(1.13)-FC5e-Ch09商务统计课件

BasicBusinessStatistics,11e©2009Prentice-Hall,Inc.Chap9-1Chapter9FundamentalsofHypothesisTesting:One-SampleTestsBusinessStatistics:AFirstCourse

5thEditionBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-2LearningObjectivesInthischapter,youlearn:

ThebasicprinciplesofhypothesistestingHowtousehypothesistestingtotestameanorproportionTheassumptionsofeachhypothesis-testingprocedure,howtoevaluatethem,andtheconsequencesiftheyareseriouslyviolatedHowtoavoidthepitfallsinvolvedinhypothesistestingTheethicalissuesinvolvedinhypothesistestingBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-3WhatisaHypothesis?Ahypothesisisaclaim (assertion)abouta populationparameter:populationmeanpopulationproportionExample:Themeanmonthlycellphonebillinthiscityisμ=$42Example:Theproportionofadultsinthiscitywithcellphonesisπ=0.68BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-4TheNullHypothesis,H0StatestheclaimorassertiontobetestedExample:TheaveragenumberofTVsetsinU.S.Homesisequaltothree()Isalwaysaboutapopulationparameter,notaboutasamplestatistic

BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-5TheNullHypothesis,H0BeginwiththeassumptionthatthenullhypothesisistrueSimilartothenotionofinnocentuntil

provenguiltyReferstothestatusquoorhistoricalvalueAlwayscontains“=”,“≤”or“

”signMayormaynotberejected(continued)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-6TheAlternativeHypothesis,H1Istheoppositeofthenullhypothesise.g.,TheaveragenumberofTVsetsinU.S.homesisnotequalto3(H1:μ≠3)ChallengesthestatusquoNevercontainsthe“=”,“≤”or“

”signMayormaynotbeprovenIsgenerallythehypothesisthattheresearcheristryingtoproveBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-7TheHypothesisTestingProcessClaim:Thepopulationmeanageis50.H0:μ=50, H1:μ≠50Samplethepopulationandfindsamplemean.PopulationSampleBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-8TheHypothesisTestingProcessSupposethesamplemeanagewasX=20.Thisissignificantlylowerthantheclaimedmeanpopulationageof50.Ifthenullhypothesisweretrue,theprobabilityofgettingsuchadifferentsamplemeanwouldbeverysmall,soyourejectthenullhypothesis.Inotherwords,gettingasamplemeanof20issounlikelyifthepopulationmeanwas50,youconcludethatthepopulationmeanmustnotbe50.(continued)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-9TheHypothesisTestingProcessSamplingDistributionofX

μ

=50If

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

thepopulationmean…X(continued)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-10TheTestStatisticand

CriticalValuesIfthesamplemeanisclosetotheassumedpopulationmean,thenullhypothesisisnotrejected.Ifthesamplemeanisfarfromtheassumedpopulationmean,thenullhypothesisisrejected.Howfaris“farenough”torejectH0?Thecriticalvalueofateststatisticcreatesa“lineinthesand”fordecisionmaking--itanswersthequestionofhowfarisfarenough.BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-11TheTestStatisticand

CriticalValuesCriticalValues“TooFarAway”FromMeanofSamplingDistributionSamplingDistributionoftheteststatisticRegionofRejectionRegionofRejectionRegionofNon-RejectionBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-12PossibleErrorsinHypothesisTestDecisionMakingTypeIError

RejectatruenullhypothesisConsideredaserioustypeoferrorTheprobabilityofaTypeIErroris

CalledlevelofsignificanceofthetestSetbyresearcherinadvanceTypeIIErrorFailuretorejectfalsenullhypothesisTheprobabilityofaTypeIIErrorisβBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-13PossibleErrorsinHypothesisTestDecisionMakingPossibleHypothesisTestOutcomesActualSituationDecisionH0TrueH0FalseDoNotRejectH0NoErrorProbability1-αTypeIIErrorProbabilityβRejectH0TypeIErrorProbabilityαNoErrorProbability1-β(continued)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-14PossibleResultsinHypothesisTestDecisionMakingTheconfidencecoefficient(1-α)istheprobabilityofnotrejectingH0whenitistrue.Theconfidencelevelofahypothesistestis(1-α)*100%.Thepowerofastatisticaltest(1-β)istheprobabilityofrejectingH0whenitisfalse.(continued)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-15TypeI&IIErrorRelationship

TypeIandTypeIIerrorscannothappenatthesametimeATypeIerrorcanonlyoccurifH0istrueATypeIIerrorcanonlyoccurifH0isfalse

IfTypeIerrorprobability(

),then TypeIIerrorprobability(β)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-16FactorsAffectingTypeIIErrorAllelseequal,

βwhenthedifferencebetweenhypothesizedparameteranditstruevalue

β when

β whenσ

β whennBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-17LevelofSignificance

andtheRejectionRegionLevelofsignificance=aThisisatwo-tailtestbecausethereisarejectionregioninbothtailsH0:μ=3H1:μ≠3CriticalvaluesRejectionRegion/20a/2aBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-18HypothesisTestsfortheMeanKnownUnknownHypothesisTestsfor(Ztest)(ttest)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-19ZTestofHypothesisfortheMean(σKnown)Convertsamplestatistic()toaZSTAT

teststatistic

XTheteststatisticis:σKnownσUnknownHypothesisTestsforKnownUnknown(Ztest)(ttest)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-20CriticalValue

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

levelofsignificance

fromatableorcomputerDecisionRule:Iftheteststatisticfallsintherejectionregion,rejectH0;otherwisedonotrejectH0

BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-21DonotrejectH0RejectH0RejectH0Therearetwocutoffvalues(criticalvalues),definingtheregionsofrejectionTwo-TailTests

/2-Zα/20H0:μ=3H1:μ

¹3+Zα/2

/2 Lowercriticalvalue Uppercriticalvalue3ZXBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-226Stepsin

HypothesisTestingStatethenullhypothesis,H0andthealternativehypothesis,H1Choosethelevelofsignificance,

,andthesamplesize,nDeterminetheappropriateteststatisticandsamplingdistributionDeterminethecriticalvaluesthatdividetherejectionandnonrejectionregionsBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-236Stepsin

HypothesisTestingCollectdataandcomputethevalueoftheteststatisticMakethestatisticaldecisionandstatethemanagerialconclusion.Iftheteststatisticfallsintothenonrejectionregion,donotrejectthenullhypothesisH0.Iftheteststatisticfallsintotherejectionregion,rejectthenullhypothesis.Expressthemanagerialconclusioninthecontextoftheproblem(continued)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-24HypothesisTestingExampleTesttheclaimthatthetruemean#ofTVsetsinUShomesisequalto3.(Assumeσ=0.8)1. Statetheappropriatenullandalternative hypothesesH0:μ=3H1:μ≠3(Thisisatwo-tailtest)2.SpecifythedesiredlevelofsignificanceandthesamplesizeSupposethat

=0.05andn=100arechosenforthistestBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-25HypothesisTestingExample3. DeterminetheappropriatetechniqueσisassumedknownsothisisaZtest.4. DeterminethecriticalvaluesFor

=0.05thecriticalZvaluesare±1.965.

CollectthedataandcomputetheteststatisticSupposethesampleresultsare n=100,X=2.84(σ=0.8isassumedknown)Sotheteststatisticis:(continued)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-26RejectH0DonotrejectH06. 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,sotheteststatisticisintherejectionregionBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-276(continued).Reachadecisionandinterprettheresult-2.0SinceZSTAT=-2.0<-1.96,rejectthenullhypothesisandconcludethereissufficientevidencethatthemeannumberofTVsinUShomesisnotequalto3HypothesisTestingExample(continued)RejectH0DonotrejectH0

=0.05/2-Zα/2=-1.960

=0.05/2RejectH0+Zα/2=+1.96BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-28p-ValueApproachtoTestingp-value:ProbabilityofobtainingateststatisticequaltoormoreextremethantheobservedsamplevaluegivenH0istrueThep-valueisalsocalledtheobservedlevelofsignificanceItisthesmallestvalueof

forwhichH0canberejectedBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-29p-ValueApproachtoTesting:

Interpretingthep-valueComparethep-valuewith

Ifp-value<

,rejectH0Ifp-value

,donotrejectH0RememberIfthep-valueislowthenH0mustgoBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-30The5Stepp-valueapproachto

HypothesisTestingStatethenullhypothesis,H0andthealternativehypothesis,H1Choosethelevelofsignificance,

,andthesamplesize,nDeterminetheappropriateteststatisticandsamplingdistributionCollectdataandcomputethevalueoftheteststatisticandthep-valueMakethestatisticaldecisionandstatethemanagerialconclusion.Ifthep-valueis<αthenrejectH0,otherwisedonotrejectH0.StatethemanagerialconclusioninthecontextoftheproblemBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-31p-valueHypothesisTestingExampleTesttheclaimthatthetruemean#ofTVsetsinUShomesisequalto3.(Assumeσ=0.8)1. Statetheappropriatenullandalternative hypothesesH0:μ=3H1:μ≠3(Thisisatwo-tailtest)2.SpecifythedesiredlevelofsignificanceandthesamplesizeSupposethat

=0.05andn=100arechosenforthistestBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-32p-valueHypothesisTestingExample3. DeterminetheappropriatetechniqueσisassumedknownsothisisaZtest.4.

Collectthedata,computetheteststatisticandthep-valueSupposethesampleresultsare n=100,X=2.84(σ=0.8isassumedknown)Sotheteststatisticis:(continued)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-33p-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)=0.0228BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-345.Isthep-value<α?Sincep-value=0.0456<α=0.05RejectH05.(continued)Statethemanagerialconclusioninthecontextofthesituation.ThereissufficientevidencetoconcludetheaveragenumberofTVsinUShomesisnotequalto3.p-valueHypothesisTestingExample(continued)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..ConnectionBetweenTwoTailTestsandConfidenceIntervalsFor

X=2.84,σ=0.8andn=100,the95%confidenceintervalis:

2.6832≤μ

≤2.9968Sincethisintervaldoesnotcontainthehypothesizedmean(3.0),werejectthenullhypothesisat

=0.05a.Atthe0.05levelofsignificance,isthereevidencethatthemeanlifeisdifferentfrom375hours?BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-36Exercise

Thequality-controlmanageratalightbulbfactoryneedstodeterminewhetherthemeanlifeofalargeshipmentoflightbulbsisequalto375hours.Thepopulationstandarddeviationis100hours.Arandomsampleof64lightbulbsindicatesasamplemeanlifeof350hours.a.b.Computethep-valueandinterpretitsmeaning.c.Constructa95%confidenceintervalestimateofthepopulationmeanlifeofthelightbulbs.d.Comparetheresultsof(a)and(c).Whatconclusionsdoyoureach?b.c.BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-37DoYouEverTrulyKnowσ?Probablynot!Invirtuallyallrealworldbusinesssituations,σisnotknown.Ifthereisasituationwhereσisknownthenµisalsoknown(sincetocalculateσyouneedtoknowµ.)Ifyoutrulyknowµtherewouldbenoneedtogatherasampletoestimateit.BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-38HypothesisTesting:

σUnknownIfthepopulationstandarddeviationisunknown,youinsteadusethesamplestandarddeviationS.Becauseofthischange,youusethetdistributioninsteadoftheZdistributiontotestthenullhypothesisaboutthemean.Whenusingthetdistributionyoumustassumethepopulationyouaresamplingfromfollowsanormaldistribution.Allothersteps,concepts,andconclusionsarethesame.BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-39tTestofHypothesisfortheMean(σUnknown)Theteststatisticis:HypothesisTestsforσKnownσUnknownKnownUnknown(Ztest)(ttest)tTestofHypothesisfortheMean(σUnknown)Convertsamplestatistic()toatSTAT

teststatistic

Theteststatisticis:HypothesisTestsforσKnownσUnknownKnownUnknown(Ztest)(ttest)

XTheteststatisticis:HypothesisTestsforσKnownσUnknownKnownUnknown(Ztest)(ttest)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-40Example:Two-TailTest

(Unknown)

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

=0.05.

(Assumethepopulationdistributionisnormal)H0:μ

=168H1:μ¹168BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-41a=0.05n

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

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

=168H1:μ¹168t24,0.025BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-42ExampleTwo-TailtTestUsingAp-valuefromExcelSincethisisat-testwecannotcalculatethep-valuewithoutsomecalculationaid.TheExceloutputbelowdoesthis:p-value>αSodonotrejectH0ExampleTwo-TailtTestUsingAp-valuefromMinitabOne-SampleTTestofmu=168vsnot=168NMeanStDevSEMean95%CITP25172.5015.403.08(166.14,178.86)1.460.157p-value>αSodonotrejectH0BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..ConnectionofTwoTailTeststoConfidenceIntervalsFor

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

=0.05BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-45ExerciseThefileFastFoodcontainstheamountthatasampleoffifteencustomersspentforlunch($)atafast-foodrestaurant:7.426.295.836.508.349.517.106.805.904.896.505.527.908.309.60b.Atthe0.05levelofsignificance,isthereevidencethatthemeanamountspentforlunchisdifferentfrom$6.50?a.Whatassumptionmustyoumakeaboutthepopulationdistributioninordertoconductthet-testin(b)?b.BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-46One-TailTestsInmanycases,thealternativehypothesisfocusesonaparticulardirectionH0:μ≥3H1:μ<3H0:μ≤3H1:μ>3Thisisalower-tailtestsincethealternativehypothesisisfocusedonthelowertailbelowthemeanof3Thisisanupper-tailtestsincethealternativehypothesisisfocusedontheuppertailabovethemeanof3BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-47RejectH0DonotrejectH0Thereisonlyonecriticalvalue,sincetherejectionareaisinonlyonetailLower-TailTestsa-Zαor-tα0μH0:μ≥3H1:μ<3ZortXCriticalvalueBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-48RejectH0DonotrejectH0Upper-TailTestsaZα

ortα0μH0:μ≤3H1:μ>3Thereisonlyonecriticalvalue,sincetherejectionareaisinonlyonetailCriticalvalueZortX_BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-49Example:Upper-TailtTest

forMean(

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

(i.e.,sufficientevidenceexiststosupportthe manager’sclaim)Formhypothesistest:BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-50RejectH0DonotrejectH0Supposethat=0.10ischosenforthistestandn=25.Findtherejectionregion:

=0.101.3180RejectH0RejectH0iftSTAT>1.318Example:FindRejectionRegion(continued)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-51ObtainsampleandcomputetheteststatisticSupposeasampleistakenwiththefollowingresults:n=25,X=53.1,andS=10

Thentheteststatisticis:Example:TestStatistic(continued)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-52RejectH0DonotrejectH0Example:Decision

=0.101.3180RejectH0DonotrejectH0sincetSTAT=0.55≤1.318thereisnotsufficientevidencethatthemeanbillisover$52tSTAT=0.55Reachadecisionandinterprettheresult:(continued)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-53Example:UtilizingThep-valueforTheTestCalculatethep-valueandcompareto

(p-valuebelowcalculatedusingexcelspreadsheetonnextpage)

RejectH0

=.10DonotrejectH01.3180RejectH0tSTAT=.55p-value=.2937DonotrejectH0sincep-value=.2937>=.10BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-54ExcelSpreadsheetCalculatingThep-valueforTheUpperTailtTestBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-55Exercise

Asurveyofnonprofitorganizationsshowedthatonlinefundraisinghasincreasedinthepastyear.Basedonarandomsampleof50nonprofitorganizations,themeanone-timegiftdonationinthepastyearwas$62,withastandarddeviationof$9.a.Atthe0.05levelofsignificance,isthereevidencethatthemeanamountspentforlunchisdifferentfrom$6.50?a.BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-56HypothesisTestsforProportionsInvolvescategoricalvariablesTwopossibleoutcomesPossessescharacteristicofinterestDoesnotpossesscharacteristicofinterestFractionorproportionofthepopulationinthecategoryofinterestisdenotedbyπBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-57ProportionsSampleproportioninthecategoryofinterestisdenotedbyp

Whenbothnπandn(1-π)areatleast5,p

canbeapproximatedbyanormaldistributionwithmeanandstandarddeviation

(continued)BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-58Thesamplingdistributionofpisapproximatelynormal,sotheteststatisticisaZSTATvalue:HypothesisTestsforProportionsnπ5andn(1-π)5HypothesisTestsforpnπ<5orn(1-π)<5NotdiscussedinthischapterBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-59Anequivalentformtothelastslide,butintermsofthenumberinthecategoryofinterest,X:ZTestforProportioninTermsofNumberinCategoryofInterestX5andn-X5HypothesisTestsforXX<5orn-X<5NotdiscussedinthischapterBasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-60Example:ZTestforProportionAmarketingcompanyclaimsthatitreceives8%responsesfromitsmailing.Totestthisclaim,arandomsampleof500weresurveyedwith25responses.Testatthe

=0.05significancelevel.Check:n

π=(500)(.08)=40n(1-π)=(500)(.92)=460

BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-61ZTestforProportion: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.-1.96BasicBusinessStatistics,11e©2009Prentice-Hall,Inc..Chap9-62DonotrejectH0RejectH0RejectH0

/2

=.0251.960Z=-2.47Calculatethep-valueandcompareto

(Foratwo-tailtestthep-valueisalwaystwo-tail)(continued)p-value=0.0136:p-ValueSolutionRejectH0sincep-value=0.0136<=0.05Z=2.47-1.9