethevalueoftheExample:IQComputersIQComputers(onebetterthanHP?)manufacturesandsellsageneralpurposemicrocomputer.Aspartofastudytoevaluatesalespersonnel,managementwantstodetermine,ata.05significancelevel,iftheannualsalesvolume(numberofunitssoldbyasalesperson)followsanormalprobabilitydistribution.GoodnessofFitTest:NormalDistributionExample:IQComputersIQ Asimplerandomsampleof30ofthesalespeoplewastakenandtheirnumbersofunitssoldarelistedbelow.Example:IQComputers(mean=71,standarddeviation=18.54)33434445525256586364646566687072737374758384858691929498102105GoodnessofFitTest:NormalDistribution Asimplerandomsampleof30HypothesesHa:Thepopulationofnumberofunitssold doesnothaveanormaldistributionwith mean71andstandarddeviation18.54.H0:Thepopulationofnumberofunitssold hasanormaldistributionwithmean71 andstandarddeviation18.54.GoodnessofFitTest:NormalDistributionHypothesesHa:ThepopulationIntervalDefinitionTosatisfytherequirementofanexpectedfrequencyofatleast5ineachintervalwewilldividethenormaldistributioninto30/5=6equalprobabilityintervals.GoodnessofFitTest:NormalDistributionIntervalDefinitionTosatIntervalDefinitionAreas=1.00/6=.16677153.0271-.43(18.54)=63.0378.9788.98=71+.97(18.54)GoodnessofFitTest:NormalDistributionIntervalDefinitionAreas7153
ObservedandExpectedFrequencies1-210-115555553063654630Lessthan53.0253.02to63.0363.03to71.0071.00to78.9778.97to88.98Morethan88.98i
fi
ei
fi-eiTotalGoodnessofFitTest:NormalDistribution ObservedandExpectedFrequenTestStatisticWith=.05andk-p-1=6-2-1=3d.f.(wherek=numberofcategoriesandp=numberofpopulationparametersestimated),RejectH0ifp-value<.05or2
>7.815.RejectionRuleGoodnessofFitTest:NormalDistributionTestStatisticWith=.0ConclusionUsingthep-ValueApproachThep-value>a.Wecannotrejectthenullhypothesis.Thereislittleevidencetosupportrejectingtheassumptionthepopulationisnormallydistributedwith=71and=18.54.Becausec2
=1.600isbetween.584and6.251intheChi-SquareDistributionTable,theareaintheuppertailofthedistributionisbetween.90and.10.AreainUpperTail.90.10.05.025.01c2Value(df=3).5846.2517.8159.34811.345GoodnessofFitTest:NormalDistributionActualp-valueis.6594ConclusionUsingthep-ValueAEndofChapter12EndofChapter12StatisticsforBusiness
andEconomicsAndersonSweeneyWilliamsSlidesbyJohnLoucksSt.Edward’sUniversityStatisticsforBusiness
andEcChapter12
TestsofGoodnessofFitandIndependence
GoodnessofFitTest:AMultinomialPopulationGoodnessofFitTest: PoissonandNormalDistributionsTestofIndependenceChapter12
TestsofGoodnessHypothesis(GoodnessofFit)Test
forProportionsofaMultinomialPopulation1.Statethenullandalternativehypotheses.H0:ThepopulationfollowsamultinomialdistributionwithspecifiedprobabilitiesforeachofthekcategoriesHa:ThepopulationdoesnotfollowamultinomialdistributionwithspecifiedprobabilitiesforeachofthekcategoriesHypothesis(GoodnessofFit)THypothesis(GoodnessofFit)Test
forProportionsofaMultinomialPopulation2.Selectarandomsampleandrecordtheobservedfrequency,fi,foreachofthekcategories.3.AssumingH0istrue,computetheexpectedfrequency,ei,ineachcategorybymultiplyingthecategoryprobabilitybythesamplesize.Hypothesis(GoodnessofFit)THypothesis(GoodnessofFit)Test
forProportionsofaMultinomialPopulation4.Computethevalueoftheteststatistic.Note:Theteststatistichasachi-squaredistributionwithk–1dfprovidedthattheexpectedfrequenciesare5ormoreforallcategories.fi=observedfrequencyforcategoryiei=expectedfrequencyforcategoryik=numberofcategorieswhere:Hypothesis(GoodnessofFit)THypothesis(GoodnessofFit)Test
forProportionsofaMultinomialPopulationwhereisthesignificancelevelandtherearek-1degreesoffreedomp-valueapproach:Criticalvalueapproach:RejectH0ifp-value<
a5.Rejectionrule:RejectH0ifHypothesis(GoodnessofFit)TMultinomialDistributionGoodnessofFitTestExample:FingerLakesHomes(A)FingerLakesHomesmanufacturesfourmodelsofprefabricatedhomes,atwo-storycolonial,alogcabin,asplit-level,andanA-frame.Tohelpinproductionplanning,managementwouldliketodetermineifpreviouscustomerpurchasesindicatethatthereisapreferenceinthestyleselected.MultinomialDistributionGoodnSplit-A-ModelColonialLogLevelFrame#Sold
30203515Thenumberofhomessoldofeachmodelfor100salesoverthepasttwoyearsisshownbelow.MultinomialDistributionGoodnessofFitTestExample:FingerLakesHomes(A)
HypothesesMultinomialDistributionGoodnessofFitTestwhere:pC=populationproportionthatpurchaseacolonialpL=populationproportionthatpurchasealogcabinpS=populationproportionthatpurchaseasplit-levelpA=populationproportionthatpurchaseanA-frameH0:pC=pL=pS=pA=.25Ha:Thepopulationproportionsarenot
pC=.25,pL=.25,pS=.25,andpA=.25HypothesesMultinomialDistribuRejectionRule27.815DoNotRejectH0RejectH0MultinomialDistributionGoodnessofFitTestWith=.05andk-1=4-1=3degreesoffreedom
RejectH0ifp-value<.05orc2>7.815.RejectionRule27.815DoNotExpectedFrequencies
TestStatisticMultinomialDistributionGoodnessofFitTeste1=.25(100)=25e2=.25(100)=25e3=.25(100)=25e4=.25(100)=25=1+1+4+4=10ExpectedFrequenciesMultinomiaMultinomialDistributionGoodnessofFitTestConclusionUsingthep-ValueApproachThep-value<
a.Wecanrejectthenullhypothesis.Becausec2
=10isbetween9.348and11.345,theareaintheuppertailofthedistributionisbetween.025and.01.AreainUpperTail.10.05.025.01.005c2Value(df=3)6.2517.8159.34811.34512.838Actualp-valueis.0186MultinomialDistributionGoodnConclusionUsingtheCriticalValueApproachMultinomialDistributionGoodnessofFitTestWereject,atthe.05levelofsignificance,theassumptionthatthereisnohomestylepreference.c2=10>7.815ConclusionUsingtheCriticalTestofIndependence:ContingencyTables1.Setupthenullandalternativehypotheses.2.Selectarandomsampleandrecordtheobservedfrequency,fij,foreachcellofthecontingencytable.3.Computetheexpectedfrequency,eij,foreachcell.H0:ThecolumnvariableisindependentoftherowvariableHa:ThecolumnvariableisnotindependentoftherowvariableTestofIndependence:ContingTestofIndependence:ContingencyTables5.Determinetherejectionrule.RejectH0ifp-value<
aor.4.Computetheteststatistic.whereisthesignificanceleveland,withnrowsandmcolumns,thereare(n-1)(m-1)degreesoffreedom.TestofIndependence:ContingEachhomesoldbyFingerLakesHomescanbeclassifiedaccordingtopriceandtostyle.FingerLakes’managerwouldliketodetermineifthepriceofthehomeandthestyleofthehomeareindependentvariables.ContingencyTable(Independence)TestExample:FingerLakesHomes(B)EachhomesoldbyFingerPriceColonialLogSplit-LevelA-FrameThenumberofhomessoldforeachmodelandpriceforthepasttwoyearsisshownbelow.Forconvenience,thepriceofthehomeislistedaseither$99,000orlessormorethan$99,000.>$99,0001214 163<$99,00018 61912ContingencyTable(Independence)TestExample:FingerLakesHomes(B)PriceColonialLHypothesesContingencyTable(Independence)TestH0:PriceofthehomeisindependentofthestyleofthehomethatispurchasedHa:PriceofthehomeisnotindependentofthestyleofthehomethatispurchasedHypothesesContingencyTable(IExpectedFrequenciesContingencyTable(Independence)TestPrice
ColonialLogSplit-LevelA-FrameTotal<$99K>$99KTotal3020351510012 1416345186191255ExpectedFrequenciesContingencRejectionRuleContingencyTable(Independence)TestWith=.05and(2-1)(4-1)=3d.f.,RejectH0ifp-value<.05or2
>7.815=.1364+2.2727+...+2.0833=9.149TestStatisticRejectionRuleContingencyTablConclusionUsingthep-ValueApproachThep-value<
a.Wecanrejectthenullhypothesis.Becausec2
=9.145isbetween7.815and9.348,theareaintheuppertailofthedistributionisbetween.05and.025.AreainUpperTail.10.05.025.01.005c2Value(df=3)6.2517.8159.34811.34512.838ContingencyTable(Independence)TestActualp-valueis.0274ConclusionUsingthep-ValueAConclusionUsingtheCriticalValueApproachContingencyTable(Independence)TestWereject,atthe.05levelofsignificance,theassumptionthatthepriceofthehomeisindependentofthestyleofhomethatispurchased.c2=9.145>7.815ConclusionUsingtheCriticalGoodnessofFitTest:PoissonDistribution1.Statethenullandalternativehypotheses.2.Selectarandomsampleand
a.RecordtheobservedfrequencyfiforeachvalueofthePoissonrandomvariable.
b.Computethemeannumberofoccurrences.H0:ThepopulationhasaPoissondistributionHa:ThepopulationdoesnothaveaPoissondistributionGoodnessofFitTest:PoissonGoodnessofFitTest:PoissonDistribution3.Computetheexpectedfrequencyofoccurrencesei
foreachvalueofthePoissonrandomvariable.MultiplythesamplesizebythePoissonprobabilityofoccurrenceforeachvalueofthePoissonrandom
variable.Iftherearefewerthanfiveexpectedoccurrences
forsomevalues,combineadjacentvaluesand
reducethenumberofcategoriesasnecessary.GoodnessofFitTest:PoissonGoodnessofFitTest:PoissonDistribution4.Computethevalueoftheteststatistic.fi=observedfrequencyforcategoryiei=expectedfrequencyforcategoryik=numberofcategorieswhere:GoodnessofFitTest:Poissonwhereisthesignificancelevelandtherearek-2degreesoffreedomp-valueapproach:Criticalvalueapproach:RejectH0ifp-value<
a5.Rejectionrule:RejectH0ifGoodnessofFitTest:PoissonDistributionwhereisthesignificancelExample:TroyParkingGarageInstudyingtheneedforanadditionalentrancetoacityparkinggarage,aconsultanthasrecommendedananalysisapproachthatisapplicableonlyinsituationswherethenumberofcarsenteringduringaspecifiedtimeperiodfollowsaPoissondistribution.GoodnessofFitTest:PoissonDistributionExample:TroyParkingGarage Arandomsampleof100one-minutetimeintervalsresultedinthecustomerarrivalslistedbelow.AstatisticaltestmustbeconductedtoseeiftheassumptionofaPoissondistributionisreasonable.GoodnessofFitTest:PoissonDistributionExample:TroyParkingGarage#Arrivals
0123456789101112Frequency014101420121298631 Arandomsampleof100one-mHypothesesHa:Numberofcarsenteringthegarageduringaone-minuteintervalisnotPoissondistributedH0:Numberofcarsenteringthegarageduringaone-minuteintervalisPoissondistributedGoodnessofFitTest:PoissonDistributionHypothesesHa:NumberofcarsEstimateofPoissonProbabilityFunctionotalArrivals=0(0)+1(1)+2(4)+...+12(1)=600Hence,Estimateof=600/100=6TotalTimePeriods=100GoodnessofFitTest:PoissonDistributionEstimateofPoissonProbabilit
ExpectedFrequenciesx
f(x)nf(x)012345613.7710.336.81100.00.1377.1033.0688.0413.0225.02011.0000789101112+Total.0025.0149.0446.0892.1339.1606.1606.251.494.468.9213.3916.0616.06x
f(x) nf(x)GoodnessofFitTest:PoissonDistribution ExpectedFrequenciesxf
ObservedandExpectedFrequencies
i
fi
ei
fi-ei-1.201.080.613.94-4.06-1.77-1.331.121.616.208.9213.3916.0616.0613.7710.336.888.39510142012129
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