AP统计学讲义PPT教学课件

统计学精讲精练主讲老师：黄逸昆*.2004APSTATISTICSFREE-RESPONSEQUESTIONS(FORMB)STATISTICSSECTIONⅡPartAQuestions1-5Spendabout65minutesonthispartoftheexam.PercentofSectionIIgrade—75*.1TheEarth'sMoonhasmanyimpactcratersthatwerecreatedwhentheinnersolarsystemwassubjectedtoheavybombardmentofsmallcelestialbodies.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)STATISTICSSECTIONⅡPartBQuestions6Spendabout25minutesonthispartoftheexam.PercentofSectionIIgrade—25*.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.*.2006APSTATISQUESTIONSSTATISTICSSECTIONⅡPartAQuestions1-5Spendabout65minutesonthispartoftheexam.PercentofSectionIIgrade—75TICSFREE-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.Astudyofthesuds'heightsforanewdishdetergentwasconducted.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.Thetimewhenthepatient'sdiagnostictreatmentbeginsisalsorecorded.*.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),describeacompletelyrandomizeddesignthatwillallowthebiologisttocomparetheshrimps'growthafter3weeks.(c)Giveonestatisticaladvantagetohavingonlytigershrimpsintheexperiment.Explainwhythisisanadvantage.(d)Giveonestatisticaldisadvantagetohavingonlytigershrimpsintheexperiment.Explainwhythisisadisadvantage.*.2006APSTATISTICSFREE-RESPONSEQUESTIONSSTATISTICSSECTIONⅡPartBQuestions6Spendabout25minutesonthispartoftheexam.PercentofSectionIIgrade—25*.6.Amanufacturerofthermostatsisconcernedthatthereadingsofitsthermostatshavebecomelessreliable(morevariable).Inthepast,thevariancehasbeen1.52degreesFahrenheit(F)squared.Arandomsampleof10recentlymanufacturedthermostatswasselectedandplacedinaroomthatwasmaintainedat68°F.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-RESPONSEQUESTIONSSTATISTICSSECTIONⅡPartAQuestions1-5Spendabout65minutesonthispartoftheexam.PercentofSectionIIgrade—75*.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(X<x)>0.5andP(X>x)>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.Thisgroupofstudentspurchased65boxesofthecerealtoinvestigatethecompany'sclaim.Thestudentsfoundatotalof11vouchersforfreevideorentalsinthe65boxes.*.Supposeitisreasonabletoassumethatthe65boxespurchasedbythestudentsarearandomsampleofallboxesofthiscereal.Basedonthissample,istheresupportforthestudents'beliefthattheproportionofboxeswithvouchersislessthan0.2?Providestatisticalevidencetosupportyouranswer.*.AsurveywillbeconductedtoexaminetheeducationallevelofadultheadsofhouseholdsintheUnitedStates.Eachrespondentinthesurveywillbeplacedintooneofthefollowingtwocategories:  Doesnothaveahighschooldiploma HasahighschooldiplomaThesurveywillbeconductedusingatelephoneinterview.Random-digitdialingwillbeusedtoselectthesample.*.(a)Forthissurvey,stateonepotentialsourceofbiasanddescribehowitmightaffecttheestimateoftheproportionofadultheadsofhouseholdsintheUnitedStateswhodonothaveahighschooldiploma.(b)Apilotsurveyindicatedthatabout22percentofthepopulationofadultheadsofhouseholdsdonothaveahighschooldiploma.Usingthisinformation,howmanyrespondentsshouldbeobtainedifthegoalofthesurveyistoestimatetheproportionofthepopulationwhodonothaveahighschooldiplomatowithin0.03with95percentconfidence?Justifyyouranswer.*.(c)Sinceeducationislargelytheresponsibilityofeachstate,theagencywantstobesurethatestimatesareavailableforeachstateaswellasforthenation.Identifyasamplingmethodthatwillachievethisadditionalgoalandbrieflydescribeawaytoselectthesurveysampleusingthismethod.*.2005APSTATISTICSFREE-RESPONSEQUESTIONSSTATISTICSSECTIONⅡPartBQuestions6Spendabout25minutesonthispartoftheexam.PercentofSectionIIgrade—25*.6.Lead,foundinsomepaints,isaneurotoxinthatcanbeespeciallyharmfultothedevelopingbrainandnervoussystemofchildren.Childrenfrequentlyputtheirhandsintheirmouthaftertouchingpaintedsurfaces,andthisisthemostcommontypeofexposuretolead.Astudywasconductedtoinvestigatewhetherthereweredifferencesinchildren'sexposuretoleadbetweensuburbanday-carecentersandurbanday-carecentersinonelargecity.Forthisstudy,researchersusedarandomsampleof20childreninsuburbanday-carecenters.Tenofthese20childrenwererandomlyselectedtoplayoutside;theremaining10childrenplayedinside.Allchildrenhadtheirhandswipedcleanbeforebeginningtheirassignedone-hourplayperiodeitheroutsideorinside.Aftertheplayperiodended,theamountofleadinmicrograms(meg)oneachchild'sdominanthandwasrecorded.*.Themeanamountofleadonthedominanthandforthechildrenplayinginsidewas3.75meg,andthemeanamountofleadforthechildrenplayingoutsidewas5.65meg.A95percentconfidenceintervalforthedifferenceinthemeanamountofleadafteronehourinsideversusonehouroutsidewascalculatedtobe(-2.46,-1.34).*.Arandomsampleof18childreninurbanday-carecentersinthesamelargecitywasselected.Forthissample,thesameprocesswasused,includingrandomlyassigningchildrentoplayinsideoroutside.Thedatafortheamount(inmeg)ofleadoneachchild'sdominanthandareshowninthetablebelow.*.(a)Usea95percentconfidenceintervaltoestimatethedifferenceinthemeanamountofleadonachild'sdominanthandafteranhourofplayinsideversusanhourofplayoutsideaturbanday-carecentersinthiscity.Besuretointerpretyourinterval.(b)Onthefigurebelow,Usingtheverticalaxisforthemeanamountoflead,plotthemeanfortheamountsofleadonthedominanthandofchildrenwhoplayedinsideatthesuburbanday-carecenterandthenplotthemeanfortheamountsofleadonthedominanthandofchildrenwhoplayedinsideattheurbanday-carecenter.*. Connectthesetwopointswithalinesegment.Plotthetwomeans(suburbanandurban)forthechildrenwhoplayedoutsideatthetwotypesofday-carecenters.Connectthesetwopointswithasecondlinesegment.*.(c)Fromthestudy,whatconclusionscanbedrawnabouttheimpactofsetting(inside,outside),environment(suburban,urb