Solution Manual for Introductory Statistics, 10th Edition

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SOLUTIONSMANUALINTRODUCTORYSTATISTICSMYLABREVISION WITHTECHUPDATESTENTHEDITIONNeil A. WeissArizona State University

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ContentsChapter 1The Nature of Statistics1Chapter 2Organizing Data21Chapter 3Descriptive Measures127Chapter 4Probability Concepts215Chapter 5Discrete Random Variables279Chapter 6The Normal Distribution335Chapter 7The Sampling Distribution of theSample Mean401Chapter 8Confidence Intervals for OnePopulation Mean461Chapter 9Hypothesis Tests for OnePopulation Mean513Chapter 10Inferences for Two Population Means591Chapter 11Inferences for PopulationStandard Deviations681Chapter 12Inferences for Population Proportions715Chapter 13Chi-Square Procedures749Chapter 14Descriptive Methods in Regressionand Correlation801Chapter 15Inferential Methods in Regressionand Correlation883Chapter 16Anaylsis of Variance (ANOVA)953

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1CHAPTER1SOLUTIONSExercises1.11.1(a)Thepopulationisthecollectionofallindividualsoritemsunderconsiderationinastatisticalstudy.(b)Asampleisthatpartofthepopulationfromwhichinformationisobtained.1.2Thetwomajortypesofstatisticsaredescriptiveandinferentialstatistics.Descriptivestatisticsconsistsofmethodsfororganizingandsummarizinginformation.Inferentialstatisticsconsistsofmethodsfordrawingandmeasuringthereliabilityofconclusionsaboutapopulationbasedoninformationobtainedfromasampleofthepopulation.1.3Descriptivemethodsareusedfororganizingandsummarizinginformationandincludegraphs,charts,tables,averages,measuresofvariation,andpercentiles.1.4Descriptivestatisticsareusedtoorganizeandsummarizeinformationfromasamplebeforeconductinganinferentialanalysis.Preliminarydescriptiveanalysisofasamplemayrevealfeaturesofthedatathatleadtotheappropriateinferentialmethod.1.5(a)Anobservationalstudyisastudyinwhichresearcherssimplyobservecharacteristicsandtakemeasurements.(b)Adesignedexperimentisastudyinwhichresearchersimposetreatmentsandcontrolsandthenobservecharacteristicsandtakemeasurements.1.6Observationalstudiescanrevealonlyassociation,whereasdesignedexperimentscanhelpestablishcausation.1.7Thisstudyisinferential.DatafromasampleofAmericansareusedtomakeanestimateof(oraninferenceabout)averageTVviewingtimeforallAmericans.1.8Thisstudyisdescriptive.Itisasummaryoftheaveragesalariesinprofessionalbaseball,basketball,andfootballfor2005and2011.1.9Thisstudyisdescriptive.ItisasummaryofinformationonallhomessoldindifferentcitiesforthemonthofSeptember2012.1.10Thisstudyisinferential.Nationalsamplesareusedtomakeestimatesof(orinferencesabout)drugusethroughouttheentirenation.1.11Thisstudyisdescriptive.ItisasummaryoftheannualfinalclosingvaluesoftheDowJonesIndustrialAverageattheendofDecemberfortheyears2004-2013.1.12Thisstudyisinferential.SurveyresultswereusedtomakepercentageestimatesonwhichcollegemajorswereindemandamongU.Sfirmsforallgraduatingcollegestudents.1.13(a)Thisstudyisinferential.ItwouldhavebeenimpossibletosurveyallU.S.adultsabouttheiropinionsonDarwinism.Therefore,thedatamusthavecomefromasample.TheninferencesweremadeabouttheopinionsofallU.S.adults.(b)ThepopulationconsistsofallU.S.adults.ThesampleconsistsonlyofthoseU.S.adultswhotookpartinthesurvey.1.14(a)ThepopulationconsistsofallU.S.adults.Thesampleconsistsofthe1000U.S.adultswhoweresurveyed.(b)Thepercentageof50%isadescriptivestatisticsinceitdescribestheopinionoftheU.S.adultswhoweresurveyed.1.15(a)Thestatementisdescriptivesinceitonlytellswhatwassaidbytherespondentsofthesurvey.

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2Chapter 1(b)ThenthestatementwouldbeinferentialsincethedatahasbeenusedtoprovideanestimateofwhatallAmericansbelieve.1.16(a)Tochangethestudytoadesignedexperiment,onewouldstartwitharandomlychosengroupofmen,thenrandomlydividethemintotwogroups,anexperimentalgroupinwhichallofthemenwouldhavevasectomiesandacontrolgroupinwhichthemenwouldnothavethem.Thiswouldenabletheresearchertomakeinferencesaboutvasectomiesbeingacauseofprostatecancer.(b)Thisexperimentisnotfeasible,since,inthevasectomygrouptherewouldbemenwhodidnotwantone,andinthecontrolgrouptherewouldbemenwhodidwantone.Sincenoonecanbeforcedtoparticipateinthestudy,thestudycouldnotbedoneasplanned.1.17Designedexperiment.Theresearchersdidnotsimplyobservethetwogroupsofchildren,butinsteadrandomlyassignedonegrouptoreceivetheSalkvaccineandtheothertogetaplacebo.1.18Observationalstudy.TheresearchersatHarvardUniversityandtheNationalInstituteofAgingsimplyobservedthetwogroups.1.19Observationalstudy.Theresearcherssimplycollecteddatafromthemenandwomeninthestudywithaquestionnaire.1.20Designedexperiment.Theresearchersdidnotsimplyobservethetwogroupsofwomen,butinsteadrandomlyassignedonegrouptoreceiveaspirinandtheothertogetaplacebo.1.21Designedexperiment.Theresearchersdidnotsimplyobservethethreegroupsofpatients,butinsteadrandomlyassignedsomepatientstoreceiveoptimalpharmacologictherapy,sometoreceiveoptimalpharmacologictherapyandapacemaker,andsometoreceiveoptimalpharmacologictherapyandapacemaker-defibrillatorcombination.1.22Observationalstudies.Theresearcherssimplycollectedavailableinformationaboutthestartingsalariesofnewcollegegraduates.1.23(a)ThisstatementisinferentialsinceitisastatementaboutallAmericansbasedonapoll.WecanbereasonablysurethatthisisthecasesincethetimeandcostofquestioningeverysingleAmericanonthisissuewouldbeprohibitive.Furthermore,bythetimeeveryonecouldbequestioned,manywouldhavechangedtheirminds.(b)Tomakeitclearthatthisisadescriptivestatement,thenewstatementcouldbe,“Of1032Americanadultssurveyed,73%favoredalawthatwouldrequireeverygunsoldintheUnitedStatestobetest-firedfirst,solawenforcementwouldhaveitsfingerprintincaseitwereeverusedinacrime.”Torephraseitasaninferentialstatement,use“Basedonasampleof1032Americanadults,itisestimatedthat73%ofAmericanadultsfavoralawthatwouldrequireeverygunsoldintheUnitedStatestobetest-firedfirst,solawenforcementwouldhaveitsfingerprintincaseitwereeverusedinacrime.”1.24Descriptivestatistics.TheU.S.NationalCenterforHealthStatisticscollectsdeathcertificateinformationfromeachstate,sotheratesshownreflectthecausesofalldeathsreportedondeathcertificates,notjustasample.1.25(a)ThepopulationconsistsofallAmericansbetweentheagesof18and29.(b)ThesampleconsistsonlyofthoseAmericanswhotookpartinthesurvey.(c)ThestatementinquotesisinferentialsinceitisastatementaboutallAmericansbasedonasurvey.(d)“BasedonasampleofAmericansbetweentheagesof18and29,itisestimatedthat59%ofAmericansopposemedicaltestingonanimals.”

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Section 1.231.26(a)The$5.36billionlobbyingexpenditurefigurewouldbeadescriptivefigureifitwasbasedontheresultsofalllobbyingexpendituresduringtheperiodfrom1998through2012.(b)The$5.36billionlobbyingexpenditurefigurewouldbeaninferentialfigureifitwasanestimatebasedontheresultsofasampleoflobbyingexpendituresduringtheperiodfrom1998through2012.Exercises1.21.27Acensusisgenerallytimeconsuming,costly,frequentlyimpractical,andsometimesimpossible.1.28Samplingandexperimentationaretwoalternativewaystoobtaininformationwithoutconductingacompletecensus.1.29Thesampleshouldberepresentativesothatitreflectsascloselyaspossibletherelevantcharacteristicsofthepopulationunderconsideration.1.30Therearemanypossibleanswers.Surveyingpeopleregardingpoliticalcandidatesastheyenterorleaveanupscalebusinesslocation,surveyingthereadersofaparticularpublicationtogetinformationaboutthepopulationingeneral,pollingcollegestudentswholiveindormitoriestoobtaininformationofinteresttoallstudentsarealllikelytoproducesamplesunrepresentativeofthepopulationunderconsideration.1.31(a)Probabilitysamplingconsistsofusingarandomizingdevicesuchastossingacoinorconsultingarandomnumbertabletodecidewhichmembersofthepopulationwillconstitutethesample.(b)No.Itispossiblefortherandomizingdevicetorandomlyproduceasamplethatisnotrepresentative.(c)Probabilitysamplingeliminatesunintentionalselectionbias,permitstheresearchertocontrolthechanceofobtaininganon-representativesample,andguaranteesthatthetechniquesofinferentialstatisticscanbeapplied.1.32(a)Simplerandomsamplingisaprocedureforwhicheachpossiblesampleofagivensizeisequallylikelytobetheoneobtained.(b)Asimplerandomsampleisonethatwasobtainedbysimplerandomsampling.(c)Randomsamplingmaybedonewithorwithoutreplacement.Insamplingwithreplacement,itispossibleforamemberofthepopulationtobechosenmorethanonce,i.e.,membersareeligibleforre-selectionaftertheyhavebeenchosenonce.Insamplingwithoutreplacement,populationmemberscanbeselectedatmostonce.1.33Simplerandomsampling.1.34Onemethodwouldbetoplacethenamesofallmembersofthepopulationunderconsiderationonindividualslipsofpaper,placetheslipsinacontainerlargeenoughtoallowthemtobethoroughlyshuffledbyshakingorspinning,andthendrawoutthedesirednumberofslipsforthesamplewhileblindfolded.Asecondmethod,whichismuchmorepracticalwhenthepopulationsizeislarge,istoassignanumbertoeachmemberofthepopulation,andthenusearandomnumbertable,randomnumbergeneratingdevice,orcomputerprogramtodeterminethenumbersofthosemembersofthepopulationwhoarechosen.1.35TheacronymusedforsimplerandomsamplingwithoutreplacementisSRS.1.36(a)123,124,125,134,135,145,234,235,245,345

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4Chapter 1(b)Thereare10samples,eachofsizethree.Eachsamplehasaonein10chanceofbeingselected.Thus,theprobabilitythatasampleofthreeis1,3,and5is1/10.(c)StartinginLine05andcolumn20,readingsingledigitnumbersdownthecolumnandthenupthenextcolumn,thefirstdigitthatisaonethroughfiveisa5.Ignoringduplicatesandskippingdigits6andaboveandalsoskippingzero,theseconddigitfoundthatisaonethroughfiveisa4.Continuingdowncolumn20andthenupcolumn21,thethirddigitfoundthatisaonethroughfiveisa1.ThustheSRSof1,4,and5isobtained.1.37(a)12,13,14,23,24,34(b)Thereare6samples,eachofsizetwo.Eachsamplehasaoneinsixchanceofbeingselected.Thus,theprobabilitythatasampleoftwois2and3is1/6.(c)StartinginLine17andcolumn07(noticethereisacolumn00),readingsingledigitnumbersdownthecolumnandthenupthenextcolumn,thefirstdigitthatisaonethroughfourisa1.Continuedowncolumn07andthenupcolumn08.Ignoringduplicatesandskippingdigits5andaboveandalsoskippingzero,theseconddigitfoundthatisaonethroughfourisa4.ThustheSRSof1and4isobtained.1.38(a)StartinginLine15andreadingtwodigitsnumbersincolumns25and26goingdownthetable,thefirsttwodigitnumberbetween01and90is06.Continuingdownthecolumnsandignoringduplicatesandnumbers91-99,thenexttwonumbersare33and61.Then,continuingupcolumns27and28,thelasttwonumbersselectedare56and20.ThereforetheSRSofsizefiveconsistsofobservations06,33,61,56,and20.(b)Therearemanypossibleanswers.1.39(a)StartinginLine10andreadingtwodigitsnumbersincolumns10and11goingdownthetable,thefirsttwodigitnumberbetween01and50is43.Continuingdownthecolumnsandignoringduplicatesandnumbers51-99,thenexttwonumbersare45and01.Then,continuingupcolumns12and13,thelastthreenumbersselectedare42,37,and47.ThereforetheSRSofsizesixconsistsofobservations43,45,01,42,37,and47.(b)Therearemanypossibleanswers.1.40Theonlinepollclearlyhasabuilt-innon-responsebias.SinceitwastakenovertheMemorialDayweekend,mostofthosewhorespondedwerepeoplewhostayedathomeandhadaccesstotheircomputers.Mostpeoplevacationingoutdoorsovertheweekendwouldnothavecarriedtheircomputerswiththemandwouldnothavebeenabletorespond.1.41Dentistsformahigh-incomegroupwhoseincomesarenotrepresentativeoftheincomesofSeattleresidentsingeneral.1.42(a)ThefivepossiblesamplesofsizeoneareG,L,S,A,andT.(b)ThereisnodifferencebetweenobtainingaSRSofsize1andselectingoneofficialatrandom.(c)TheonepossiblesampleofsizefiveisGLSAT.(d)ThereisnodifferencebetweenobtainingaSRSofsize5andtakingacensusofthefiveofficials.1.43(a)GLS,GLA,GLT,GSA,GST,GAT,LSA,LST,LAT,SAT.(b)Thereare10samples,eachofsizethree.Eachsamplehasaonein10chanceofbeingselected.Thus,theprobabilitythatasampleofthreeofficialsisthefirstsampleonthelistpresentedinpart(a)is1/10.Thesameistrueforthesecondsampleandforthetenthsample.

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Section 1.251.44(a)E,ME,AM,LP,LL,AE,PE,BM,AP,AL,BE,LM,PM,BP,BA,B(b)Oneprocedurefortakingarandomsampleoftworepresentativesfromthesixistowritetheinitialsoftherepresentativesonsixseparatepiecesofpaper,placethesixslipsofpaperintoabox,andthen,whileblindfolded,picktwooftheslipsofpaper.Or,numbertherepresentatives1-6,anduseatableofrandomnumbersorarandom-numbergeneratortoselecttwodifferentnumbersbetween1and6.(c)1/15;1/151.45(a)E,M,P,LE,M,L,BE,P,A,BM,P,A,BE,M,P,AE,M,A,BE,L,A,BM,L,A,BE,M,P,BE,P,L,AM,P,L,AP,L,A,BE,M,L,AE,P,L,BM,P,L,B(b)Oneprocedurefortakingarandomsampleoffourrepresentativesfromthesixistowritetheinitialsoftherepresentativesonsixseparatepiecesofpaper,placethesixslipsofpaperintoabox,andthen,whileblindfolded,pickfouroftheslipsofpaper.Or,numbertherepresentatives1-6,anduseatableofrandomnumbersorarandom-numbergeneratortoselectfourdifferentnumbersbetween1and6.(c)1/15;1/151.46(a)E,M,PE,P,AM,P,LM,A,BE,M,LE,P,BM,P,AP,L,AE,M,AE,L,AM,P,BP,L,BE,M,BE,L,BM,L,AP,A,BE,P,LE,A,BM,L,BL,A,B(b)Oneprocedurefortakingarandomsampleofthreerepresentativesfromthesixistowritetheinitialsoftherepresentativesonsixseparatepiecesofpaper,placethesixslipsofpaperintoabox,andthen,whileblindfolded,pickthreeoftheslipsofpaper.Or,numbertherepresentatives1-6,anduseatableofrandomnumbersorarandom-numbergeneratortoselectthreedifferentnumbersbetween1and6.(c)1/20;1/201.47(a)F,TF,GF,HF,LF,BF,AT,GT,HT,LT,BT,AG,HG,LG,BG,AH,LH,BH,AL,BL,AB,A(b)1/21;1/211.48(a)IamusingTableItoobtainalistof20differentrandomnumbersbetween1and80asfollows.Istartatthetwodigitnumberinlinenumber5andcolumnnumbers31-32,whichisthenumber86.SinceIwantnumbersbetween1and80only,Ithrowoutnumbersbetween81and99,inclusive.Ialsodiscardthenumber00.Inowgodownthetableandrecordthetwo-digitnumbersappearingdirectlybeneath86.Afterskipping86,Irecord39,03,skip97,record28,58,59,skip81,record09,36,skip81,record52,skip94,record24and78.

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6Chapter 1NowthatI'vereachedthebottomofthetable,Imovedirectlyrightwardtotheadjacentcolumnoftwo-digitnumbersandgoup.Iskip84,record57,40,skip89,record69,25,skip95,record51,20,42,77,skip89,skip40(duplicate),record14,and34.I'vefinishedrecordingthe20randomnumbers.Insummary,theseare3903285859093652247857406925512042771434(b)WecanuseMinitabtogeneraterandomnumbers.FollowingtheinstructionsinTheTechnologyCenter,ourresultsare55,47,66,2,72,56,10,31,5,19,39,57,44,60,23,34,43,9,49,and62.Yourresultmaybedifferentfromours.1.49(a)IamusingTableItoobtainalistof10randomnumbersbetween1and500asfollows.Istartatthethreedigitnumberinlinenumber14andcolumnnumbers10-12,whichisthenumber452.Inowgodownthetableandrecordthethree-digitnumbersappearingdirectlybeneath452.SinceIwantnumbersbetween1and500only,Ithrowoutnumbersbetween501and999,inclusive.Ialsodiscardthenumber000.After452,Iskip667,964,593,534,andrecord016.NowthatI'vereachedthebottomofthetable,Imovedirectlyrightwardtotheadjacentcolumnofthree-digitnumbersandgoup.Irecord343,242,skip748,755,record428,skip852,794,596,record378,skip890,record163,skip892,847,815,729,911,745,record182,293,and422.I'vefinishedrecordingthe10randomnumbers.Insummary,theseare:452016343242428378163182293422(b)WecanuseMinitabtogeneraterandomnumbers.FollowingtheinstructionsinTheTechnologyCenter,ourresultsare489,451,61,114,389,381,364,166,221,and437.Yourresultmaybedifferentfromours.1.50(a)Firstassignthedigits0though9tothetencitiesaslistedintheexercise.SelectarandomstartingpointinTableIofAppendixAandreadinapre-selecteddirectionuntilyouhaveencountered5differentdigits.Forexample,ifwestartatthetopofthefifthcolumnofdigitsandreaddown,weencounterthedigits4,1,5,2,5,6.Weignorethesecond‘5’.ThusoursampleoffivecitiesconsistsofOsaka,Tokyo,Miami,SanFrancisco,andNewYork.Youranswermaybedifferentfromthisone.(b)WecanuseMinitabtogeneraterandomnumbers.FollowingtheinstructionsinTheTechnologyCenter,ourresultsare3,8,6,5,9.Thusoursampleof5citiesisLosAngeles,Manila,NewYork,Miami,andLondon.Yourresultmaybedifferentfromours.1.51(a)Firstre-assigntheelements93though118aselements01to26.SelectarandomstartingpointinTableIofAppendixAandreadinapre-selecteddirectionuntilyouhaveencountered8differentelements.Forexample,ifwestartatthetopofthecolumn10andreadtwodigitnumbersdownandthenupinthefollowingcolumns,weencounter

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Section 1.37theelements04,01,03,08,11,18,22,and15.ThiscorrespondstoasampleoftheelementsCm,Np,Am,Fm,Lr,Ds,Fl,andBh.Youranswermaybedifferentfromthisone.(b)WecanuseMinitabtogeneraterandomnumbers.FollowingtheinstructionsinTheTechnologyCenter,ourresultsare8,2,9,20,24,19,21,and13.Thusoursampleof8elementsisFm,Pu,Md,Cn,Lv,Rg,Uut,andDb.Yourresultmaybedifferentfromours.1.52(a)Oneofthebiggestreasonsforundercoverageinhouseholdsurveysisthatrespondentsdonotcorrectlyindicateallwhoarelivinginahouseholdmaybeduetodeliberateconcealmentorirregularhouseholdstructureorlivingarrangements.Thehouseholdresidentsareonlypartiallylisted.(b)AtelephonesurveyofAmericansfromaphonebookwilllikelyhavebiasduetoundercoveragebecausemanypeoplehaveunlistedphonenumbersandalsoitisbecomingmorepopularthatmanypeopledonotevenhavehomephones.Thiswouldcausethephonebooktobeanincompletelistofthepopulation.1.53(a)Oneofthedangersofnonresponseisthattheindividualswhodonotrespondmayhaveadifferentobservedvaluethantheindividualsthatdorespondcausinganonresponsebiasintheestimate.Nonresponsebiasmaymakethemeasuredvaluetoosmallortoolarge.(b)Thelowertheresponserate,themorelikelythereisanonresponsebiasintheestimate.Thereforetheestimatewilleitherunderoroverestimatethegeneralizedresultstotheentirepopulation.1.54(a)Therespondentmaywishtopleasethequestionerbyansweringwhatismorallyorlegallyright.Therespondentmightnotbewillingtoadmittothequestionerthattheysmokemarijuanaandthemeasuredvalueofthepercentageofpeoplethatsmokemarijuanawouldthenbeunderestimatedduetoresponsebias.(b)Anothersituationthatmightbeconducivetoresponsebiasisperhapsawomanquestioningmenontheiropinionofdomesticviolence,oranenvironmentalistquestioningpeopleontheirrecyclinghabits.(c)Thewordingofaquestioncouldleadtoresponsebias.Whetherthesurveyisanonymousornotcouldleadtoresponsebias.Thecharacteristicsofthequestionercouldleadtoresponsebias.Itcouldalsohappenifthequestionerobviouslyfavorsandispushingforoneparticularanswer.Exercises1.31.55Systematicrandomsamplingiseasiertoexecutethansimplerandomsamplingandusuallyprovidescomparableresults.Theexceptionisthepresenceofsomekindofcyclicalpatterninthelistingofthemembersofthepopulation.1.56Ideally,inclustersampling,eachclustershouldpatterntheentirepopulation.1.57Ideally,instratifiedsampling,themembersofeachstratumshouldbehomogeneousrelativetothecharacteristicunderconsideration.1.58Surveysthatcombineoneormoreofsimplerandomsampling,systematicrandomsampling,clustersampling,andstratifiedsamplingemploywhatiscalledmultistagesampling.1.59(a)Answerswillvary,buthereistheprocedure:(1)Dividethepopulationsize,372,bythesamplesize,5,androunddowntothenearestwholenumberifnecessary;thisgives74.Useatableofrandomnumbers(orasimilardevice)toselectanumberbetween1and74,callitk.(3)Listevery74thnumber,startingwithk,until5numbersareobtained;

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8Chapter 1thus,thefirstnumberoftherequiredlistof5numbersisk,thesecondisk+74,thethirdisk+148,andsoforth.(b)Followingpart(a)withk=10,thefirstnumberofthesampleis10,thesecondis10+74=84.Theremainingthreenumbersinthesamplewouldbe158,232,and306.Thus,thesampleof5wouldbe10,84,158,232,and306.1.60(a)Answerswillvary,buthereistheprocedure:(1)Dividethepopulationsize,500,bythesamplesize,9,androunddowntothenearestwholenumberifnecessary;thisgives55.Useatableofrandomnumbers(orasimilardevice)toselectanumberbetween1and55,callitk.(3)Listevery55thnumber,startingwithk,until9numbersareobtained;thus,thefirstnumberoftherequiredlistof9numbersisk,thesecondisk+55,thethirdisk+110,andsoforth.(b)Followingpart(a)withk=48,thefirstnumberofthesampleis48,thesecondis48+55=103.Theremainingsevennumbersinthesamplewouldbe158,213,268,323,378,433,and488.Thus,thesampleof9wouldbe48,103,158,213,268,323,378,433,and488.1.61(a)Answerswillvary,buthereistheprocedure:(1)Thepopulationofsize50isalreadydividedintofiveclustersofsize10.(2)Sincetherequiredsamplesizeis20,wewillneedtotakeaSRSof2clusters.Useatableofrandomnumbers(orasimilardevice)toselecttwonumbersbetween1and5.Thesearethetwoclustersthatareselected.(3)Useallthemembersofeachclusterselectedinpart(2)asthesample.(b)Followingpart(a)withclusters#1and#3selected,wewouldselectallthemembersincluster1,whichare1–10,andallthemembersincluster3,whichare21–30.1.62(a)Answerswillvary,buthereistheprocedure:(1)Thepopulationofsize100isalreadydividedintotenclustersofsize10.(2)Sincetherequiredsamplesizeis30,wewillneedtotakeaSRSof3clusters.Useatableofrandomnumbers(orasimilardevice)toselectthreenumbersbetween1and10.Thesearethethreeclustersthatareselected.(3)Useallthemembersofeachclusterselectedinpart(2)asthesample.(b)Followingpart(a)withclusters#2,#6,and#9selected,wewouldselectallthemembersincluster2(11-20),allthemembersincluster6(51-60),andallthemembersincluster9(81-90).Therefore,oursamplewouldconsistof11-20,51-60,and81-90.1.63(a)Fromeachstrata,weneedtoobtainaSRSofasizeproportionaltothesizeofthestratum.Therefore,sincestrata#1is30%ofthepopulation,aSRSequalto30%of20,or6,shouldbesampledfromstrata#1.Sincestrata#2is20%ofthepopulation,aSRSequalto20%of20,or4,shouldbesampledfromstrata#2.Similarly,aSRSofsize8shouldbesampledfromstrata#3andaSRSofsize2shouldbesampledfromstrata#4.Thesamplesizesfromstratum#1through#4are6,4,8,and2respectively.(b)Answerswillvaryfollowingtheprocedureinpart(a).1.64(a)Fromeachstrata,weneedtoobtainaSRSofasizeproportionaltothesizeofthestratum.Therefore,sincestrata#1is40%ofthepopulation,aSRSequalto40%of10,or4,shouldbesampledfromstrata#1.Sincestrata#2is30%ofthepopulation,aSRSequalto30%of10,or3,shouldbesampledfromstrata#2.Similarly,aSRSofsize3shouldbesampledfromstrata#3.Thesamplesizesfromstratum#1through#3are4,3,and3respectively.(b)Answerswillvaryfollowingtheprocedureinpart(a).

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Section 1.391.65StratifiedSampling.Theentirepopulationisnaturallydividedintosubpopulations,onefromeachlake,andrandomsamplingisdonefromeachlake.Thestratifiedsamplingisnotwithproportionalallocationsincethatwouldrequireknowinghowmanyfishwereineachlake.1.66StratifiedSampling.Theentirepopulationisnaturallydividedintofoursubpopulations,andrandomsamplingisdonefromeachandthencombinedintoasinglesample.1.67SystematicRandomSampling.Kennedyselectedhissampleusingthefixedperiodicintervalofevery50thletter,whichisthesimilartothemethodpresentedinprocedure1.1.1.68ClusterSampling.Theclustersofthissamplingdesignarethe1285journals.Arandomsampleof26clusterswasselectedandthenallarticlesfromtheselectedjournalsforaparticularyearwereexamined.1.69ClusterSampling.Theclustersofthissamplingdesignarethe46schools.Arandomsampleof10clusterswasselectedandthenalloftheparentsofthenonimmunizedchildrenatthe10selectedschoolsweresentaquestionnaire.1.70SystematicRandomSampling.Thissamplingdesignfollowsprocedure1.1.First,dividingthepopulationsizeof8493by30,theyarrivedatk=283.Then,therandomlyselectedstartingpointwasm=10.Then,thesampledstickerswerem=10,m+k=293,m+2k=576,etc.1.71(a)Answerswillvary,buthereistheprocedure:(1)Dividethepopulationsize,500,bythesamplesize,10,androunddowntothenearestwholenumberifnecessary;thisgives50.(2)Useatableofrandomnumbers(orasimilardevice)toselectanumberbetween1and50,callitk.(3)Listevery50th,startingwithk,until10numbersareobtained;thus,thefirstnumberontherequiredlistof10numbersisk,thesecondisk+50,thethirdisk+100,andsoforth(e.g.,ifk=6,thenthenumbersonthelistare6,56,106,...).(b)Systematicrandomsamplingiseasier.(c)Theanswerdependsonthepurposeofthesampling.IfthepurposeofsamplingisnotrelatedtothesizeofthesalesoutsidetheU.S.,systematicsamplingwillwork.However,sincethelistingisarankingbyamountofsales,ifkislow(say2),thenthesamplewillcontainfirmsthat,ontheaverage,havehighersalesoutsidetheU.S.thanthepopulationasawhole.Ifthekishigh,(say49)thenthesamplewillcontainfirmsthat,ontheaverage,havelowersalesthanthepopulationasawhole.Ineitherofthosecases,thesamplewouldnotberepresentativeofthepopulationinregardtotheamountofsalesoutsidetheU.S.1.72(a)Answerswillvary,buthereistheprocedure:(1)Dividethepopulationsize,80,bythesamplesize,20,androunddowntothenearestwholenumberifnecessary;thisgives4.(2)Useatableofrandomnumbers(orasimilardevice)toselectanumberbetween1and4,callitk.(3)Listevery4thnumber,startingwithk,until20numbersareobtained;thusthefirstnumberontherequiredlistof20numbersisk,thesecondisk+4,thethirdisk+8,andsoforth(e.g.,ifk=3,thenthenumbersonthelistare3,7,11,15,...).(b)Systematicrandomsamplingiseasier.(c)No.InKeno,youwanteverysetof20ballstohavethesamechanceofbeingchosen.Systematicsamplingwouldgiveeachof4setsofballs[(1,5,9,...,77),(2,6,10,...,78),(3,7,11,...,79)and(4,8,12,...,80)],a1/4chanceofoccurring,whilealloftheotherpossiblesetsofballswouldhavenochanceofoccurring.

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10Chapter 11.73(a)Numberthesuitesfrom1to48,useatableofrandomnumberstorandomlyselectthreeofthe48suites,andtakeasthesamplethe24dormitoryresidentslivinginthethreesuitesobtained.(b)Probablynot,sincefriendsaremorelikelytohavesimilaropinionsthanarestrangers.(c)Thereare384studentsintotal.Freshmenmakeup1/3ofthem.Sophomoresmakeup7/24ofthem,Juniors1/4,andSeniors1/8.Multiplyingeachofthesefractionsby24yieldstheproportionalallocation,whichdictatesthatthenumberoffreshmen,sophomores,juniors,andseniorsselectedshouldbe,respectively,8,7,6,and3.Thusastratifiedsampleof24dormitoryresidentscanbeobtainedasfollows:Numberthefreshmendormitoryresidentsfrom1to128anduseatableofrandomnumberstorandomlyselect8ofthe128freshmandormitoryresidents;numberthesophomoredormitoryresidentsfrom1to112anduseatableofrandomnumberstorandomlyselect7ofthe112sophomoredormitoryresidents;andsoforth.1.74(a)Eachcategoryof“Percentfreelunch”shouldberepresentedinthesampleinthesameproportionthatitispresentinthepopulationoftop100rankedhighschools.Thus50/100ofthesampleof25schoolsshouldbefromthe0tounder10%freelunchcategory,18/100fromthesecondcategory,11/100fromthethird,8/100fromthefourth,and13/100fromthelast.Multiplyingeachofthesefractionsby25givesusthesamplesizesfromeachcategory.Thesesamplesizeswillnotnecessarilybeintegers,sowewillneedtomakesomeminoradjustmentsoftheresults.Thefirstcategoryshouldhave(50/100)(25)=12.5.Thesecondshouldhave(18/100)(25)=4.5.Similarly,thethird,fourth,andfifthcategoriesshouldhave2.75,2,and3.25fortheirsamplesizes.Weroundthethirdandfifthsamplesizeseachto3.Afterflippingacoin,weroundthefirsttwocategoriesto12and5.ThusthesamplesizesforthefivePercentfreelunchcategoriesshouldbe12,5,3,2,and3respectively.Wewouldnowusearandomnumbergeneratortoselect12outofthe50inthefirstcategory,5outofthe18inthesecond,3outofthe11inthethird,2ofthe8inthefourth,and3ofthe13inthelastcategory.(b)Frompart(a),twoschoolswouldbeselectedfromthestratawithapercentfreelunchvalueof30-under40.1.75(a)Answerswillvary,buthereistheprocedure:(1)Dividethepopulationsize,435,bythesamplesize,15,androunddowntothenearestwholenumberifnecessary;thisgives29.Useatableofrandomnumbers(orasimilardevice)toselectanumberbetween1and29,callitk.(3)Listevery29thnumber,startingwithk,until15numbersareobtained;thus,thefirstnumberoftherequiredlistof15numbersisk,thesecondisk+29,thethirdisk+58,andsoforth.(b)Followingpart(a)withk=12,thefirstnumberofthesampleis12,thesecondis12+29=41.Thethirdnumberselectedis12+58=70.Theremainingtwelvenumbersaresimilarlyselected.Thus,thesampleof15wouldbe12,41,70,99,128,157,186,215,244,273,302,331,360,389,and418.1.76(a)Eachcategoryof“Region”shouldberepresentedinthesampleinthesameproportionthatitispresentinthepopulation.Thus43%ofthesampleof50shouldbevolunteersservinginAfrica,21%fromLatinAmerica,15%fromEasternEurope/CentralAsia,10%fromAsia,4%fromtheCaribbean,4%fromNorthAfrica/MiddleEast,and3%fromthePacificIsland.Findingeachoftheseproportionsof50givesusthesamplesizesfromeachcategory.Thesesamplesizeswillnotnecessarilybeintegers,sowewillneedtomakesomeminoradjustmentsoftheresults.VolunteersfromAfricashouldhave(0.43)(50)=21.5.VolunteersfromLatinAmericashouldhave(0.21)(50)=10.5.

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Section 1.311Similarly,theremainingcategoriesshouldhave7.5,5,2,2,and1.5fortheirsamplesizes.Afterflippingacoin,weroundthefirsttwocategorieseitherupordown.Thusthesamplesizesforthecategoriesshouldbe21,11,7,5,2,2,and2respectively.Wewouldnowusearandomnumbergeneratortoselectthevolunteersfromeachcategory.(b)Frompart(a),twovolunteerswouldbeselectedfromthestratawithvolunteersservingintheCaribbean.1.77(a)ThisisapolltakenbycallingrandomlyselectedU.S.adults.Thus,thesamplingdesignappearstobesimplerandomsampling,althoughitispossiblethatamorecomplexdesignwasusedtoensurethatvariouspolitical,religious,educational,orothertypesofgroupswereproportionatelyrepresentedinthesample.(b)Thesamplesizeforthesecondquestionwas78%of1010or788.(c)Thesamplesizeforthethirdquestionwas28%of788or221.1.78No.Inyourtext,Example1.10,only48differentsamplesarepossible.Asamplecontainingstudents5,6,and7isnotpossibleatall.Whilethe48possiblesamplesareequallylikely,thereareothersamplesthatcouldbeobtainedthroughsimplerandomsamplingthatarenotpossibleatallinsystematicsampling.Thusnotallpossiblesamplesareequallylikely.Nevertheless,ifthereisnopatternorcycletothedata,thismethodwilltendtogiveaboutthesameresultsassimplerandomsampling.1.79(a)Itisalsotrueforsystematicrandomsamplingifthepopulationsizedividedbythesamplesizeresultsinanintegerform.Thechanceforeachmembertobeselectedisthenstillequaltothesamplesizedividedbythepopulationsize.Forexample,supposethepopulationsizeisN=10andthesamplesizeisn=2.Thechancethateachmemberinsimplerandomsamplingtobeselectedis2/10=1/5.Insystematicrandomsamplingforthesameexample,m=5.Thepossiblesamplesofsizetwoare1and6,2and7,3and8,4and9,and5and10.Therefore,thechancethatamemberisselectedisequaltothechanceofoneofthosefivesamplesbeingselected,whichisthesameassimplerandomsamplingof1/5.(b)Itisnottrueforsystematicrandomsamplingifthepopulationsizedividedbythesamplesizedoesnotresultinanintegerform.Forexample,supposethepopulationsizeisN=15andthesamplesizeisn=2.Afterdividingthepopulationsizebythesamplesizeandroundingdowntothenearestwholenumber,wegetm=7.Youwouldselectevery7thmemberafterarandomstartingplacek,between1and7,isdetermined.Ifk=1,youwouldselectthefirstandeighthmember.Ifk=7,youwouldselecttheseventhandfourteenthmember.Inthissituation,thelastmember(fifteenth)canneverbeselected.Therefore,thelastmemberofthesampledoesnothavethesamechanceofbeingselectedasanyothermemberinthepopulation.1.80Refertoexample1.14.Ifweapproachedthisproblemasasimplerandomsampleeachmemberwouldhaveachanceofbeingselectedequaltothesamplesizedividedbythepopulationsize:20/250,or2/25.Ifweapproachedthissameexampleasastratifiedsamplewithproportionalallocation,wewouldselect2outof25householdsintheupperincomegroup,14outofthe175householdsinthemiddleincomegroup,and4outof50householdsinthelowerincomegroup.Thusthechancethatanupperincomehouseholdisselectedis2/25.Thechancethatamiddleincomehouseholdisselectedis14/175=2/25.Finally,thechancethatalowerincomehouseholdisselectedis4/50=2/25.Thus,thechancethateachmemberisselectedisthesameasasimplerandomsample.

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12Chapter 1Exercises1.41.81(a)Experimentalunitsaretheindividualsoritemsonwhichtheexperimentisperformed.(b)Whentheexperimentalunitsarehumans,wecallthemsubjects.1.82Thethreebasicprinciplesofexperimentaldesignarecontrol,randomization,andreplication.Control:Twoormoretreatmentsshouldbecompared.Randomization:Theexperimentalunitsshouldberandomlydividedintogroupstoavoidunintentionalselectionbiasinconstitutingthegroups.Replication:Asufficientnumberofexperimentalunitsshouldbeusedtoensurethatrandomizationcreatesgroupsthatresembleeachothercloselyandtoincreasethechancesofdetectingdifferencesamongthetreatments.1.83(a)Theresponsevariableisthecharacteristicoftheexperimentaloutcomethatistobemeasuredorobserved.(b)Afactorisavariablewhoseeffectontheresponsevariableisofinterestintheexperiment.(c)Thelevelsarethepossiblevaluesofthefactor.(d)Foraone-factorexperiment,thetreatmentsarethelevelsofthefactor.Formultifactorexperiments,thetreatmentsarethecombinationsoflevelsofthefactors.1.84Onetypeofstatisticaldesignisacompletelyrandomizeddesign.Inacompletelyrandomizeddesign,alltheexperimentalunitsareassignedrandomlyamongallthetreatments.Thesecondtypeofstatisticaldesignisarandomizedblockdesign.Inarandomizedblockdesign,theexperimentalunitsareassignedrandomlyamongallthetreatmentsseparatelywithineachblock.1.85Inaone-factorexperiment,thenumberoftreatmentsisequaltothenumberoflevelsofthefactor.Therefore,therearefourtreatments.1.86Inaone-factorexperiment,thenumberoftreatmentsisequaltothenumberoflevelsofthefactor.Therefore,therearefivetreatments.1.87(a)Bb1b2b3b4Aa1a1b1a1b2a1b3a1b4a2a2b1a2b2a2b3a2b4a3a3b1a3b2a3b3a3b4(b)Therearetwelvecombinationsofthelevelsofthefactors.Therefore,therearetwelvetreatments.(c)Yes,youcouldhavemultipliedthenumberoflevelsineachfactor.TherearethreelevelsoffactorAandfourlevelsoffactorB.Therefore,thereare(3)(4)=12treatments.

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Section 1.4131.88(a)Bb1b2Aa1a1b1a1b2a2a2b1a2b2a3a3b1a3b2a4a4b1a4b2(b)Thereareeightcombinationsofthelevelsofthefactors.Therefore,thereareeighttreatments.(c)Yes,youcouldhavemultipliedthenumberoflevelsineachfactor.TherearefourlevelsoffactorAandtwolevelsoffactorB.Therefore,thereare(4)(2)=8treatments.1.89Youcanmultiplythenumberoflevelsineachfactor.Therearemlevelsinthefirstfactorandnlevelsinthesecondfactor.Therefore,thereare(m)(n)=mntreatments.1.90(a)Thetreatmentgroupconsistedofthe2444patientswhotookProzac.(b)Thecontrolgroupconsistedofthe1331patientswhoreceivedaplacebo.(c)ThetreatmentswereadministeringProzacandadministeringtheplacebo.1.91(a)Therewerethreetreatments.(b)Thefirstgroup,theonereceivingonlythepharmacologictherapy,wouldbeconsideredthecontrolgroup.(c)Therewerethreetreatmentgroups.Thefirstreceivedonlypharmacologictherapy,thesecondreceivedpharmacologictherapyplusapacemaker,andthethirdreceivedpharmacologictherapyplusapacemaker-defibrillatorcombination.(d)Thefirstgroup(control)contained1/5ofthe1520patientsor304.Theothertwogroupseachcontained2/5ofthe1520patientsor608.(e)Eachpatientcouldberandomlyassignedanumberfrom1to1520.Anypatientassignedanumberbetween1and304wouldbeassignedtothecontrolgroup;anypatientassignedtothenext608numbers(305to912)wouldbeassignedtoreceivethepharmacologictherapyplusapacemaker;andanypatientassignedanumberbetween913and1520wouldreceivepharmacologictherapyplusapacemaker-defibrillatorcombination.Eachrandomnumberwouldbeusedonlyoncetoensurethattheresultingtreatmentgroupswereoftheintendedsizes.1.92(a)Experimentalunits:batchesoftheproductbeingsold(b)Responsevariable:thenumberofunitsoftheproductsold(c)Factors:twofactors-displaytypeandpricingscheme(d)Levelsofeachfactor:threetypesofdisplayoftheproductandthreepricingschemes(e)Treatments:theninedifferentcombinationsofdisplaytypeandpriceresultingfromtestingeachofthethreepricingschemeswitheachofthethreedisplaytypes1.93(a)Experimentalunits:thedrivers(b)Responsevariable:thedetectiondistance,infeet(c)Factors:twofactors–signsizeandsignmaterial

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