商务统计学课件英文版BSFC7e-CH09

FundamentalsofHypothesisTesting:One-SampleTestsChapter9ObjectivesInthischapter,youlearn:

ThebasicprinciplesofhypothesistestingHowtousehypothesistestingtotestameanorproportionTheassumptionsofeachhypothesis-testingprocedure,howtoevaluatethem,andtheconsequencesiftheyareseriouslyviolatedPitfallsðicalissuesinvolvedinhypothesistestingHowtoavoidthepitfallsinvolvedinhypothesistestingWhatisaHypothesis?Ahypothesisisaclaim (assertion)abouta populationparameter:populationmeanpopulationproportionExample:Themeanmonthlycellphonebillinthiscityisμ=$42Example:Theproportionofadultsinthiscitywithcellphonesisπ=0.68DCOVATheNullHypothesis,H0StatestheclaimorassertiontobetestedExample:Themeandiameterofamanufacturedboltis30mm()Isalwaysaboutapopulationparameter,notaboutasamplestatistic

DCOVATheNullHypothesis,H0BeginwiththeassumptionthatthenullhypothesisistrueSimilartothenotionofinnocentuntil

provenguiltyReferstothestatusquoorhistoricalvalueAlwayscontains“=“,or“≤”,or“≥”signMayormaynotberejected(continued)DCOVATheAlternativeHypothesis,H1Istheoppositeofthenullhypothesise.g.,Themeandiameterofamanufacturedboltisnotequalto30mm(H1:μ≠30)ChallengesthestatusquoNevercontainsthe“=“,or“≤”,or“≥”signMayormaynotbeprovenIsgenerallythehypothesisthattheresearcheristryingtoproveDCOVATheHypothesisTestingProcessClaim:Thepopulationmeanageis50.H0:μ=50, H1:μ≠50Samplethepopulationandfindthesamplemean.PopulationSampleDCOVATheHypothesisTestingProcessSupposethesamplemeanagewasX=20.Thisissignificantlylowerthantheclaimedmeanpopulationageof50.Ifthenullhypothesisweretrue,theprobabilityofgettingsuchadifferentsamplemeanwouldbeverysmall,soyourejectthenullhypothesis.Inotherwords,gettingasamplemeanof20issounlikelyifthepopulationmeanwas50,youconcludethatthepopulationmeanmustnotbe50.(continued)DCOVATheHypothesisTestingProcess

μ

=50If

H0istrueIfitisunlikelythatyouwouldgetasamplemeanofthisvalue......thenyourejectthenullhypothesisthatμ=50.20...Wheninfactthiswere

thepopulationmean…SamplingDistributionofXX(continued)DCOVATheTestStatisticand

CriticalValuesIfthesamplemeanisclosetothestatedpopulationmean,thenullhypothesisisnotrejected.Ifthesamplemeanisfarfromthestatedpopulationmean,thenullhypothesisisrejected.Howfaris“farenough”torejectH0?Thecriticalvalueofateststatisticcreatesa“lineinthesand”fordecisionmaking--itanswersthequestionofhowfarisfarenough.DCOVATheTestStatisticand

CriticalValuesCriticalValues“TooFarAway”FromMeanofSamplingDistributionSamplingDistributionoftheteststatisticRegionofRejectionRegionofRejectionRegionofNon-RejectionDCOVARisksinDecisionMakingUsingHypothesisTestingTypeIError

RejectatruenullhypothesisAtypeIerrorisa“falsealarm”TheprobabilityofaTypeIErroris

CalledlevelofsignificanceofthetestSetbyresearcherinadvanceTypeIIErrorFailuretorejectafalsenullhypothesisTypeIIerrorrepresentsa“missedopportunity”TheprobabilityofaTypeIIErrorisβDCOVAPossibleErrorsinHypothesisTestDecisionMakingPossibleHypothesisTestOutcomesActualSituationDecisionH0TrueH0FalseDoNotRejectH0NoErrorProbability1-αTypeIIErrorProbabilityβRejectH0TypeIErrorProbabilityαNoErrorPower1-β(continued)DCOVAPossibleErrorsinHypothesisTestDecisionMakingTheconfidencecoefficient(1-α)istheprobabilityofnotrejectingH0whenitistrue.Theconfidencelevelofahypothesistestis(1-α)*100%.Thepowerofastatisticaltest(1-β)istheprobabilityofrejectingH0whenitisfalse.(continued)DCOVATypeI&IIErrorRelationship

TypeIandTypeIIerrorscannothappenatthesametimeATypeIerrorcanonlyoccurifH0istrueATypeIIerrorcanonlyoccurifH0isfalse

IfTypeIerrorprobability(

),then TypeIIerrorprobability(β)DCOVAFactorsAffectingTypeIIErrorAllelseequal,

βwhenthedifferencebetweenhypothesizedparameteranditstruevalue

β when

β whenσ

β whennDCOVALevelofSignificance

andtheRejectionRegionLevelofsignificance=aThisisatwo-tailtestbecausethereisarejectionregioninbothtailsH0:μ=30H1:μ≠30CriticalvaluesRejectionRegion/230a/2aDCOVAHypothesisTestsfortheMeanKnownUnknownHypothesisTestsfor(Ztest)(ttest)DCOVAZTestofHypothesisfortheMean(σKnown)Convertsamplestatistic()toaZSTAT

teststatistic

XσUnknownHypothesisTestsforUnknownTheteststatisticis:σKnownKnown(Ztest)(ttest)DCOVACriticalValue

ApproachtoTestingForatwo-tailtestforthemean,σknown:Convertsamplestatistic()toteststatistic(ZSTAT)DeterminethecriticalZvaluesforaspecified

levelofsignificance

fromatableorbyusingcomputersoftwareDecisionRule:Iftheteststatisticfallsintherejectionregion,rejectH0;otherwisedonotrejectH0

DCOVADonotrejectH0RejectH0RejectH0Therearetwocutoffvalues(criticalvalues),definingtheregionsofrejectionTwo-TailTests

/2-Zα/20H0:μ=30H1:μ

¹30+Zα/2

/2 Lowercriticalvalue Uppercriticalvalue30ZXDCOVA6Stepsin

HypothesisTestingStatethenullhypothesis,H0andthealternativehypothesis,H1Choosethelevelofsignificance,

,andthesamplesize,n.ThelevelofsignificanceisbasedontherelativeimportanceofTypeIandTypeIIerrorsDeterminetheappropriateteststatisticandsamplingdistributionDeterminethecriticalvaluesthatdividetherejectionandnonrejectionregionsDCOVA6Stepsin

HypothesisTestingCollectdataandcomputethevalueoftheteststatisticMakethestatisticaldecisionandstatethemanagerialconclusion.Iftheteststatisticfallsintothenonrejectionregion,donotrejectthenullhypothesisH0.Iftheteststatisticfallsintotherejectionregion,rejectthenullhypothesis.Expressthemanagerialconclusioninthecontextoftheproblem(continued)DCOVAHypothesisTestingExampleTesttheclaimthatthetruemeandiameterofamanufacturedboltis30mm.(Assumeσ=0.8)1. Statetheappropriatenullandalternative hypothesesH0:μ=30H1:μ≠30(Thisisatwo-tailtest)2.SpecifythedesiredlevelofsignificanceandthesamplesizeSupposethat

=0.05andn=100arechosenforthistestDCOVAHypothesisTestingExample3. DeterminetheappropriatetechniqueσisassumedknownsothisisaZtest.4. DeterminethecriticalvaluesFor

=0.05thecriticalZvaluesare±1.965.

CollectthedataandcomputetheteststatisticSupposethesampleresultsare n=100,X=29.84(σ=0.8isassumedknown)Sotheteststatisticis:(continued)DCOVARejectH0DonotrejectH06.Istheteststatisticintherejectionregion?/2=0.025-Zα/2=-1.960RejectH0ifZSTAT<-1.96orZSTAT>1.96;otherwisedonotrejectH0HypothesisTestingExample(continued)/2=0.025RejectH0+Zα/2=+1.96Here,ZSTAT=-2.0<-1.96,sotheteststatisticisintherejectionregionDCOVA6(continued).Reachadecisionandinterprettheresult-2.0SinceZSTAT=-2.0<-1.96,rejectthenullhypothesisandconcludethereissufficientevidencethatthemeandiameterofamanufacturedboltisnotequalto30HypothesisTestingExample(continued)RejectH0DonotrejectH0

=0.05/2-Zα/2=-1.960

=0.05/2RejectH0+Zα/2=+1.96DCOVAp-ValueApproachtoTestingp-value:ProbabilityofobtainingateststatisticequaltoormoreextremethantheobservedsamplevaluegivenH0istrueThep-valueisalsocalledtheobservedlevelofsignificanceItisthesmallestvalueof

forwhichH0canberejectedDCOVAp-ValueApproachtoTesting:

Interpretingthep-valueComparethep-valuewith

Ifp-value<

,rejectH0Ifp-value

,donotrejectH0RememberIfthep-valueislowthenH0mustgoDCOVAThe5Stepp-valueapproachto

HypothesisTestingStatethenullhypothesis,H0andthealternativehypothesis,H1Choosethelevelofsignificance,

,andthesamplesize,n.ThelevelofsignificanceisbasedontherelativeimportanceoftherisksofatypeIandatypeIIerror.DeterminetheappropriateteststatisticandsamplingdistributionCollectdataandcomputethevalueoftheteststatisticandthep-valueMakethestatisticaldecisionandstatethemanagerialconclusion.Ifthep-valueis<αthenrejectH0,otherwisedonotrejectH0.StatethemanagerialconclusioninthecontextoftheproblemDCOVAp-valueHypothesisTestingExampleTesttheclaimthatthetruemeandiameterofamanufacturedboltis30mm.(Assumeσ=0.8)1. Statetheappropriatenullandalternative hypothesesH0:μ=30H1:μ≠30(Thisisatwo-tailtest)2.SpecifythedesiredlevelofsignificanceandthesamplesizeSupposethat

=0.05andn=100arechosenforthistestDCOVAp-valueHypothesisTestingExample3. DeterminetheappropriatetechniqueσisassumedknownsothisisaZtest.4.

Collectthedata,computetheteststatisticandthep-valueSupposethesampleresultsare n=100,X=29.84(σ=0.8isassumedknown)Sotheteststatisticis:(continued)DCOVAp-ValueHypothesisTestingExample:

Calculatingthep-value4.(continued)Calculatethep-value.HowlikelyisittogetaZSTATof-2(orsomethingfurtherfromthemean(0),ineitherdirection)ifH0istrue?p-value=0.0228+0.0228=0.0456P(Z<-2.0)=0.02280-2.0Z2.0P(Z>2.0)=0.0228DCOVA(continued)5.Isthep-value<α?Sincep-value=0.0456<α=0.05RejectH05.(continued)Statethemanagerialconclusioninthecontextofthesituation.Thereissufficientevidencetoconcludethemeandiameterofamanufacturedboltisnotequalto30mm.p-valueHypothesisTestingExample(continued)DCOVAConnectionBetweenTwoTailTestsandConfidenceIntervalsFor

X=29.84,σ=0.8andn=100,the95%confidenceintervalis:

29.6832≤μ

≤29.9968Sincethisintervaldoesnotcontainthehypothesizedmean(30),werejectthenullhypothesisat

=0.05DCOVADoYouEverTrulyKnowσ?Probablynot!Invirtuallyallrealworldbusinesssituations,σisnotknown.Ifthereisasituationwhereσisknownthenµisalsoknown(sincetocalculateσyouneedtoknowµ.)Ifyoutrulyknowµtherewouldbenoneedtogatherasampletoestimateit.DCOVAHypothesisTesting:

σUnknownIfthepopulationstandarddeviationisunknown,youinsteadusethesamplestandarddeviationS.Becauseofthischange,youusethetdistributioninsteadoftheZdistributiontotestthenullhypothesisaboutthemean.Whenusingthetdistributionyoumustassumethepopulationyouaresamplingfromfollowsanormaldistribution.Allothersteps,concepts,andconclusionsarethesame.DCOVAtTestofHypothesisfortheMean(σUnknown)Convertsamplestatistic()toatSTAT

teststatistic

XTheteststatisticis:HypothesisTestsforσKnownσUnknownKnownUnknown(Ztest)(ttest)Theteststatisticis:HypothesisTestsforσKnownσUnknownKnownUnknown(Ztest)(ttest)Theteststatisticis:HypothesisTestsforσKnownσUnknownKnownUnknown(Ztest)(ttest)DCOVAExample:Two-TailTest

(Unknown)

TheaveragecostofahotelroominNewYorkissaidtobe$168pernight.Todetermineifthisistrue,arandomsampleof25hotelsistakenandresultedinanXof$172.50andanSof$15.40.Testtheappropriatehypothesesat

=0.05.

(Assumethepopulationdistributionisnormal)H0:μ

=168H1:μ¹168DCOVAa=0.05n

=25,df=25-1=24isunknown,souseatstatisticCriticalValue:±t24,0.025=±2.0639ExampleSolution:

Two-TailtTestDonotrejectH0:insufficientevidencethattruemeancostisdifferentfrom$168RejectH0RejectH0a/2=.025-t24,0.025DonotrejectH00a/2=.025-2.06392.06391.46H0:μ

=168H1:μ¹168t24,0.025DCOVAToUsethet-testMustAssumethePopulationIsNormalAslongasthesamplesizeisnotverysmallandthepopulationisnotveryskewed,thet-testcanbeused.Toevaluatethenormalityassumption:Determinehowcloselysamplestatisticsmatchthenormaldistribution’stheoreticalproperties.Constructahistogramorstem-and-leafdisplayorboxplotoranormalprobabilityplot.Section6.3hasmoredetailsonevaluatingnormality.DCOVAExampleTwo-TailtTestUsingAp-valuefromExcelSincethisisat-testwecannotcalculatethep-valuewithoutsomecalculationaid.TheExceloutputbelowdoesthis:p-value>αSodonotrejectH0DCOVAExampleTwo-TailtTestUsingAp-valuefromMinitabDCOVAOne-SampleTTestofmu=168vsnot=168NMeanStDevSEMean95%CITP25172.5015.403.08(166.14,178.86)1.460.157p-value>αSodonotrejectH01234ConnectionofTwoTailTeststoConfidenceIntervalsFor

X=172.5,S=15.40andn=25,the95%confidenceintervalforµis:172.5-(2.0639)15.4/25to172.5+(2.0639)15.4/25

166.14≤μ

≤178.86SincethisintervalcontainstheHypothesizedmean(168),wedonotrejectthenullhypothesisat

=0.05DCOVAOne-TailTestsInmanycases,thealternativehypothesisfocusesonaparticulardirectionH0:μ≥3H1:μ<3H0:μ≤3H1:μ>3Thisisalower-tailtestsincethealternativehypothesisisfocusedonthelowertailbelowthemeanof3Thisisanupper-tailtestsincethealternativehypothesisisfocusedontheuppertailabovethemeanof3DCOVARejectH0DonotrejectH0Thereisonlyonecriticalvalue,sincetherejectionareaisinonlyonetailLower-TailTestsa-Zαor-tα0μH0:μ≥3H1:μ<3ZortXCriticalvalueDCOVARejectH0DonotrejectH0Upper-TailTestsaZα

ortα0μH0:μ≤3H1:μ>3Thereisonlyonecriticalvalue,sincetherejectionareaisinonlyonetailCriticalvalueZortX_DCOVAExample:Upper-TailtTest

forMean(

unknown)Aphoneindustrymanagerthinksthatcustomermonthlycellphonebillshaveincreased,andnowaverageover$52permonth.Thecompanywishestotestthisclaim.(Assumeanormalpopulation)H0:μ≤52themeanisnotover$52permonthH1:μ>52themeanisgreaterthan$52permonth

(i.e.,sufficientevidenceexiststosupportthe manager’sclaim)Formhypothesistest:DCOVARejectH0DonotrejectH0Supposethat=0.10ischosenforthistestandn=25.Findtherejectionregion:

=0.101.3180RejectH0RejectH0iftSTAT>1.318Example:FindRejectionRegion(continued)DCOVAObtainsampleandcomputetheteststatisticSupposeasampleistakenwiththefollowingresults:n=25,X=53.1,andS=10

Thentheteststatisticis:Example:TestStatistic(continued)DCOVARejectH0DonotrejectH0Example:Decision

=0.101.3180RejectH0DonotrejectH0sincetSTAT=0.55<1.318thereisnotsufficientevidencethatthemeanbillisover$52tSTAT=0.55Reachadecisionandinterprettheresult:(continued)DCOVAExample:UtilizingThep-valueforTheTestCalculatethep-valueandcompareto

(p-valuebelowcalculatedusingexcelspreadsheetonnextpage)

RejectH0

=.10DonotrejectH01.3180RejectH0tSTAT=.55p-value=.2937DonotrejectH0sincep-value=.2937>=.10DCOVAExcelSpreadsheetCalculatingThep-valueforTheUpperTailtTestDCOVAUsingMinitabtocalculateThep-valueforTheUpperTailtTestDCOVA1234One-SampleTTestofmu=52vs>52 95%LowerNMeanStDevSEMeanBoundTP2553.1010.002.00 49.680.550.294p-value>αSodonotrejectH0HypothesisTestsforProportionsInvolvescategoricalvariablesTwopossibleoutcomesPossessescharacteristicofinterestDoesnotpossesscharacteristicofinterestFractionorproportionofthepopulationinthecategoryofinterestisdenotedbyπDCOVAProportionsSampleproportioninthecategoryofinterestisdenotedbyp

Whenbothnπandn(1-π)areatleast5,p

canbeapproximatedbyanormaldistributionwithmeanandstandarddeviation

(continued)DCOVAThesamplingdistributionofpisapproximatelynormal,sotheteststatisticisaZSTATvalue:HypothesisTestsforProportionsnπ5andn(1-π)5HypothesisTestsforpnπ<5orn(1-π)<5NotdiscussedinthischapterDCOVAAnequivalentformtothelastslide,butintermsofthenumberinthecategoryofinterest,X:ZTestforProportioninTermsofNumberinCategoryofInterestX5andn-X5HypothesisTestsforXX<5orn-X<5NotdiscussedinthischapterDCOVAExample:ZTestforProportionAmarketingcompanyclaimsthatitreceives8%responsesfromitsmailing.Totestthisclaim,arandomsampleof500weresurveyedwith25responses.Testatthe

=0.05significancelevel.Check:n

π=(500)(.08)=40n(1-π)=(500)(.92)=460

DCOVAZTestforProportion:Solutiona

=0.05n=500,p=0.05RejectH0at

=0.05H0:π=0.08H1:π

¹0.08CriticalValues:±1.96TestStatistic:Decision:Conclusion:z0RejectReject.025.0251.96-2.47Thereissufficientevidencetorejectthecompany’sclaimof8%responserate.-1.96DCOVADonotrejectH0RejectH0RejectH0

/2

=.0251.960Z=-2.47Calculatethep-valueandcompareto

(Foratwo-tailtestthep-valueisalways