Statistics-(I)

PAGE\*MERGEFORMAT#Statistics(I)FinalExamJan.11,2002解答：ddada,ccddcMultipleChoiceQuestions(30%)Whichofthefollowingisnotoneofthesamplingdistributionofthesamplemean?a.Themeanofthesamplingdistributionisequaltothepopulationmean.Thestandarddeviationofthesamplingdistributionisequaltothestandarddeviationofthepopulationdividedbythesquarerootofn.Theshapeofthesamplingdistributionisapproximatelynormalifthesamplesizeissufficientlylarge.Alloftheabovearepropertiesofthesamplingdistributionofthesamplemean.WhyistheCentralLimitTheoremsoimportanttothestudyofsamplingdistributions?Itallowsustodisregardthesizeofthesampleselectedwhenthepopulationisapproximatelynormal.Itallowsustodisregardtheshapeofthesamplingdistributionwhenthesizeofpopulationislarge.Itallowsustodisregardthesizeofthepopulationwearesamplingfrom.Itallowsustodisregardtheshapeofpopulationwhennislarge.Asamplethatdoesnotprovideagoodrepresentationofthepopulationfromwhichitwascollectedisreferredtoasa(n)_sample.a.Biasedb.Empiricalc.Statisticd.InferentialSupposea95%confidenceintervalfor仙turnsouttobe(1,000,2,100).Basedontheinterval,doyoubelievetheaverageisequalto2,200?Yes,andIam95%sureofit.Yes,andIam100%sureofit.No,andIam95%sureofit.1,200isapossiblevalueforthemean.Intheconstructionofconfidenceintervals,ifallotherquantitiesareunchanged,anincreaseinthesamplesizewillleadtoainterval.a.Narrowerb.Widerc.Lesssignificantd.BiasedThesamplesizeformulasareusefulfor.CalculatingavalueoftheteststatisticwhentestingnCalculatingap-valueforalargesampletestofn.Calculatingthesamplesizetoestimateaparametertowithinaspecifiedbound.Comparingdifferentconfidenceintervalsconstructedusingdifferentsamplesizes.SupposewewishtotestH:仙=47vsHa：仙>47.Whatwillresultifweconcludethatthemeanisgreaterthan47whenitstruevalueisreally52?WehavemadeaTypeIIerror.WehavemadeaTypeIerror.Wehavemadeacorrectdecision.Noneoftheabovearecorrect.WhichofthefollowingstatementsisNottrueaboutthelevelofsignificanceinatestofhypothesis?Thelargerthelevelofsignificance,themorelikelyyouaretorejectthenullhypothesis.ThelevelofsignificanceisthemaximumriskwearewillingtoacceptinmakingaTypeIerror.Thelevelofsignificanceisalsocalledthealphalevel.Thelevelofsignificanceisanothernameforap-value.Whichpartofthetestofhypothesisproceduredeterminesthevalueofthep-value?ThealternativehypothesisTheteststatisticThesamplingdistributionoftheteststatisticAllofaboveWehavecreateda95%confidenceintervalforwiththeresult(10,15).WhatconclusionwillwemakeifwetestHo:仙=16vsHa：仙>16ata=0.05?RejectH0infavorofHa.AcceptH0infavorofH0.FailtorejectH0.Wecannottellwhatourdecisionwillbewiththeinformationgiven.EssayQuestionsAlocalbankreportedtothefederalgovernmentthatits5,246savingsaccountshaveameanbalanceof$1,000andastandarddeviationof$240.Governmentauditorshaveaskedtorandomlysample64ofthebank'saccountstoassessthereliabilityofthemeanbalancereportedbythebank.Ifthebank'sinformationiscorrect,findtheprobabilitythatthesamplemeanbalancewouldbelessthan$1,072.(8%)答：N=5246（有限母H随械檬本非完全力蜀立建H数要做修正）n=64X~N(1000,240-J-N—n)107210002405246648\5245)=P(Z<2.4145)=0.99264.N1_x1000P(x<1072)=P(2405246648■.5245LetXandYbeindependentrandomvariableswithE(X)=3,E(Y)=5,Var(X)=Var(Y)=o-2,findthevalueofMwhichmakesM(X2-Y2)+Y2beanunbiasedestimateofo-2.(12%)答：使M(X2丫2)+丫2之期望值等於2E(X)=3,E(Y)=5Var(X)=Var(Y)=2E{M(X2Y2)+Y2}=ME(X2)(1M)E(Y2)其中，Var(X)=E(X2)[E(X)]22=E(X2)9E(X2)=29Var(Y)=E(Y2)[E(Y)]22=E(Y2)25E(Y2)=225E{M(X2Y2)+Y2}=ME(X2)(1M)E(Y2)=2(不偏估^元)M[29](1M)[225]=2216M25=216M=25M=25/16Anormalpopulationhasunknownvariance,o-2.If(1.48,8.5)isa95%confidence1222intervalforo-basedonarandomsampleofsize12,findthevalueof(xix).i1(10%)答：2=(1.48,8.5)1.48<2<8.5推想2信赖^之言十算方法_2_(n1)S2(n-1)n=12,=0.05信之建立:-2X)12(0.025;11)2(0.975;11)2(11)(122)Si^--3.8162(11)12—2(XiX)i121.9212(Xii12X)2——3.81612—2(XiX)2i13.8168.512—2一一一4，一一一一一(XiX)2=8.53.816=32.43612—2(XiX)2i121.921.48122(XiX)2或^^~~2——21.9212_o(XiX)2=1.4821.92=32.4416i1Thebranchmanagerofanoutletofalarge,nationwidebookstorechainwantstostudycharacteristicsofcustomersofherstore,whichislocatednearthecampusofalargestateuniversity.Inparticular,shedecidestofocusontwovariables:theamountofmoneyspentbycustomersandwhetherthecustomerswouldconsiderpurchasingeducationalvideotapesrelatingtospecificcoursessuchasstatistics,accounting,orcalculus.Theresultsfromasampleof70customersareasfollows:Amountspent:X$28.52,s$11.39.28customersstatedthattheywouldconsiderpurchasingeducationalvideotapes.Setupa95%confidenceintervalestimateofthepopulationaverageamountspentinthebookstore.(7%)Setupa90%confidenceintervalestimateofthepopulationproportionofcustomerswhowouldconsiderpurchasingeducationalvideotapes.(7%)1139a.28.521.96=(25.8517,31.1883),70b.竺1.645J强7070(0.3037,0.4693)The“betacoefficient"ofastockisameasureofthestock'svolatility(orrisk)relativetothemarketasawhole.Stockswithbetacoefficientsgreaterthan1generallybeargreaterriskthanthemarket,whereasstockswithbetacoefficientslessthan1arelessriskythantheoverallmarket.Arandomsampleof15high-technologystockswasselectedattheendof1996,andthemeanandstandarddeviationofthebetacoefficientswerecalculated:X=1.23,s=0.37Setuptheappropriatenullandalternativehypothesestotestwhethertheaveragehigh-technologystockisriskierthanthemarketasawhole.(4%)Establishtheappropriateteststatisticandrejectionregionforthetest.(8%)Whatassumptionsarenecessarytoensurethevalidityofthetest?(4%)Calculatetheteststatisticandstateyourconclusion.(4%)Whatistheapproximatep-valueassociatedwiththistest?Interpretit.(6%)答：(a)H。：=1Hi：>1(b)梅T定统舒量t=x一-~t(n-1)=t(14)s\n拒余色域：{tIt>t(0.05;14)=1.761}(c)觐察值来自常熊分配231t=2.408>1.761拒余色H0,(结谕自己JW…)0.3715p-value=P(t(14)>2.41),介於0.01典0.025之因悬小於=0.05,故拒^H0