[经济学]计量经济学基础第3章课件
Chapter3IntervalEstimationandHypothesisTestingWalterR.PaczkowskiRutgersUniversity\n3.1IntervalEstimation3.2HypothesisTests3.3RejectionRegionsforSpecificAlternatives3.4ExamplesofHypothesisTests3.5Thep-value3.6LinearCombinationsofParametersChapterContents\n3.1IntervalEstimation\nTherearetwotypesofestimatesPointestimatesTheestimateb2isapointestimateoftheunknownpopulationparameterintheregressionmodel.IntervalestimatesIntervalestimationproposesarangeofvaluesinwhichthetrueparameterislikelytofallProvidingarangeofvaluesgivesasenseofwhattheparametervaluemightbe,andtheprecisionwithwhichwehaveestimateditSuchintervalsareoftencalledconfidenceintervals.Weprefertocallthemintervalestimatesbecausetheterm‘‘confidence’’iswidelymisunderstoodandmisused3.1IntervalEstimation\nThenormaldistributionofb2,theleastsquaresestimatorofβ2,isAstandardizednormalrandomvariableisobtainedfromb2bysubtractingitsmeananddividingbyitsstandarddeviation:3.1.1Thet-DistributionEq.3.13.1IntervalEstimation\nWeknowthat:Substituting:Rearranging:3.1IntervalEstimation3.1.1Thet-Distribution\nThetwoend-pointsprovideanintervalestimator.Inrepeatedsampling95%oftheintervalsconstructedthiswaywillcontainthetruevalueoftheparameterβ2.ThiseasyderivationofanintervalestimatorisbasedontheassumptionSR6andthatweknowthevarianceoftheerrortermσ2.3.1IntervalEstimation3.1.1Thet-Distribution\nReplacingσ2withcreatesarandomvariablet:Theratiohasat-distributionwith(N–2)degreesoffreedom,whichwedenoteas:Eq.3.23.1IntervalEstimation3.1.1Thet-Distribution\nIngeneralwecansay,ifassumptionsSR1-SR6holdinthesimplelinearregressionmodel,thenThet-distributionisabellshapedcurvecenteredatzeroItlookslikethestandardnormaldistribution,exceptitismorespreadout,withalargervarianceandthickertailsTheshapeofthet-distributioniscontrolledbyasingleparametercalledthedegreesoffreedom,oftenabbreviatedasdfEq.3.33.1IntervalEstimation3.1.1Thet-Distribution\nWecanfinda“criticalvalue”fromat-distributionsuchthatwhereαisaprobabilityoftentakentobeα=0.01orα=0.05.Thecriticalvaluetcfordegreesoffreedommisthepercentilevaluet(1-α/2,m)3.1.2ObtainingIntervalEstimates3.1IntervalEstimation\nFigure3.1Criticalvaluesfromat-distribution.3.1IntervalEstimation3.1.1Thet-Distribution\nEachshaded‘‘tail’’areacontainsα/2oftheprobability,sothat1-αoftheprobabilityiscontainedinthecenterportion.Consequently,wecanmaketheprobabilitystatementororEq.3.4Eq.3.53.1IntervalEstimation3.1.2ObtainingIntervalEstimates\nWhenbkandse(bk)areestimatedvalues(numbers),basedonagivensampleofdata,thenbk±tcse(bk)iscalleda100(1-α)%intervalestimateofbk.Equivalentlyitiscalleda100(1-α)%confidenceinterval.Usuallyα=0.01orα=0.05,sothatweobtaina99%confidenceintervalora95%confidenceinterval.3.1IntervalEstimation3.1.2ObtainingIntervalEstimates\nTheinterpretationofconfidenceintervalsrequiresagreatdealofcareThepropertiesoftheintervalestimationprocedurearebasedonthenotionofrepeatedsamplingAnyoneintervalestimate,basedononesampleofdata,mayormaynotcontainthetrueparameterβk,andbecauseβkisunknown,wewillneverknowwhetheritdoesordoesnotWhen‘‘confidenceintervals’’arediscussed,rememberthatourconfidenceisintheprocedureusedtoconstructtheintervalestimate;itisnotinanyoneintervalestimatecalculatedfromasampleofdata3.1IntervalEstimation3.1.2ObtainingIntervalEstimates\nForthefoodexpendituredataThecriticalvaluetc=2.024,whichisappropriatefor=.05and38degreesoffreedomToconstructanintervalestimatefor2weusetheleastsquaresestimateb2=10.21anditsstandarderror3.1.3AnIllustrationEq.3.63.1IntervalEstimation\nA“95%confidenceintervalestimate”for2:Whentheprocedureweusedisappliedtomanyrandomsamplesofdatafromthesamepopulation,then95%ofalltheintervalestimatesconstructedusingthisprocedurewillcontainthetrueparameter3.1IntervalEstimation3.1.3AnIllustration\nIsβ2actuallyintheinterval[5.97,14.45]?Wedonotknow,andwewillneverknowWhatwedoknowisthatwhentheprocedureweusedisappliedtomanyrandomsamplesofdatafromthesamepopulation,then95%ofalltheintervalestimatesconstructedusingthisprocedurewillcontainthetrueparameterTheintervalestimationprocedure‘‘works’’95%ofthetimeWhatwecansayabouttheintervalestimatebasedonouronesampleisthat,giventhereliabilityoftheprocedure,wewouldbe‘‘surprised’’ifβ2isnotintheinterval[5.97,14.45].3.1IntervalEstimation3.1.3AnIllustration\nWhatistheusefulnessofanintervalestimateofβ2?Whenreportingregressionresultswealwaysgiveapointestimate,suchasb2=10.21However,thepointestimatealonegivesnosenseofitsreliabilityThus,wemightalsoreportanintervalestimateIntervalestimatesincorporateboththepointestimateandthestandarderroroftheestimate,whichisameasureofthevariabilityoftheleastsquaresestimator3.1IntervalEstimation3.1.3AnIllustration\n3.1.4TheRepeatedSamplingContextTable3.1LeastSquaresEstimatesfrom10RandomSamples3.1IntervalEstimation\nTable3.2IntervalEstimatesfrom10RandomSamples3.1IntervalEstimation3.1.4TheRepeatedSamplingContext\n3.2HypothesisTests\n3.2HypothesisTestsHypothesistestingprocedurescompareaconjecturewehaveaboutapopulationtotheinformationcontainedinasampleofdataGivenaneconomicandstatisticalmodel,hypothesesareformedabouteconomicbehavior.ThesehypothesesarethenrepresentedasstatementsaboutmodelparametersHypothesistestsusetheinformationaboutaparameterthatiscontainedinasampleofdata,itsleastsquarespointestimate,anditsstandarderror,todrawaconclusionaboutthehypothesis\nAnullhypothesisH0AnalternativehypothesisH1AteststatisticArejectionregionAconclusion3.2HypothesisTestsCOMPONENTSOFHYPOTHESISTESTS\nAnullhypothesisisthebeliefwewillmaintainuntilweareconvincedbythesampleevidencethatitisnottrue,inwhichcasewerejectthenullhypothesisThenullhypothesisisstatedasH0:βk=c,wherecisaconstant,andisanimportantvalueinthecontextofaspecificregressionmodelAcommonvalueforcis03.2.1TheNullHypothesis3.2HypothesisTests\nPairedwitheverynullhypothesisisalogicalalternativehypothesisH1thatwewillacceptifthenullhypothesisisrejectedThealternativehypothesisisflexibleanddependstosomeextentoneconomictheory3.2.2TheAlternativeHypothesis3.2HypothesisTests\nPossibleAlternativehypothesesare:H1:βk>cH1:βk
c,rejectthenullhypothesisandacceptthealternativehypothesisift≥t(1-α;N-2)3.3RejectionRegionsforSpecificAlternatives\nFigure3.2Rejectionregionforaone-tailtestofH0:βk=cagainstH1:βk>c3.3RejectionRegionsforSpecificAlternatives3.3.1One-tailTestwithAlternative“GreaterThan”\n3.3.2One-tailTestwithAlternative“LessThan”WhentestingthenullhypothesisH0:βk=cagainstthealternativehypothesisH1:βk0TheteststatisticisEq.3.7Inthiscasec=0,sot=b2/se(b2)~t(N–2)ifthenullhypothesisistrueSelectα=0.05Thecriticalvaluefortheright-tailrejectionregionisthe95thpercentileofthet-distributionwithN–2=38degreesoffreedom,t(0.95,38)=1.686.Thuswewillrejectthenullhypothesisifthecalculatedvalueoft≥1.686.Ift<1.686,wewillnotrejectthenullhypothesis.3.4ExamplesofHypothesisTests\n3.4.1aOne-tailTestofSignificanceUsingthefoodexpendituredata,wefoundthatb2=10.21withstandarderrorse(b2)=2.09Thevalueoftheteststatisticis:Sincet=4.88>1.686,werejectthenullhypothesisthatβ2=0andacceptthealternativethatβ2>0Thatis,werejectthehypothesisthatthereisnorelationshipbetweenincomeandfoodexpenditure,andconcludethatthereisastatisticallysignificantpositiverelationshipbetweenhouseholdincomeandfoodexpenditure3.4ExamplesofHypothesisTests\n3.4.1bOne-tailTestofanEconomicHypothesisThenullhypothesisisH0:β2≤5.5ThealternativehypothesisisH1:β2>5.5Theteststatisticist=(b2-5.5)/se(b2)~t(N–2)ifthenullhypothesisistrueSelectα=0.01Thecriticalvaluefortheright-tailrejectionregionisthe99thpercentileofthet-distributionwithN–2=38degreesoffreedom,t(0.99,38)=2.429Thuswewillrejectthenullhypothesisifthecalculatedvalueoft≥2.429Ift<2.429,wewillnotrejectthenullhypothesis3.4ExamplesofHypothesisTests\nUsingthefoodexpendituredata,wefoundthatb2=10.21withstandarderrorse(b2)=2.09Thevalueoftheteststatisticis:Sincet=2.25<2.429wedonotrejectthenullhypothesisthatβ2≤5.5Wearenotabletoconcludethatthenewsupermarketwillbeprofitableandwillnotbeginconstruction3.4.1bOne-tailTestofanEconomicHypothesis3.4ExamplesofHypothesisTests\n3.4.2Left-tailTestsThenullhypothesisisH0:β2≥15ThealternativehypothesisisH1:β2<15Theteststatisticist=(b2-15)/se(b2)~t(N–2)ifthenullhypothesisistrueSelectα=0.05Thecriticalvaluefortheleft-tailrejectionregionisthe5thpercentileofthet-distributionwithN–2=38degreesoffreedom,t(0.05,38)=-1.686.Thuswewillrejectthenullhypothesisifthecalculatedvalueoft≤-1.686Ift>-1.686,wewillnotrejectthenullhypothesis3.4ExamplesofHypothesisTests\nUsingthefoodexpendituredata,wefoundthatb2=10.21withstandarderrorse(b2)=2.09Thevalueoftheteststatisticis:Sincet=-2.29<-1.686werejectthenullhypothesisthatβ2≥15andacceptthealternativethatβ2<15Weconcludethathouseholdsspendlessthan$15fromeachadditional$100incomeonfood3.4.2Left-tailTests3.4ExamplesofHypothesisTests\n3.4.3aTwo-tailTestofanEconomicHypothesisThenullhypothesisisH0:β2=7.5ThealternativehypothesisisH1:β2≠7.5Theteststatisticist=(b2–7.5)/se(b2)~t(N–2)ifthenullhypothesisistrueSelectα=0.05Thecriticalvalueforthetwo-tailrejectionregionisthe2.5thpercentileofthet-distributionwithN–2=38degreesoffreedom,t(0.025,38)=-2.024andthe97.5thpercentilet(0.975,38)=2.024Thuswewillrejectthenullhypothesisifthecalculatedvalueoft≥2.024orift≤-2.0243.4ExamplesofHypothesisTests\nUsingthefoodexpendituredata,wefoundthatb2=10.21withstandarderrorse(b2)=2.09ThevalueoftheteststatisticisSince-2.0242.024werejectthenullhypothesisthatβ2=0Weconcludethatthereisastatisticallysignificantrelationshipbetweenincomeandfoodexpenditure3.4.3bTwo-tailTestofSignificance3.4ExamplesofHypothesisTests\n3.4.3bTwo-tailTestofSignificanceFromtheSTATAoutput,wecaneasilyfindthecalculatedtvalueusedinthisexample3.4ExamplesofHypothesisTests\n3.5Thep-Value\n3.5Thep-ValueWhenreportingtheoutcomeofstatisticalhypothesistests,ithasbecomestandardpracticetoreportthep-value(anabbreviationforprobabilityvalue)ofthetest.Ifwehavethep-valueofatest,p,wecandeterminetheoutcomeofthetestbycomparingthep-valuetothechosenlevelofsignificance,α,withoutlookinguporcalculatingthecriticalvalues.Thisismuchmoreconvenient\nRejectthenullhypothesiswhenthep-valueislessthan,orequalto,thelevelofsignificanceα.Thatis,ifp≤αthenrejectH0.Ifp>αthendonotrejectH0.p-VALUERULE3.5Thep-Value\nIftisthecalculatedvalueofthet-statistic,then:ifH1:βK>cp=probabilitytotherightoftifH1:βK5.5Thep-valueis3.5.1p-ValueforaRight-tailTest3.5Thep-Value\n3.5.1p-ValueforaRight-tailTestFigure3.5Thep-valueforaright-tailtest.3.5Thep-Value=T.DIST.RT(2.25,38)=T.INV.2T(0.02,38)\nFromSection3.4.2,wehaveThenullhypothesisisH0:β2≥15ThealternativehypothesisisH1:β2<15Thep-valueis3.5.2p-ValueforaLeft-tailTest3.5Thep-Value\nFigure3.6Thep-valueforaleft-tailtest.3.5.2p-ValueforaLeft-tailTest3.5Thep-Value\nFromSection3.4.3a,wehaveThenullhypothesisisH0:β2=7.5ThealternativehypothesisisH1:β2≠7.5Thep-valueis3.5.3p-ValueforaTwo-tailTest3.5Thep-Value\nFigure3.7Thep-valueforatwo-tailtestofsignificance.3.5.3p-ValueforaTwo-tailTest3.5Thep-Value\nFromSection3.4.3b,wehaveThenullhypothesisisH0:β2=0ThealternativehypothesisisH1:β2≠0Thep-valueis3.5.4p-ValueforaTwo-tailTestofSignificance3.5Thep-Value\nFromtypicalEviewsoutput,wecaneasilyfindthecalculatedp-valueusedinthisexample3.5.4p-ValueforaTwo-tailTestofSignificance3.5Thep-Value\n3.6LinearCombinationsofParameters\n3.6LinearCombinationsofParametersWemaywishtoestimateandtesthypothesesaboutalinearcombinationofparametersλ=c1β1+c2β2,wherec1andc2areconstantsthatwespecifyUnderassumptionsSR1–SR5theleastsquaresestimatorsb1andb2arethebestlinearunbiasedestimatorsofβ1andβ2Itisalsotruethat=c1b1+c2b2isthebestlinearunbiasedestimatorofλ=c1β1+c2β2\nAsanexampleofalinearcombination,ifweletc1=1andc2=x0,thenwehavewhichisjustoutbasicmodel3.6LinearCombinationsofParameters\nTheestimatorisunbiasedbecause3.6LinearCombinationsofParameters\nThevarianceofiswherethevariancesandcovariancesaregiveninEq.2.20-2.22Eq.3.83.6LinearCombinationsofParameters\nWeestimatebyreplacingtheunknownvariancesandcovarianceswiththeirestimatedvariancesandcovariancesinEq.2.20-2.22Eq.3.93.6LinearCombinationsofParameters\nThestandarderrorofisthesquarerootoftheestimatedvarianceEq.3.103.6LinearCombinationsofParameters\nIfinadditionSR6holds,orifthesampleislarge,theleastsquaresestimatorsb1andb2havenormaldistributions.Itisalsotruethatlinearcombinationsofnormallydistributedvariablesarenormallydistributed,sothat3.6LinearCombinationsofParameters\nyearoutputlaborcapital1899100100100190010110510719011121101141902122118122190312412313119041221161381905143125149190615213316319071511381761908126121185190915514019819101591442081911153145216191217715222619131841542361914169149244191518915426619162251822981917227196335191822320036619192181933871920231193407192117914741719222401614313.6LinearCombinationsofParametersEmpiricalAnalysisonCDFunction(Data)Source:JesusFelipe&F.GerardAdams(2005)."ATheoryofProduction”TheEstimationoftheCobb-DouglasFunction:ARetrospectiveView.EasternEconomicJournal31(3),427-445.\nclearalllogusing"G:\EnvironClass\Cobb-DouglassRegression.smcl"use"G:\EnvironClass\Cobb-DouglassData.dta",cleargenlnq=ln(output)genlnl=ln(labor)genlnk=ln(capital)sumlnqlnllnkscatterlnqlnl,mlabel(year)regresslnQlnLlnKoutreg2usingcobb-doug,bdec(3)tstatalpha(0.01,0.05,0.1)symbol(***,**,*)level(95)exceltestlnL+lnK=1*summarizelnqlnllnk*predictyhat,xb*predictehat,residual*histogramehatlogcloseview3.6.1EstimatingExpectedFoodExpenditure3.6LinearCombinationsofParametersEmpiricalAnalysisonCDFunction(Code)\nEmpiricalAnalysisonCDFunction(Results)α=0.807,representingoutputelasticityw.r.tlaborinputβ=0.233,representingoutputelasticityw.r.tcapital;specifically,a1%ofcapitalinputwillincreaseoutputby0.233%α+β=1.04，chi-squareteststatisticfailstorejectthenullhypothesisofCRSR2=0.975implyingthatvariationsofcapitalandlaborcanexplain97.5%outputvariation.Source:JesusFelipe&F.GerardAdams(2005)."ATheoryofProduction”TheEstimationoftheCobb-DouglasFunction:ARetrospectiveView.EasternEconomicJournal31(3),427-445.\nWecanestimatetheaverage(orexpected)expenditureonfoodas:Ifthehouseholdincomeis$2000,whichis20sinceincomeismeasuredin$100unitsinthisexample,thentheaverageexpenditureis:Weestimatethattheexpectedfoodexpenditurebyahouseholdwith$2,000incomeis$287.61perweek3.6.1EstimatingExpectedFoodExpenditure3.6LinearCombinationsofParametersAnotherExamplefromtheBook\nThet-statisticforthelinearcombinationis:3.6.2AnIntervalEstimateofExpectedFoodExpenditureEq.3.113.6LinearCombinationsofParameters\nSubstitutingthetvalueintoP(-tc≤t≤tc)=1–α,weget:sothatthe(1–α)%intervalis3.6.2AnIntervalEstimateofExpectedFoodExpenditure3.6LinearCombinationsofParameters\nForourexample,theestimatedvariancesandcovarianceis:3.6.2AnIntervalEstimateofExpectedFoodExpenditureCIncomeC1884.442-85.9032Income-85.90324.38183.6LinearCombinationsofParameters\nTheestimatedvarianceofourexpectedfoodexpenditureis:andthecorrespondingstandarderroris:3.6.2AnIntervalEstimateofExpectedFoodExpenditure3.6LinearCombinationsofParameters\nThe95%intervalisthen:orWeestimatewith95%confidencethattheexpectedfoodexpenditurebyahouseholdwith$2,000incomeisbetween$258.91and$316.313.6.2AnIntervalEstimateofExpectedFoodExpenditure3.6LinearCombinationsofParameters\nAgenerallinearhypothesisinvolvesbothparameters,β1andβ2andmaybestatedas:or,equivalently,3.6.3TestingaLinearCombinationofParametersEq.3.12aEq.3.12b3.6LinearCombinationsofParameters\nThealternativehypothesismightbeanyoneofthefollowing:3.6.3TestingaLinearCombinationofParameters3.6LinearCombinationsofParameters\nThet-statisticis:ifthenullhypothesisistrueTherejectionregionsfortheone-andtwo-tailalternatives(i)–(iii)arethesameasthosedescribedinSection3.3,andconclusionsareinterpretedthesamewayaswell3.6.3TestingaLinearCombinationofParametersEq.3.133.6LinearCombinationsofParameters\nSupposeweconjecturethat:Usethisasthealternativehypothesis:or3.6.4TestingExpectedFoodExpenditure3.6LinearCombinationsofParameters\nThenullhypothesisisthelogicalalternative:orThenullandalternativehypothesisareinthesameformasthegenerallinearhypothesiswithc1=1,c2=20,andc0=2503.6.4TestingExpectedFoodExpenditure3.6LinearCombinationsofParameters\nThet-statisticis3.6.4TestingExpectedFoodExpenditure3.6LinearCombinationsofParameters\nSincet=2.65>tc=1.686,werejectthenullhypothesisthatahouseholdwithweeklyincomeof$2,000willspend$250perweekorlessonfood,andconcludethattheconjecturethatsuchhouseholdsspendmorethan$250iscorrect,withtheprobabilityofTypeIerror0.053.6.4TestingExpectedFoodExpenditure3.6LinearCombinationsofParameters\nKeyWords\nalternativehypothesisconfidenceintervalscriticalvaluedegreesoffreedomhypotheseshypothesistestingInferenceKeywordsintervalestimationlevelofsignificancelinearhypothesisnullhypothesisone-tailtestspointestimatesprobabilityvaluep-valuerejectionregiontestofsignificanceteststatistictwo-tailtestsTypeIerrorTypeIIerror\nAppendices\n3ADerivationofthet-distribution3BDistributionofthet-statisticunderH13CMonteCarloSimulation\n3ADerivationofthet-DistributionConsiderthenormaldistributionofb2,theleastsquaresestimatorofβ2,whichwedenoteas:Thestandardizednormalis:Eq.3A.1\nIfalltherandomerrorsareindependent,thenSincethetruerandomerrorsareunobservable,wereplacethembytheirsamplecounterparts,theleastsquaresresidualstoobtain:Eq.3A.3Eq.3A.23ADerivationofthet-Distribution\nTherefore,Althoughwehavenotestablishedthefactthatthechi-squarerandomvariableVisstatisticallyindependentoftheleastsquaresestimatorsb1andb2,itisConsequently,VandthestandardnormalrandomvariableZinEq.3A.1areindependentEq.3A.43ADerivationofthet-Distribution\nAt-randomvariableisformedbydividingastandardnormalrandomvariable,Z~N(0,1),bythesquarerootofanindependentchi-squarerandomvariable,V~χ2(m),thathasbeendividedbyitsdegreesoffreedom,mThatis:3ADerivationofthet-Distribution\nUsingZandVfromEq.3A.1andEq.3A.4,respectively,wehave:Eq.3A.53ADerivationofthet-Distribution\n3BDistributionofthet-StatisticunderH1Toexaminethedistributionofthet-statisticinEq.3.7whenthenullhypothesisisnottrue,supposethatthetrueβ2=1Wecanshowthat:\nIfβ2=1andweincorrectlyhypothesizethatβ2=c,thenthenumeratorinEq.3A.5thatisusedinformingEq.3.7hasthedistribution:Sinceitsmeanisnotzero,thedistributionofthevariableinEq.3B.1isnotstandardnormal,asrequiredintheformationofat-randomvariableEq.3B.13BDistributionofthet-StatisticunderH1\n3CMonteCarloSimulationWhenstudyingtheperformanceofhypothesistestsandintervalestimatorsitisnecessarytouseenoughMonteCarlosamplessothatthepercentagesinvolvedareestimatedpreciselyenoughtobeusefulFortestswithprobabilityofTypeIerrorα=0.05weshouldobservetruenullhypothesesbeingrejected5%ofthetimeFor95%intervalestimatorsweshouldobservethat95%oftheintervalestimatescontainthetrueparametervaluesWeuseM=10,000MonteCarlosamplessothattheexperimentalerrorisverysmall\n3C.1RepeatedSamplingPropertiesofIntervalEstimatorsTable3C.1Resultsof10000MonteCarloSimulations3CMonteCarloSimulation\nThelessonis,thatinmanyrepeatedsamplesfromthedatagenerationprocess,andifassumptionsSR1–SR6hold,theprocedureforconstructing95%intervalestimates‘‘works’’95%ofthetime3C.1RepeatedSamplingPropertiesofIntervalEstimators3CMonteCarloSimulation\nThelessonisthatinmanyrepeatedsamplesfromthedatagenerationprocess,andifassumptionsSR1–SR6hold,theprocedurefortestingatruenullhypothesisatsignificancelevelα=0.05rejectsthetruenullhypothesis5%ofthetimeOr,statedpositively,thetestproceduredoesnotrejectthetruenullhypothesis95%ofthetime3C.2RepeatedSamplingPropertiesofHypothesisTests3CMonteCarloSimulation\nThepointisthatiffewerMonteCarlosamplesarechosenthe‘‘noise’’intheMonteCarloexperimentcanleadtoapercentofsuccessesorrejectionsthathastoowideamarginoferrorforustotellwhetherthestatisticalprocedure,intervalestimation,orhypothesistesting,is‘‘working’’properlyornot3C.3ChoosingteNumberofMonteCarloSamples3CMonteCarloSimulation
