Design-and-analysis-of-surveys-Homework-4-Solution

1.(4.18)Resourcemanagersofforestgamelandswereconcernedaboutthesizeofthedeerandrabbitpopulationsduringthewintermonthsinaparticularforest.Asanestimateofpopulationsize,theyproposedusingtheaveragenumberofpelletgroupsforrabbitsanddeerper30-foot-squareplots.Fromanaerialphotograph,theforestwasdividedintoN=10,00030-foot-squaregrids.ASRSof500plotswastaken,andthenumberofpelletgroupswasobservedforrabbitsandfordeer.Theresultsaresummarizedbelow.DeerRabbitsSamplemean2.304.52Samplevariance0.650.97a)Estimatemu1andmu2,thepopulationaveragenumberofpelletgroupsfordeerandrabbits,respectively,per30-foot-squareplots.PlaceboundsontheerrorsofestimationSolutionThepopulationaveragenumberofpelletgroupsfordeerandrabbits,respectively,per30-foot-squareplotsis111ˆ()2.30Eyyand222ˆ()4.52EyyTheboundsontheerrorsofestimationare21110.65500ˆ2()2(1)2(1)0.0750010000snBVynN22220.97500ˆ2()2(1)2(1)0.08650010000snBVynNb)Estimatethedifferenceinthemeansizeofpelletgroupsperplotforthetwoanimals(i.e.mu2-mu1),withanappropriatemarginoferror.Solution1212ˆˆ()2()()yyVyVyThatis,0.655000.97500(2.304.52)2(1)(1)2.220.115001000050010000Thuswehave95%confidentthatthetruedifferenceinmeansizeofpelletgroupsperplotforthetwoanimalslieswithin-2.220.11.2.(4.22)Acompanyauditorwasinterestedinestimatingthetotalnumberoftravelvouchersthatwereincorrectlyfiled.InaSRSofn=50voucherstakenfromagroupof250,twentywerefiledincorrectly.Estimatethetotalnumberofvouchersthathavebeenfiledincorrectlyandplaceaboundontheerrorofestimation.HINT:Ifpisthepopulationproportionofincorrectvouchers,thentau=Np.SolutionWeknowthat250N,50nand20ˆ0.450pTheestimatedtotalnumberofvouchersthathavebeenfiledincorrectlyisˆˆ250(0.4)100NpAlso,theestimatedvarianceis220.4(10.4)50ˆˆˆˆˆˆ()()()250(1)244.898501250VVNpNVpTherefore,theboundontheerrorofestimationisˆˆ2()2244.8982(15.65)31.3BVNp3.(4.36)Asurveyof811registeredvotersinFlorida,takenattheendofSeptember1994,reportedthat57%oftherespondentsareagainstlegalizedcasinogamblinginFlorida.Thearticlereportingthesurveystatesthatthesamplingerroris3.5%.a)Isthisthecorrectsamplingerrortoreport?Ifnot,computethecorrectsamplingerror.SolutionYes,thisisthecorrectsamplingerrortoreport.Weknowthatthesamplingerrorcanbemeasuredbyˆˆ(1)ˆˆ2()2(1)1ppnVpnN.SincethepopulationinFloridashouldbeverylarge(N20n=1622),(1)nNiscloseto0.So,thesamplingerrorisˆˆ(1)0.57(10.57)ˆˆ2()220.03483.5%18111ppVpnb)Canavalidstatisticalargumentbemadethat“mostFloridiansdisapprovedoflegalizingcasinosinthestateatthattime?”Explain.SolutionYes,itisavalidstatisticalargument.The95%ConfidentIntervalfortheproportionofFloridiansdisapprovedoflegalizingcasinosinthestateatthattimeisˆˆˆ2()0.570.035pVp,thatis,53.5%60.5%pTherefore,weconcludethatatleast53.5%ofFloridiansdisapprovedoflegalizingcasinosinthestateatthattime.b)Proponentsofthecasinosconductedapollthatshowedthatonlyabout33%ofFloridiansopposedcasinos.Whatcouldcausethisdifferencebetweenthetwopollresults?SolutionProponentsofthecasinosmightconductthepollindifferentplaces,e.g.aroundoratthecasinos.Respondentstherearemorelikelytosupportcasinos.4.(5.1)Achainofdepartmentstoresisinterestedinestimatingtheproportionofaccountsreceivablethataredelinquent.Thechainconsistsoffourstores.Toreducethecostofsampling,stratifiedrandomsamplingisused,witheachstoreasastratum.Becausenoinformationonpopulationproportionsisavailablebeforesampling,proportionalallocationisused.Fromtheaccompanyingtable,estimatep,theproportionofdelinquentaccountsforthechain,andplaceaboundontheestimationerror.Stratum1Stratum2Stratum3Stratum4Numberofaccountsreceivable65429325Samplesize149216Samplenumberofdelinquentaccounts4281SolutionWeknowthatthenumberofaccountsreceivableinstratumiare165N,242N,393Nand425N.Also,thesamplesizefromeachstratumis114n,29n,321nand46n.Theestimatedproportionofdelinquentaccountsforeachstratumis14ˆ0.28614p,22ˆ0.2229p,38ˆ0.38121pand41ˆ0.1676pThenumberofsamplingunitsinthepopulationis41225iiNNTheestimatedproportionofdelinquentaccountsforthechainis4111ˆˆ(65(0.286)42(0.222)93(0.381)25(0.167))0.3225stiiipNpNTheboundontheerrorofestimationis42211ˆˆˆˆ2()2()stiiiBVpNVpN,whereˆˆ(1)ˆˆ()(1)1iiiiiippnVpnNThatis,1ˆˆ()0.0123Vp,2ˆˆ()0.017Vp,3ˆˆ()0.009Vpand4ˆˆ()0.021VpSo,0.1173B5.(5.5)Acorporationwishestoobtaininformationontheeffectivenessofabusinessmachine.Anumberofdivisionheadswillbeinterviewedbytelephoneandaskedtoratetheequipmentonanumericalscale.ThedivisionsarelocatedinNorthAmerica,Europe,andAsia.Hence,stratifiedsamplingwillbeused.ItismoreexpensivetointerviewdivisionheadslocatedoutsideofNorthAmerica.Thetablebelowgivesthecostsperinterview,variancesoftheratings,andthestratumsizes.ThecorporationwantstoestimatetheaverageratingwithV[ybarst]=0.1.Determinethesamplesizenthatsatisfiesthisvarianceandappropriatelyallocatethesampletothe3strata.NorthAmericaEuropeAsiaCost($)92536Populationvariance2.253.243.24Stratumsize1126839SolutionWeknowthatthecostsperinterview19c,225cand336c.Also,thepopulationvarianceoftheratingsforeach