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ap统计学讲义
统计学精讲精练主讲老师：黄逸昆\n2004APSTATISTICSFREE-RESPONSEQUESTIONS(FORMB)STATISTICSSECTIONⅡPartAQuestions1-5Spendabout65minutesonthispartoftheexam.PercentofSectionIIgrade—75\n1TheEarth'sMoonhasmanyimpactcratersthatwerecreatedwhentheinnersolarsystemwassubjectedtoheavybombardmentofsmallcelestialbodies.Scientistsstudied11impactcratersontheMoontodeterminewhethertherewasanyrelationshipbetweentheageofthecraters(basedonradioactivedatingoflunarrocks)andtheimpactrate(asdeducedfromthedensityofthecraters).Thedataaredisplayedinthescatterplotbelow.\n(a)Describethenatureoftherelationshipbetweenimpactrateandage.\nPriortofittingalinearregressionmodel,theresearcherstransformedbothimpactrateandagebyusinglogarithms.Thefollowingcomputeroutputandresidualplotwereproduced.\n(b)Interpretthevalueofr2.(c)Commentontheappropriatenessofthislinearregressionformodelingtherelationshipbetweenthetransformedvariables.\n2Atacertainuniversity,studentswholiveinthedormitorieseatatacommondininghall.Recently,somestudentshavebeencomplainingaboutthequalityofthefoodservedthere.Thedininghallmanagerdecidedtodoasurveytoestimatetheproportionofstudentslivinginthedormitorieswhothinkthatthequalityofthefoodshouldbeimproved.Oneevening,themanageraskedthefirst100studentsenteringthedininghalltoanswerthefollowingquestion.\n(a)Inthissetting,explainhowbiasmayhavebeenintroducedbasedonthewaythisconveniencesamplewasselectedandsuggesthowthesamplecouldhavebeenselecteddifferentlytoavoidthatbias.(b)Inthissetting,explainhowbiasmayhavebeenintroducedbasedonthewaythequestionwaswordedandsuggesthowitcouldhavebeenwordeddifferentlytoavoidthatbias.\nTrainscarrybauxiteorefromamineinCanadatoanaluminumprocessingplantinnorthernNewYorkstateinhoppercars.Fillingequipmentisusedtoloadoreintothehoppercars.Whenfunctioningproperly,theactualweightsoforeloadedintoeachcarbythefillingequipmentatthemineareapproximatelynormallydistributedwithameanof70tonsandastandarddeviationof0.9ton.Ifthemeanisgreaterthan70tons,theloadingmechanismisoverfilling.\n(a)Ifthefillingequipmentisfunctioningproperly,whatistheprobabilitythattheweightoftheoreinarandomlyselectedcarwillbe70.7tonsormore?Showyourwork.(b)Supposethattheweightoforeinarandomlyselectedcaris70.7tons.Wouldthatfactmakeyoususpectthattheloadingmechanismisoverfillingthecars?Justifyyouranswer.(c)Ifthefillingequipmentisfunctioningproperly,whatistheprobabilitythatarandomsampleof10carswillhaveameanoreweightof70.7tonsormore?Showyourwork.(d)Basedonyouranswerinpart(c),ifarandomsampleof10carshadameanoreweightof70.7tons,wouldyoususpectthattheloadingmechanismwasoverfillingthecars?Justifyyouranswer.\n4.TheprincipalatCrestMiddleSchool,whichenrollsonlysixth-gradestudentsandseventh-gradestudents,isinterestedindetermininghowmuchtimestudentsatthatschoolspendonhomeworkeachnight.Thetablebelowshowsthemeanandstandarddeviationoftheamountoftimespentonhomeworkeachnight(inminutes)forarandomsampleof20sixth-gradestudentsandaseparaterandomsampleof20seventh-gradestudentsatthisschool.\nBasedondotplotsofthesedata,itisnotunreasonabletoassumethatthedistributionoftimesforeachgradewereapproximatelynormallydistributed.(a)Estimatethedifferenceinmeantimesspentonhomeworkforallsixth-andseventh-gradestudentsinthisschoolusinganinterval.Besuretointerpretyourinterval.\n(b)Anassistantprincipalreasonedthatamuchnarrowerconfidenceintervalcouldbeobtainedifthestudentswerepairedbasedontheirresponses;forexample,pairingthesixth-gradestudentandtheseventh-gradestudentwiththehighestnumberofminutesspentonhomework,thesixth-gradestudentandseventh-gradestudentwiththenexthighestnumberofminutesspentonhomework,andsoon.Istheassistantprincipalcorrectinthinkingthatmatchingstudentsinthiswayandthencomputingamatched-pairsconfidenceintervalforthemeandifferenceintimespentonhomeworkisabetterprocedurethantheoneusedinpart(a)?Explainwhyorwhynot.\n5.AresearcherthinksthatmodernThaidogsmaybedescendantsofgoldenjackals.Arandomsampleof16animalswascollectedfromeachofthetwopopulations.Thelength(inmillimeters)ofthemandible(jawbone)wasmeasuredforeachanimal.Thelowerquartile,median,andupperquartileforeachsampleareshowninthetablebelow,alongwithallvaluesbelowthelowerquartileandallvaluesabovetheupperquartile.\n(a)Displayparallelboxplotsofmandiblelengths(showingoutliers,ifany)forthemodernThaidogsandthegoldenjackalsonthegridbelow.\nBasedontheboxplots,writeafewsentencescomparingthedistributionsofmandiblelengthsforthetwotypesofdogs.(b)IsitreasonabletousethesampleofmandiblelengthsofmodernThaidogstoconstructanintervalestimateofthemeanmandiblelengthforthepopulationofmodernThaidogs?Justifyyouranswer.(Note:Youdonothavetocomputetheinterval.)(c)IsitreasonabletousethesampledataofmandiblelengthsofmodernThaidogsandthesampledataofmandiblelengthsofgoldenjackalstoperformatwo-samplef-testforthedifferenceinmeanmandiblelengthsforthetwotypesofdogs?Justifyyouranswer.(Note:Youdonothavetoconductthetest.)\n2004APSTATISTICSFREE-RESPONSEQUESTIONS(FORMB)STATISTICSSECTIONⅡPartBQuestions6Spendabout25minutesonthispartoftheexam.PercentofSectionIIgrade—25\n6.Inordertomonitorthepopulationsofbirdsofaparticularspeciesontwoislands,thefollowingprocedurewasimplemented.Researcherscapturedaninitialsampleof200birdsofthespeciesonIslandA;theyattachedlegbandstoeachofthebirds,andthenreleasedthebirds.Similarly,asampleof250birdsofthesamespeciesonIslandBwascaptured,banded,andreleased.Sufficienttimewasallowedforthebirdstoreturntotheirnormalroutineandlocation.Subsequentsamplesofbirdsofthespeciesofinterestwerethentakenfromeachisland.Thenumberofbirdscapturedandthenumberofbirdswithlegbandswererecorded.Theresultsaresummarizedinthefollowingtable.\nAssumethatboththeinitialsampleandthesubsequentsamplesthatweretakenoneachislandcanberegardedasrandomsamplesfromthepopulationofbirdsofthisspecies.(a)Dothedatafromthesubsequentsamplesindicatethatthereisadifferenceinproportionsofthebandedbirdsonthesetwoislands?Givestatisticalevidencetosupportyouranswer.\n(b)Researcherscanestimatethetotalnumberofbirdsofthisspeciesonanislandbyusinginformationonthenumberofbirdsintheinitialsampleandtheproportionofbandedbirdsinthesubsequentsample.UsethisinformationtoestimatethetotalnumberofbirdsofthisspeciesonIslandA.Showyourwork.(c)Theanalysesinparts(a)and(b)assumethatthesamplesofbirdscapturedinboththeinitialandsubsequentsamplescanberegardedasrandomsamplesofthepopulationofbirdsofthisspeciesthatliveontherespectiveislands.Thisisacommonassumptionmadebywildliferesearchers.Describetwoconcernsthatshouldbeaddressedbeforemakingthisassumption.\n2006APSTATISQUESTIONSSTATISTICSSECTIONⅡPartAQuestions1-5Spendabout65minutesonthispartoftheexam.PercentofSectionIIgrade—75TICSFREE-RESPONSE\n1.Twoparentshaveeachbuiltatoycatapultforuseinagameatanelementaryschoolfair.Toplaythegame,studentswillattempttolaunchPing-Pongballsfromthecatapultssothattheballslandwithina5-centimeterband.Atargetlinewillbedrawnthroughthemiddleoftheband,asshowninthefigurebelow.Allpointsonthetargetlineareequidistantfromthelaunchinglocation.\nIfaballlandswithintheshadedband,thestudentwillwinaprize.Theparentshaveconstructedthetwocatapultsaccordingtoslightlydifferentplans.Theywanttotestthesecatapultsbeforebuildingadditionalones.Underidenticalconditions,theparentslaunch40Ping-Pongballsfromeachcatapultandmeasurethedistancethattheballtravelsbeforelanding.Distancestothenearestcentimeteraregraphedinthedotplotsbelow.\n(a)CommentonanysimilaritiesandanydifferencesinthetwodistributionsofdistancestraveledbyballslaunchedfromcatapultAandcatapultB.\n(b)IftheparentswanttomaximizetheprobabilityofhavingthePing-Pongballslandwithintheband,whichoneofthetwocatapults,AorB,wouldbebettertousethantheother?Justifyyourchoice.(c)Usingthecatapultthatyouchoseinpart(b),howmanycentimetersfromthetargetlineshouldthiscatapultbeplaced?Explainwhyyouchosethisdistance.\n2.Amanufacturerofdishdetergentbelievestheheightofsoapsudsinthedishpandependsontheamountofdetergentused.Astudyofthesuds'heightsforanewdishdetergentwasconducted.Sevenpansofwaterwereprepared.Allpanswereofthesamesizeandtypeandcontainedthesameamountofwater.Thetemperatureofthewaterwasthesameforeachpan.Anamountofdishdetergentwasassignedatrandomtoeachpan,andthatamountofdetergentwasaddedtothepan.Thenthewaterinthedishpanwasagitatedforasetamountoftime,andtheheightoftheresultingsudswasmeasured.\nAplotofthedataandthecomputeroutputfromfittingaleastsquaresregressionlinetothedataareshownbelow.\nWritetheequationofthefittedregressionline.Defineanyvariablesusedinthisequation.(b)Notethats=1.99821inthecomputeroutput.Interpretthisvalueinthecontextofthisstudy.(c)Identifyandinterpretthestandarderroroftheslope.\n3.ThedepthfromthesurfaceofEarthtoarefractinglayerbeneaththesurfacecanbeestimatedusingmethodsdevelopedbyseismologists.Onemethodisbasedonthetimerequiredforvibrationstotravelfromadistantexplosiontoareceivingpoint.Thedepthmeasurement(M)isthesumofthetruedepth(D)andtherandommeasurementerror(E).Thatis,M-D+E.Themeasurementerror(E)isassumedtobenormallydistributedwithmean0feetandstandarddeviation1.5feet.\n(a)Ifthetruedepthatacertainpointis2feet,whatistheprobabilitythatthedepthmeasurementwillbenegative?(b)Supposethreeindependentdepthmeasurementsaretakenatthepointwherethetruedepthis2feet.Whatistheprobabilitythatatleastoneofthesemeasurementswillbenegative?(c)Whatistheprobabilitythatthemeanofthethreeindependentdepthmeasurementstakenatthepointwherethetruedepthis2feetwillbenegative?\n4.Patientswithheart-attacksymptomsarriveatanemergencyroomeitherbyambulanceorself-transportationprovidedbythemselves,family,orfriends.Whenapatientarrivesattheemergencyroom,thetimeofarrivalisrecorded.Thetimewhenthepatient'sdiagnostictreatmentbeginsisalsorecorded.\nAnadministratorofalargehospitalwantedtodeterminewhetherthemeanwaittime(timebetweenarrivalanddiagnostictreatment)forpatientswithheart-attacksymptomsdiffersaccordingtothemodeoftransportation.Arandomsampleof150patientswithheart-attacksymptomswhohadreportedtotheemergencyroomwasselected.Foreachpatient,themodeoftransportationandwaittimewererecorded.Summarystatisticsforeachmodeoftransportationareshowninthetablebelow.\n(a)Usea99percentconfidenceintervaltoestimatethedifferencebetweenthemeanwaittimesforambulance-transportedpatientsandself-transportedpatientsatthisemergencyroom.(b)Basedonlyonthisconfidenceinterval,doyouthinkthedifferenceinthemeanwaittimesisstatisticallysignificant?Justifyyouranswer.\n5.Abiologistisinterestedinstudyingtheeffectofgrowth-enhancingnutrientsanddifferentsalinity(salt)levelsinwateronthegrowthofshrimps.Thebiologisthasorderedalargeshipmentofyoungtigershrimpsfromasupplyhouseforuseinthestudy.Theexperimentistobeconductedinalaboratorywhere10tigershrimpsareplacedrandomlyintoeachof12similartanksinacontrolledenvironment.Thebiologistisplanningtouse3differentgrowth-enhancingnutrients(A,B,andC)andtwodifferentsalinitylevels(lowandhigh).(a)Listthetreatmentsthatthebiologistplanstouseinthisexperiment.\n(b)Usingthetreatmentslistedinpart(a),describeacompletelyrandomizeddesignthatwillallowthebiologisttocomparetheshrimps'growthafter3weeks.(c)Giveonestatisticaladvantagetohavingonlytigershrimpsintheexperiment.Explainwhythisisanadvantage.(d)Giveonestatisticaldisadvantagetohavingonlytigershrimpsintheexperiment.Explainwhythisisadisadvantage.\n2006APSTATISTICSFREE-RESPONSEQUESTIONSSTATISTICSSECTIONⅡPartBQuestions6Spendabout25minutesonthispartoftheexam.PercentofSectionIIgrade—25\n6.Amanufacturerofthermostatsisconcernedthatthereadingsofitsthermostatshavebecomelessreliable(morevariable).Inthepast,thevariancehasbeen1.52degreesFahrenheit(F)squared.Arandomsampleof10recentlymanufacturedthermostatswasselectedandplacedinaroomthatwasmaintainedat68°F.Thereadingsforthose10thermostatsaregiveninthetablebelow.\n(a)Statethenullandalternativehypothesesthatthemanufacturerisinterestedintesting.(b)Calculatethevalueof(n-1)S2/1.52forthesedata.\n(c)Assumethatthepopulationofthermostattemperaturesfollowsanormaldistribution.Usetheteststatistic(n-1)S2/1.52frompart(b)andthechi-squaredistributiontotestthehypothesesinpart(a).(d)Forthetestconductedinpart(c),whatisthesmallestvalueoftheteststatisticthatwouldhaveledtotherejectionofthenullhypothesisatthe5percentsignificancelevel?\nMarkthisvalueoftheteststatisticonthegraphofthechi-squaredistributionbelow.Indicatetheregionthatcontainsallofthevaluesthatwouldhaveledtotherejectionofthenullhypothesis.\n(e)Usingsimulation,1,000samples,eachofsize10,wererandomlygeneratedfrom3populationswithdifferentvariances.Eachpopulationwasnormallydistributedwithmean68andvariancegreaterthan1.52.Thehistogramsbelowshowthesimulatedsamplingdistributionof(n-1)S2/1.52foreachpopulation.Marktheregionidentifiedinpart(d)oneachofthehistogramsbelow.\n2006APSTATISTICSFREE-RESPONSEQUESTIONS\n(f)Basedontheregionsthatyoumarkedinpart(e),identifythesimulatedsamplingdistributionthatcorrespondstothepopulationwiththelargestvariance.Thenidentifythesimulatedsamplingdistributionthatcorrespondstothepopulationwiththesmallestvariance.Justifyyourchoices.\n2005APSTATISTICSFREE-RESPONSEQUESTIONSSTATISTICSSECTIONⅡPartAQuestions1-5Spendabout65minutesonthispartoftheexam.PercentofSectionIIgrade—75\n1.ThegoalofanutritionalstudywastocomparethecaloricintakeofadolescentsIrvinginruralareasoftheUnitedStateswiththecaloricintakeofadolescentslivinginurbanareasoftheUnitedStates.Arandomsampleofninth-gradestudentsfromonehighschoolinaruralareawasselected.Anotherrandomsampleofninthgradersfromonehighschoolinanurbanareawasalsoselected.Eachstudentineachsamplekeptrecordsofallthefoodheorsheconsumedinoneday.\nTheback-to-backstemplotbelowdisplaysthenumberofcaloriesoffoodconsumedperkilogramofbodyweightforeachstudentonthatday.\n(a)Writeafewsentencescomparingthedistributionofthedailycaloricintakeofninth-gradestudentsintheruralhighschoolwiththedistributionofthedailycaloricintakeofninth-gradestudentsintheurbanhighschool.(b)Isitreasonabletogeneralizethefindingsofthisstudytoallruralandurbanninth-gradestudentsintheUnitedStates?Explain.\nResearcherswhowanttoconductasimilarstudyaredebatingwhichofthefollowingtwoplanstouse.PlanI:Haveeachstudentinthestudyrecordallthefoodheorsheconsumedinoneday.Thenresearcherswouldcomputethenumberofcaloriesoffoodconsumedperkilogramofbodyweightforeachstudentforthatday.PlanII:Haveeachstudentinthestudyrecordallthefoodheorsheconsumedoverthesame7-dayperiod.Thenresearcherswouldcomputetheaveragedailynumberofcaloriesoffoodconsumedperkilogramofbodyweightforeachstudentduringthat7-dayperiod.Assumingthatthestudentskeepaccuraterecords,whichplan,IorII,wouldbettermeetthegoalofthestudy?Justifyyouranswer.\n2.LettherandomvariableXrepresentthenumberoftelephonelinesinusebythetechnicalsupportcenterofasoftwaremanufactureratnooneachday.TheprobabilitydistributionofXisshowninthetablebelow.(a)Calculatetheexpectedvalue(themean)ofX.\n(b)Usingpastrecords,thestaffatthetechnicalsupportcenterrandomlyselected20daysandfoundthatanaverageof1.25telephonelineswereinuseatnoononthosedays.Thestaffproposestoselectanotherrandomsampleof1,000daysandcomputetheaveragenumberoftelephonelinesthatwereinuseatnoononthosedays.Howdoyouexpecttheaveragefromthisnewsampletocomparetothatofthefirstsample?Justifyyourresponse.\n(c)ThemedianofarandomvariableisdefinedasanyvaluexsuchthatP(X查看更多