Hypothesis_Testing(统计学假设检验).ppt

Chapter9,INTRODUCTIONTOHYPOTHESISTESTING,StatisticsforBusiness(ENV),1,HypothesisTesting,9.1NullandAlternativeHypothesesandErrorsinTesting9.2zTestsaboutaPopulationwithknowns9.3tTestsaboutaPopulationwithunknowns,2,Hypothesistesting-1,Researchersusuallycollectdatafromasampleandthenusethesampledatatohelpanswerquestionsaboutthepopulation.Hypothesistestingisaninferentialstatisticalprocessthatuseslimitedinformationfromthesampledataastoreachageneralconclusionaboutthepopulation.,3,Ahypothesistestisaformalizedprocedurethatfollowsastandardseriesofoperations.Inthisway,researchershaveastandardizedmethodforevaluatingtheresultsoftheirresearchstudies.,4,Hypothesistesting-2,5,Thebasicexperimentalsituationforusinghypothesistestingispresentedhere.Itisassumedthattheparameterisknownforthepopulationbeforetreatment.Thepurposeoftheexperimentistodeterminewhetherornotthetreatmenthasaneffect.Isthepopulationmeanaftertreatmentthesameasordifferentfromthemeanbeforetreatment?Asampleisselectedfromthetreatedpopulationtohelpanswerthisquestion.,Proceduresofhypothesis-testing,6,1.First,westateahypothesisaboutapopulation.Usuallythehypothesisconcernsthevalueofapopulationparameter.Forexample,wemighthypothesizethatthemeanIQforUICstudentsism=110.2.Next,weobtainarandomsamplefromthepopulation.Forexample,wemightselectarandomsampleofn=100UICstudents.3.Finally,wecomparethesampledatawiththehypothesis.Ifthedataareconsistentwiththehypothesis,wewillconcludethatthehypothesisisreasonable.Butifthereisabigdiscrepancybetweenthedataandthehypothesis,wewilldecidethatthehypothesisiswrong.,NullandAlternativeHypotheses,Thenullhypothesis,denotedH0,isastatementofthebasicpropositionbeingtested.Itgenerallyrepresentsthestatusquo(astatementof“noeffect”or“nodifference”,orastatementofequality)andisnotrejectedunlessthereisconvincingsampleevidencethatitisfalse.The(scientificor)alternativehypothesis,denotedHa(orH1),isanalternative(tothenullhypothesis)statementthatwillbeacceptedonlyifthereisconvincingsampleevidencethatitistrue.Thesetwohypothesesaremutuallyexclusiveandexhaustive.,7,8,Determinedbythelevelofsignificanceorthealphalevel,9,Alphalevelof.05-theprobabilityofrejectingthenullhypothesiswhenitistrueisnomorethan5%.,Z,10,Thelocationsofthecriticalregionboundariesforthreedifferentlevelsofsignificance,11,Example:Alcoholappearstobeinvolvedinavarietyofbirthdefects,includinglowbirthweightandretardedgrowth.Aresearcherwouldliketoinvestigatetheeffectofprenatalalcoholonbirthweight.Arandomsampleofn=16pregnantratsisobtained.Themotherratsaregivendailydosesofalcohol.Atbirth,onepupisselectedfromeachlittertoproduceasampleofn=16newbornrats.Theaverageweightforthesampleis15grams.Theresearcherwouldliketocomparethesamplewiththegeneralpopulationofrats.Itisknownthatregularnewbornrats(notexposedtoalcohol)haveanaverageweightofm=18grams.Thedistributionofweightsisnormalwithsd=4.,12,H0:=18,13,1.StatethehypothesesThenullhypothesisstatesthatexposuretoalcoholhasnoeffectonbirthweight.Thealternativehypothesisstatesthatalcoholexposuredoesaffectbirthweight.2.SelecttheLevelofSignificance(alpha)levelWewilluseanalphalevelof.05.Thatis,wearetakinga5%riskofcommittingaTypeIerror,or,theprobabilityofrejectingthenullhypothesiswhenitistrueisnomorethan5%.3.Setthedecisioncriteriabylocatingthecriticalregion,14,Alphalevelof.05-theprobabilityofrejectingthenullhypothesiswhenitistrueisnomorethan5%.,Z,15,4.COLLECTDATAandCOMPUTESAMPLESTATISTICSThesamplemeanisthenconvertedtoaz-score,whichisourteststatistic.,5.ArriveatadecisionRejectthenullhypothesis,HypothesisTesting,AlternativeHypothesisH1:AstatementthatisacceptedifH0isfalseWithout“=”signSay,“2”or“m0H1:mCriticalz,ComputedzCriticalzOrComputedzCriticalz,One-TailedTestofSignificance,.,IfH0:0istrue,itisveryunlikelythatthecomputedzvalueissolarge.,25,26,H0:0,Computedz1.96,31,Step4:ConcludeWecanseethatz=1.8971.96,thusourteststatisticisnotintherejectionregion.Thereforewefailtorejectthenullhypothesis.Wecannotconcludeanythingstatisticallysignificantfromthistest,andcannottelltheinsurancecompanywhetherornottheyshouldbeconcernedabouttheircurrentpolicies.,ExampleOneTailed(UpperTailed),32,Tryingtoencouragepeopletostopdrivingtocampus,theuniversityclaimsthatonaverageittakespeople30minutestofindaparkingspaceoncampus.Johndoesnotthinkittakessolongtofindaspot.Hecalculatedthemeantimetofindaparkingspaceoncampusforthelastfivetimesandfoundittobe20minutes.Assumingthatthetimeittakestofindaparkingspotisnormallydistributed,andthatthepopulationstandarddeviation=6minutes,performahypothesistestwithlevelofsignificancealpha=0.10toseeifhisclaimiscorrect.,Example:OneTailed(LowerTailed),33,Step1:Setthenullandalternativehypotheses,Example:OneTailed(LowerTailed),Step2:Calculatetheteststatistic,Step3:SetRejectionRegionLookingatthepicturebelow,weneedtoputallofalphainthelefttail.Thus,R:Z-1.28,34,Example:OneTailed(LowerTailed),Step4:ConcludeWecanseethatz=-3.727Criticalzof1.65,pof.007830,withtheunderlyingdistributionmayormaynotbeNormal(ii)smallsamplen2.353andt1.96orz1.833,53,Computedtof3.162Criticaltof1.833pof.0058alphaof.05RejectHo,Thep-valueis0.0058.(obtainedfromt,needasoftwaretofindit.),Step5Makeadecisionandinterprettheresults.,Themeannumberoffusesproducedismorethan250perhour.,54,Ifthep-valueislessthanalpha,thenrejectthenullhypothesis.,AmountoftimeUICstudentsspendinlibraryfromsurveyMean41.72minutesStandarddeviation40.179minutesNumberofcases294Nationalsurveyfindsuniversitylibraryusersspendmeanof38minutesIspopulationmeanforUICLibraryusersdifferentfromnationalmean?,Example:One-samplehypothesistestformean,NullhypothesisH0:=0=38AlternativeorresearchhypothesisHa:038,Step1.Hypotheses,Step2.Levelofsignificance,ProbabilityoferrorinmakingdecisiontorejectnullhypothesisForthistestchoose=0.05,Step3.Teststatistic,n=294sousecriticaltvaluesfromtableforinfinity.,CannotrejectthenullhypothesisCannotconcludethatpopulationmeanisdifferentfrom38minutes,4.Decision,95%confidenceIntervalinthisexample:E=1.96*=4.5941.72-4.59,41.72+4.59or37.13,46.31,Confidenceintervalfortimespentinlibraryis37.1346.31Hypothesizedvalueof38minutesfallswithinconfidenceintervalThereforewecannotsaythatpopulationmeanisnotequalto38minutes,cannotrejectthenullhypothesis,Confidenceintervalandhypothesistestforlibraryexample,ForparametersforasinglesampleOne-samplehypothesistestinvolvescomparisonwithpre-specifiedvalueWhichisoftenartificialSoconfidenceintervalmostappropriateforreportingresultsForparametersfortwosamplesDifferenceinparametersisofinterestHypothesistestexaminesdirectlyConfidenceintervallessintuitive,Usingconfidenceintervalsorhypothesistests,ConfidenceintervalorHypothesistest?,Hypothesistestsarebetterwhenthechiefissueistomakeayes/nodecisionaboutwhetherapatternexistsinapopulation.Confidenceintervalsarebetterwhenthechiefissueistomakeabestguessofapopulationparameter.,63,Whenreadingascientificjournal,youtypicallywillnotbetoldexplicitlythattheresearcherevaluatedthedatausingaz-scoreasateststatisticwithanalphalevelof.05.Norwillyoubetoldthat“thenullhypothesisisrejected.”Instead,youwillseeastatementsuchas:Thetreatmentwithmedicationhadasignificanteffectonpeoplesdepressionscores,z=3.85,p.05.Letusexaminethisstatementpiecebypiece.First,whatismeantbythetermsignificant?Instatisticaltests,thiswordindicatesthattheresultisdifferentfromwhatwouldbeexpectedduetochance.Asignificantresultmeansthatthenullhypothesishasbeenrejected.Thatis,thedataaresignificantbecausethesamplemeanfallsinthecriticalregionandisnotwhatwewouldhaveexpectedtoobtainifH0weretrue.Next,whatisthemeaningofz=3.85?Thezindicatesthataz-scorewasusedastheteststatistictoevaluatethesampledataandthatitsvalueis3.85.,64,Finally,whatismeantbyp.05.Inthiscase,wearesayingthattheobtainedresult,z=1.30,isnotunusual(notinthecriticalregion)andisrelativelylikelytooccurbychance(theprobabilityisgreaterthan.05).Thus,H0wasnotrejected.,Usingthep-ValueinHypothesisTesting,Ifthep-Valuea,H0cannotberejected.Ifthep-Value10ThesampleisarandomsampleofsomesortThevariableisadiscrete