AP统计学讲义

统计学精讲精练,主讲老师：黄逸昆,2004APSTATISTICSFREE-RESPONSEQUESTIONS(FORMB)STATISTICSSECTIONPartAQuestions1-5Spendabout65minutesonthispartoftheexam.PercentofSectionIIgrade75,1TheEarthsMoonhasmanyimpactcratersthatwerecreatedwhentheinnersolarsystemwassubjectedtoheavybombardmentofsmallcelestialbodies.Scientistsstudied11impactcratersontheMoontodeterminewhethertherewasanyrelationshipbetweentheageofthecraters(basedonradioactivedatingoflunarrocks)andtheimpactrate(asdeducedfromthedensityofthecraters).Thedataaredisplayedinthescatterplotbelow.,(a)Describethenatureoftherelationshipbetweenimpactrateandage.,Priortofittingalinearregressionmodel,theresearcherstransformedbothimpactrateandagebyusinglogarithms.Thefollowingcomputeroutputandresidualplotwereproduced.,(b)Interpretthevalueofr2.(c)Commentontheappropriatenessofthislinearregressionformodelingtherelationshipbetweenthetransformedvariables.,2Atacertainuniversity,studentswholiveinthedormitorieseatatacommondininghall.Recently,somestudentshavebeencomplainingaboutthequalityofthefoodservedthere.Thedininghallmanagerdecidedtodoasurveytoestimatetheproportionofstudentslivinginthedormitorieswhothinkthatthequalityofthefoodshouldbeimproved.Oneevening,themanageraskedthefirst100studentsenteringthedininghalltoanswerthefollowingquestion.,(a)Inthissetting,explainhowbiasmayhavebeenintroducedbasedonthewaythisconveniencesamplewasselectedandsuggesthowthesamplecouldhavebeenselecteddifferentlytoavoidthatbias.(b)Inthissetting,explainhowbiasmayhavebeenintroducedbasedonthewaythequestionwaswordedandsuggesthowitcouldhavebeenwordeddifferentlytoavoidthatbias.,TrainscarrybauxiteorefromamineinCanadatoanaluminumprocessingplantinnorthernNewYorkstateinhoppercars.Fillingequipmentisusedtoloadoreintothehoppercars.Whenfunctioningproperly,theactualweightsoforeloadedintoeachcarbythefillingequipmentatthemineareapproximatelynormallydistributedwithameanof70tonsandastandarddeviationof0.9ton.Ifthemeanisgreaterthan70tons,theloadingmechanismisoverfilling.,(a)Ifthefillingequipmentisfunctioningproperly,whatistheprobabilitythattheweightoftheoreinarandomlyselectedcarwillbe70.7tonsormore?Showyourwork.(b)Supposethattheweightoforeinarandomlyselectedcaris70.7tons.Wouldthatfactmakeyoususpectthattheloadingmechanismisoverfillingthecars?Justifyyouranswer.(c)Ifthefillingequipmentisfunctioningproperly,whatistheprobabilitythatarandomsampleof10carswillhaveameanoreweightof70.7tonsormore?Showyourwork.(d)Basedonyouranswerinpart(c),ifarandomsampleof10carshadameanoreweightof70.7tons,wouldyoususpectthattheloadingmechanismwasoverfillingthecars?Justifyyouranswer.,4.TheprincipalatCrestMiddleSchool,whichenrollsonlysixth-gradestudentsandseventh-gradestudents,isinterestedindetermininghowmuchtimestudentsatthatschoolspendonhomeworkeachnight.Thetablebelowshowsthemeanandstandarddeviationoftheamountoftimespentonhomeworkeachnight(inminutes)forarandomsampleof20sixth-gradestudentsandaseparaterandomsampleof20seventh-gradestudentsatthisschool.,Basedondotplotsofthesedata,itisnotunreasonabletoassumethatthedistributionoftimesforeachgradewereapproximatelynormallydistributed.(a)Estimatethedifferenceinmeantimesspentonhomeworkforallsixth-andseventh-gradestudentsinthisschoolusinganinterval.Besuretointerpretyourinterval.,(b)Anassistantprincipalreasonedthatamuchnarrowerconfidenceintervalcouldbeobtainedifthestudentswerepairedbasedontheirresponses;forexample,pairingthesixth-gradestudentandtheseventh-gradestudentwiththehighestnumberofminutesspentonhomework,thesixth-gradestudentandseventh-gradestudentwiththenexthighestnumberofminutesspentonhomework,andsoon.Istheassistantprincipalcorrectinthinkingthatmatchingstudentsinthiswayandthencomputingamatched-pairsconfidenceintervalforthemeandifferenceintimespentonhomeworkisabetterprocedurethantheoneusedinpart(a)?Explainwhyorwhynot.,5.AresearcherthinksthatmodernThaidogsmaybedescendantsofgoldenjackals.Arandomsampleof16animalswascollectedfromeachofthetwopopulations.Thelength(inmillimeters)ofthemandible(jawbone)wasmeasuredforeachanimal.Thelowerquartile,median,andupperquartileforeachsampleareshowninthetablebelow,alongwithallvaluesbelowthelowerquartileandallvaluesabovetheupperquartile.,(a)Displayparallelboxplotsofmandiblelengths(showingoutliers,ifany)forthemodernThaidogsandthegoldenjackalsonthegridbelow.,Basedontheboxplots,writeafewsentencescomparingthedistributionsofmandiblelengthsforthetwotypesofdogs.(b)IsitreasonabletousethesampleofmandiblelengthsofmodernThaidogstoconstructanintervalestimateofthemeanmandiblelengthforthepopulationofmodernThaidogs?Justifyyouranswer.(Note:Youdonothavetocomputetheinterval.)(c)IsitreasonabletousethesampledataofmandiblelengthsofmodernThaidogsandthesampledataofmandiblelengthsofgoldenjackalstoperformatwo-samplef-testforthedifferenceinmeanmandiblelengthsforthetwotypesofdogs?Justifyyouranswer.(Note:Youdonothavetoconductthetest.),2004APSTATISTICSFREE-RESPONSEQUESTIONS(FORMB)STATISTICSSECTIONPartBQuestions6Spendabout25minutesonthispartoftheexam.PercentofSectionIIgrade25,6.Inordertomonitorthepopulationsofbirdsofaparticularspeciesontwoislands,thefollowingprocedurewasimplemented.Researcherscapturedaninitialsampleof200birdsofthespeciesonIslandA;theyattachedlegbandstoeachofthebirds,andthenreleasedthebirds.Similarly,asampleof250birdsofthesamespeciesonIslandBwascaptured,banded,andreleased.Sufficienttimewasallowedforthebirdstoreturntotheirnormalroutineandlocation.Subsequentsamplesofbirdsofthespeciesofinterestwerethentakenfromeachisland.Thenumberofbirdscapturedandthenumberofbirdswithlegbandswererecorded.Theresultsaresummarizedinthefollowingtable.,Assumethatboththeinitialsampleandthesubsequentsamplesthatweretakenoneachislandcanberegardedasrandomsamplesfromthepopulationofbirdsofthisspecies.(a)Dothedatafromthesubsequentsamplesindicatethatthereisadifferenceinproportionsofthebandedbirdsonthesetwoislands?Givestatisticalevidencetosupportyouranswer.,(b)Researcherscanestimatethetotalnumberofbirdsofthisspeciesonanislandbyusinginformationonthenumberofbirdsintheinitialsampleandtheproportionofbandedbirdsinthesubsequentsample.UsethisinformationtoestimatethetotalnumberofbirdsofthisspeciesonIslandA.Showyourwork.(c)Theanalysesinparts(a)and(b)assumethatthesamplesofbirdscapturedinboththeinitialandsubsequentsamplescanberegardedasrandomsamplesofthepopulationofbirdsofthisspeciesthatliveontherespectiveislands.Thisisacommonassumptionmadebywildliferesearchers.Describetwoconcernsthatshouldbeaddressedbeforemakingthisassumption.,2006APSTATISQUESTIONSSTATISTICSSECTIONPartAQuestions1-5Spendabout65minutesonthispartoftheexam.PercentofSectionIIgrade75TICSFREE-RESPONSE,1.Twoparentshaveeachbuiltatoycatapultforuseinagameatanelementaryschoolfair.Toplaythegame,studentswillattempttolaunchPing-Pongballsfromthecatapultssothattheballslandwithina5-centimeterband.Atargetlinewillbedrawnthroughthemiddleoftheband,asshowninthefigurebelow.Allpointsonthetargetlineareequidistantfromthelaunchinglocation.,Ifaballlandswithintheshadedband,thestudentwillwinaprize.Theparentshaveconstructedthetwocatapultsaccordingtoslightlydifferentplans.Theywanttotestthesecatapultsbeforebuildingadditionalones.Underidenticalconditions,theparentslaunch40Ping-Pongballsfromeachcatapultandmeasurethedistancethattheballtravelsbeforelanding.Distancestothenearestcentimeteraregraphedinthedotplotsbelow.,(a)CommentonanysimilaritiesandanydifferencesinthetwodistributionsofdistancestraveledbyballslaunchedfromcatapultAandcatapultB.,(b)IftheparentswanttomaximizetheprobabilityofhavingthePing-Pongballslandwithintheband,whichoneofthetwocatapults,AorB,wouldbebettertousethantheother?Justifyyourchoice.(c)Usingthecatapultthatyouchoseinpart(b),howmanycentimetersfromthetargetlineshouldthiscatapultbeplaced?Explainwhyyouchosethisdistance.,2.Amanufacturerofdishdetergentbelievestheheightofsoapsudsinthedishpandependsontheamountofdetergentused.Astudyofthesudsheightsforanewdishdetergentwasconducted.Sevenpansofwaterwereprepared.Allpanswereofthesamesizeandtypeandcontainedthesameamountofwater.Thetemperatureofthewaterwasthesameforeachpan.Anamountofdishdetergentwasassignedatrandomtoeachpan,andthatamountofdetergentwasaddedtothepan.Thenthewaterinthedishpanwasagitatedforasetamountoftime,andtheheightoftheresultingsudswasmeasured.,Aplotofthedataandthecomputeroutputfromfittingaleastsquaresregressionlinetothedataareshownbelow.,Writetheequationofthefittedregressionline.Defineanyvariablesusedinthisequation.(b)Notethats=1.99821inthecomputeroutput.Interpretthisvalueinthecontextofthisstudy.(c)Identifyandinterpretthestandarderroroftheslope.,3.ThedepthfromthesurfaceofEarthtoarefractinglayerbeneaththesurfacecanbeestimatedusingmethodsdevelopedbyseismologists.Onemethodisbasedonthetimerequiredforvibrationstotravelfromadistantexplosiontoareceivingpoint.Thedepthmeasurement(M)isthesumofthetruedepth(D)andtherandommeasurementerror(E).Thatis,M-D+E.Themeasurementerror(E)isassumedtobenormallydistributedwithmean0feetandstandarddeviation1.5feet.,(a)Ifthetruedepthatacertainpointis2feet,whatistheprobabilitythatthedepthmeasurementwillbenegative?(b)Supposethreeindependentdepthmeasurementsaretakenatthepointwherethetruedepthis2feet.Whatistheprobabilitythatatleastoneofthesemeasurementswillbenegative?(c)Whatistheprobabilitythatthemeanofthethreeindependentdepthmeasurementstakenatthepointwherethetruedepthis2feetwillbenegative?,4.Patientswithheart-attacksymptomsarriveatanemergencyroomeitherbyambulanceorself-transportationprovidedbythemselves,family,orfriends.Whenapatientarrivesattheemergencyroom,thetimeofarrivalisrecorded.Thetimewhenthepatientsdiagnostictreatmentbeginsisalsorecorded.,Anadministratorofalargehospitalwantedtodeterminewhetherthemeanwaittime(timebetweenarrivalanddiagnostictreatment)forpatientswithheart-attacksymptomsdiffersaccordingtothemodeoftransportation.Arandomsampleof150patientswithheart-attacksymptomswhohadreportedtotheemergencyroomwasselected.Foreachpatient,themodeoftransportationandwaittimewererecorded.Summarystatisticsforeachmodeoftransportationareshowninthetablebelow.,(a)Usea99percentconfidenceintervaltoestimatethedifferencebetweenthemeanwaittimesforambulance-transportedpatientsandself-transportedpatientsatthisemergencyroom.(b)Basedonlyonthisconfidenceinterval,doyouthinkthedifferenceinthemeanwaittimesisstatisticallysignificant?Justifyyouranswer.,5.Abiologistisinterestedinstudyingtheeffectofgrowth-enhancingnutrientsanddifferentsalinity(salt)levelsinwateronthegrowthofshrimps.Thebiologisthasorderedalargeshipmentofyoungtigershrimpsfromasupplyhouseforuseinthestudy.Theexperimentistobeconductedinalaboratorywhere10tigershrimpsareplacedrandomlyintoeachof12similartanksinacontrolledenvironment.Thebiologistisplanningtouse3differentgrowth-enhancingnutrients(A,B,andC)andtwodifferentsalinitylevels(lowandhigh).(a)Listthetreatmentsthatthebiologistplanstouseinthisexperiment.,(b)Usingthetreatmentslistedinpart(a),describeacompletelyrandomizeddesignthatwillallowthebiologisttocomparetheshrimpsgrowthafter3weeks.(c)Giveonestatisticaladvantagetohavingonlytigershrimpsintheexperiment.Explainwhythisisanadvantage.(d)Giveonestatisticaldisadvantagetohavingonlytigershrimpsintheexperiment.Explainwhythisisadisadvantage.,2006APSTATISTICSFREE-RESPONSEQUESTIONSSTATISTICSSECTIONPartBQuestions6Spendabout25minutesonthispartoftheexam.PercentofSectionIIgrade25,6.Amanufacturerofthermostatsisconcernedthatthereadingsofitsthermostatshavebecomelessreliable(morevariable).Inthepast,thevariancehasbeen1.52degreesFahrenheit(F)squared.Arandomsampleof10recentlymanufacturedthermostatswasselectedandplacedinaroomthatwasmaintainedat68F.Thereadingsforthose10thermostatsaregiveninthetablebelow.,(a)Statethenullandalternativehypothesesthatthemanufacturerisinterestedintesting.,(b)Calculatethevalueof(n-1)S2/1.52forthesedata.,(c)Assumethatthepopulationofthermostattemperaturesfollowsanormaldistribution.Usetheteststatistic(n-1)S2/1.52frompart(b)andthechi-squaredistributiontotestthehypothesesinpart(a).(d)Forthetestconductedinpart(c),whatisthesmallestvalueoftheteststatisticthatwouldhaveledtotherejectionofthenullhypothesisatthe5percentsignificancelevel?,Markthisvalueoftheteststatisticonthegraphofthechi-squaredistributionbelow.Indicatetheregionthatcontainsallofthevaluesthatwouldhaveledtotherejectionofthenullhypothesis.,(e)Usingsimulation,1,000samples,eachofsize10,wererandomlygeneratedfrom3populationswithdifferentvariances.Eachpopulationwasnormallydistributedwithmean68andvariancegreaterthan1.52.Thehistogramsbelowshowthesimulatedsamplingdistributionof(n-1)S2/1.52foreachpopulation.Marktheregionidentifiedinpart(d)oneachofthehistogramsbelow.,2006APSTATISTICSFREE-RESPONSEQUESTIONS,(f)Basedontheregionsthatyoumarkedinpart(e),identifythesimulatedsamplingdistributionthatcorrespondstothepopulationwiththelargestvariance.Thenidentifythesimulatedsamplingdistributionthatcorrespondstothepopulationwiththesmallestvariance.Justifyyourchoices.,2005APSTATISTICSFREE-RESPONSEQUESTIONSSTATISTICSSECTIONPartAQuestions1-5Spendabout65minutesonthispartoftheexam.PercentofSectionIIgrade75,1.ThegoalofanutritionalstudywastocomparethecaloricintakeofadolescentsIrvinginruralareasoftheUnitedStateswiththecaloricintakeofadolescentslivinginurbanareasoftheUnitedStates.Arandomsampleofninth-gradestudentsfromonehighschoolinaruralareawasselected.Anotherrandomsampleofninthgradersfromonehighschoolinanurbanareawasalsoselected.Eachstudentineachsamplekeptrecordsofallthefoodheorsheconsumedinoneday.,Theback-to-backstemplotbelowdisplaysthenumberofcaloriesoffoodconsumedperkilogramofbodyweightforeachstudentonthatday.,(a)Writeafewsentencescomparingthedistributionofthedailycaloricintakeofninth-gradestudentsintheruralhighschoolwiththedistributionofthedailycaloricintakeofninth-gradestudentsintheurbanhighschool.(b)Isitreasonabletogeneralizethefindingsofthisstudytoallruralandurbanninth-gradestudentsintheUnitedStates?Explain.,Researcherswhowanttoconductasimilarstudyaredebatingwhichofthefollowingtwoplanstouse.PlanI:Haveeachstudentinthestudyrecordallthefoodheorsheconsumedinoneday.Thenresearcherswouldcomputethenumberofcaloriesoffoodconsumedperkilogramofbodyweightforeachstudentforthatday.PlanII:Haveeachstudentinthestudyrecordallthefoodheorsheconsumedoverthesame7-dayperiod.Thenresearcherswouldcomputetheaveragedailynumberofcaloriesoffoodconsumedperkilogramofbodyweightforeachstudentduringthat7-dayperiod.Assumingthatthestudentskeepaccuraterecords,whichplan,IorII,wouldbettermeetthegoalofthestudy?Justifyyouranswer.,2.LettherandomvariableXrepresentthenumberoftelephonelinesinusebythetechnicalsupportcenterofasoftwaremanufactureratnooneachday.TheprobabilitydistributionofXisshowninthetablebelow.,(a)Calculatetheexpectedvalue(themean)ofX.,(b)Usingpastrecords,thestaffatthetechnicalsupportcenterrandomlyselected20daysandfoundthatanaverageof1.25telephonelineswereinuseatnoononthosedays.Thestaffproposestoselectanotherrandomsampleof1,000daysandcomputetheaveragenumberoftelephonelinesthatwereinuseatnoononthosedays.Howdoyouexpecttheaveragefromthisnewsampletocomparetothatofthefirstsample?Justifyyourresponse.,(c)ThemedianofarandomvariableisdefinedasanyvaluexsuchthatP(X0.5andP(Xx)0.5.Fortheprobabilitydistributionshowninthetableabove,determinethemedianofX.(d)Inasentenceortwo,commentontherelationshipbetweenthemeanandthemedianrelativetotheshapeofthisdistribution.,3.TheGreatPlainsRailroadisinterestedinstudyinghowfuelconsumptionisrelatedtothenumberofrailcarsforitstrainsonacertainroutebetweenOklahomaCityandOmaha.Arandomsampleof10trainsonthisroutehasyieldedthedatainthetablebelow.,Ascatterplot,aresidualplot,andtheoutputfromtheregressionanalysisforthesedataareshownbelow.,(a)Isalinearmodelappropriateformodelingthesedata?Clearlyexplainyourreasoning.,(b)Supposethefuelconsumptioncostis$25perunit.Giveapointestimate(singlevalue)forthechangeintheaveragecostoffuelpermileforeachadditionalrailcarattachedtoatrain.Showyourwork.(c)Interpretthevalueofr2inthecontextofthisproblem.(d)Woulditbereasonabletousethefittedregressionequationtopredictthefuelconsumptionforatrainonthisrouteifthetrainhad65railcars?Explain.,4Someboxesofacertainbrandofbreakfastcerealincludeavoucherforafreevideorentalinsidethebox.Thecompanythatmakesthecerealclaimsthatavouchercanbefoundin20percentoftheboxes.However,basedontheirexperienceseatingthiscerealathome,agroupofstudentsbelievesthattheproportionofboxeswithvouchersislessthan0.2.Thisgroupofstudentspurchased65boxesofthecerealtoinvestigatethecompanysclaim.Thestudentsfoundatotalof11vouchersforfreevideorentalsinthe65boxes.,Supposeitisreasonabletoassumethatthe65boxespurchasedbythestudentsarearandomsampleofallboxesofthiscereal.Basedonthissample,istheresupportforthestudentsbeliefthattheproportionofboxeswithvouchersislessthan0.2?Providestatisticalevidencetosupportyouranswer.,AsurveywillbeconductedtoexaminetheeducationallevelofadultheadsofhouseholdsintheUnitedStates.Eachrespondentinthesurveywillbeplacedintooneofthefollowingtwocategories:DoesnothaveahighschooldiplomaHasahighschooldiplomaThesurveywillbeconductedusingatelephoneinterview.Random-digitdialingwillbeusedtoselectthesample.,(a)Forthissurvey,stateonepotentialsourceofbiasanddescribehowitmightaffecttheestimateoftheproportionofadultheadsofhouseholdsintheUnitedStateswhodonothaveahighschooldiploma.(b)Apilotsurveyindicatedthatabout22percentofthepopulationofadultheadsofhouseholdsdonothaveahighschooldiploma.Usingthisinformation,howmanyrespondentsshouldbeobtainedifthegoalofthesurveyistoestimatetheproportionofthepopulationwhodonothaveahighschooldiplomatowithin0.03with95percentconfidence?Justifyyouranswer.,(c)Sinceeducationislargelytheresponsibilityofeachstate,theagencywantstobesurethatestimatesareavailableforeachstateaswellasforthenation.Identifyasamplingmethodthatwillachievethisadditionalgoalandbrieflydescribeawaytoselectthesurveysampleusingthismethod.,2005APSTATISTICSFREE-RESPONSEQUESTIONSSTATISTICSSECTIONPartBQuestions6Spendabout25minutesonthispartoftheexam.PercentofSectionIIgrade25,6.Lead,foundinsomepaints,isaneurotoxinthatcanbeespeciallyharmfultothedevelopingbrainandnervoussystemofchildren.Childrenfrequentlyputtheirhandsintheirmoutha