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1、AP统计学讲义2004 AP STATISTICS FREE-RESPONSE QUESTIONS(FORM B)STATISTICS SECTION Part A Questions 1-5 Spend about 65 minutes on this part of the exam.Percent of Section II grade75(b)Interpretthevalueofr 2.(c)Commentontheappropriatenessofthislinearregressionformodelingtherelationshipbetweenthetransformedv
2、ariables.2Atacertainuniversity,studentswholiveinthedormitorieseatatacommondininghall.Recently,somestudentshavebeencomplainingaboutthequalityofthefoodservedthere.Thedininghallmanagerdecidedtodoasurveytoestimatetheproportionofstudentslivinginthedormitorieswhothinkthatthequalityofthefoodshouldbeimprove
3、d.Oneevening,themanageraskedthefirst100studentsenteringthedininghalltoanswerthefollowingquestion.(a)Inthissetting,explainhowbiasmayhavebeenintroducedbasedonthewaythisconveniencesamplewasselectedandsuggesthowthesamplecouldhavebeenselecteddifferentlytoavoidthatbias.(b)Inthissetting,explainhowbiasmayha
4、vebeenintroducedbasedonthewaythequestionwaswordedandsuggesthowitcouldhavebeenwordeddifferentlytoavoidthatbias.3TrainscarrybauxiteorefromamineinCanadatoanaluminumprocessingplantinnorthernNewYorkstateinhoppercars.Fillingequipmentisusedtoloadoreintothehoppercars.Whenfunctioningproperly,theactualweights
5、oforeloadedintoeachcarbythefillingequipmentatthemineareapproximatelynormallydistributedwithameanof70tonsandastandarddeviationof0.9ton.Ifthemeanisgreaterthan70tons,theloadingmechanismisoverfilling.(a)Ifthefillingequipmentisfunctioningproperly,whatistheprobabilitythattheweightoftheoreinarandomlyselect
6、edcarwillbe70.7tonsormore?Showyourwork.(b)Supposethattheweightoforeinarandomlyselectedcaris70.7tons.Wouldthatfactmakeyoususpectthattheloadingmechanismisoverfillingthecars?Justifyyouranswer.(c)Ifthefillingequipmentisfunctioningproperly,whatistheprobabilitythatarandomsampleof10carswillhaveameanoreweig
7、htof70.7tonsormore?Showyourwork.(d)Basedonyouranswerinpart(c),ifarandomsampleof10carshadameanoreweightof70.7tons,wouldyoususpectthattheloadingmechanismwasoverfillingthecars?Justifyyouranswer.4.TheprincipalatCrestMiddleSchool,whichenrollsonlysixth-gradestudentsandseventh-gradestudents,isinterestedind
8、etermininghowmuchtimestudentsatthatschoolspendonhomeworkeachnight.Thetablebelowshowsthemeanandstandarddeviationoftheamountoftimespentonhomeworkeachnight(inminutes)forarandomsampleof20sixth-gradestudentsandaseparaterandomsampleof20seventh-gradestudentsatthisschool.Basedondotplotsofthesedata,itisnotun
9、reasonabletoassumethatthedistributionoftimesforeachgradewereapproximatelynormallydistributed.(a)Estimatethedifferenceinmeantimesspentonhomeworkforallsixth-andseventh-gradestudentsinthisschoolusinganinterval.Besuretointerpretyourinterval.(b)Anassistantprincipalreasonedthatamuchnarrowerconfidenceinter
10、valcouldbeobtainedifthestudentswerepairedbasedontheirresponses;forexample,pairingthesixth-gradestudentandtheseventh-gradestudentwiththehighestnumberofminutesspentonhomework,thesixth-gradestudentandseventh-gradestudentwiththenexthighestnumberofminutesspentonhomework,andsoon.Istheassistantprincipalcor
11、rectinthinkingthatmatchingstudentsinthiswayandthencomputingamatched-pairsconfidenceintervalforthemeandifferenceintimespentonhomeworkisabetterprocedurethantheoneusedinpart(a)?Explainwhyorwhynot.5.AresearcherthinksthatmodernThaidogsmaybedescendantsofgoldenjackals.Arandomsampleof16animalswascollectedfr
12、omeachofthetwopopulations.Thelength(inmillimeters)ofthemandible(jawbone)wasmeasuredforeachanimal.Thelowerquartile,median,andupperquartileforeachsampleareshowninthetablebelow,alongwithallvaluesbelowthelowerquartileandallvaluesabovetheupperquartile.(a)Displayparallelboxplotsofmandiblelengths(showingou
13、tliers,ifany)forthemodernThaidogsandthegoldenjackalsonthegridbelow.Basedontheboxplots,writeafewsentencescomparingthedistributionsofmandiblelengthsforthetwotypesofdogs.(b)IsitreasonabletousethesampleofmandiblelengthsofmodernThaidogstoconstructanintervalestimateofthemeanmandiblelengthforthepopulationo
14、fmodernThaidogs?Justifyyouranswer.(Note:Youdonothavetocomputetheinterval.)(c)IsitreasonabletousethesampledataofmandiblelengthsofmodernThaidogsandthesampledataofmandiblelengthsofgoldenjackalstoperformatwo-samplef-testforthedifferenceinmeanmandiblelengthsforthetwotypesofdogs?Justifyyouranswer.(Note:Yo
15、udonothavetoconductthetest.)2004 AP STATISTICS FREE-RESPONSE QUESTIONS(FORM B)STATISTICS SECTION Part B Questions 6 Spend about 25 minutes on this part of the exam.Percent of Section II grade256.Inordertomonitorthepopulationsofbirdsofaparticularspeciesontwoislands,thefollowingprocedurewasimplemented
16、.Researcherscapturedaninitialsampleof200birdsofthespeciesonIslandA;theyattachedlegbandstoeachofthebirds,andthenreleasedthebirds.Similarly,asampleof250birdsofthesamespeciesonIslandBwascaptured,banded,andreleased.Sufficienttimewasallowedforthebirdstoreturntotheirnormalroutineandlocation.Subsequentsamp
17、lesofbirdsofthespeciesofinterestwerethentakenfromeachisland.Thenumberofbirdscapturedandthenumberofbirdswithlegbandswererecorded.Theresultsaresummarizedinthefollowingtable.Assumethatboththeinitialsampleandthesubsequentsamplesthatweretakenoneachislandcanberegardedasrandomsamplesfromthepopulationofbird
18、softhisspecies.(a)Dothedatafromthesubsequentsamplesindicatethatthereisadifferenceinproportionsofthebandedbirdsonthesetwoislands?Givestatisticalevidencetosupportyouranswer.(b)Researcherscanestimatethetotalnumberofbirdsofthisspeciesonanislandbyusinginformationonthenumberofbirdsintheinitialsampleandthe
19、proportionofbandedbirdsinthesubsequentsample.UsethisinformationtoestimatethetotalnumberofbirdsofthisspeciesonIslandA.Showyourwork.(c)Theanalysesinparts(a)and(b)assumethatthesamplesofbirdscapturedinboththeinitialandsubsequentsamplescanberegardedasrandomsamplesofthepopulationofbirdsofthisspeciesthatli
20、veontherespectiveislands.Thisisacommonassumptionmadebywildliferesearchers.Describetwoconcernsthatshouldbeaddressedbeforemakingthisassumption.2006 AP STATISQUESTIONS2006 AP STATISQUESTIONSSTATISTICS STATISTICS SECTION SECTION Part A Part A Questions 1-5 Questions 1-5 Spend about 65 minutes on this pa
21、rt Spend about 65 minutes on this part of the exam.of the exam.Percent of Section II grade75Percent of Section II grade75TICS FREE-RESPONSETICS FREE-RESPONSE1.Twoparentshaveeachbuiltatoycatapultforuseinagameatanelementaryschoolfair.Toplaythegame,studentswillattempttolaunchPing-Pongballsfromthecatapu
22、ltssothattheballslandwithina5-centimeterband.Atargetlinewillbedrawnthroughthemiddleoftheband,asshowninthefigurebelow.Allpointsonthetargetlineareequidistantfromthelaunchinglocation.Ifaballlandswithintheshadedband,thestudentwillwinaprize.Theparentshaveconstructedthetwocatapultsaccordingtoslightlydiffe
23、rentplans.Theywanttotestthesecatapultsbeforebuildingadditionalones.Underidenticalconditions,theparentslaunch40Ping-Pongballsfromeachcatapultandmeasurethedistancethattheballtravelsbeforelanding.Distancestothenearestcentimeteraregraphedinthedotplotsbelow.(a)Commentonanysimilaritiesandanydifferencesint
24、hetwodistributionsofdistancestraveledbyballslaunchedfromcatapultAandcatapultB.(b)IftheparentswanttomaximizetheprobabilityofhavingthePing-Pongballslandwithintheband,whichoneofthetwocatapults,AorB,wouldbebettertousethantheother?Justifyyourchoice.(c)Usingthecatapultthatyouchoseinpart(b),howmanycentimet
25、ersfromthetargetlineshouldthiscatapultbeplaced?Explainwhyyouchosethisdistance.2.Amanufacturerofdishdetergentbelievestheheightofsoapsudsinthedishpandependsontheamountofdetergentused.Astudyofthesudsheightsforanewdishdetergentwasconducted.Sevenpansofwaterwereprepared.Allpanswereofthesamesizeandtypeandc
26、ontainedthesameamountofwater.Thetemperatureofthewaterwasthesameforeachpan.Anamountofdishdetergentwasassignedatrandomtoeachpan,andthatamountofdetergentwasaddedtothepan.Thenthewaterinthedishpanwasagitatedforasetamountoftime,andtheheightoftheresultingsudswasmeasured.Aplotofthedataandthecomputeroutputfr
27、omfittingaleastsquaresregressionlinetothedataareshownbelow.(a)Writetheequationofthefittedregressionline.Defineanyvariablesusedinthisequation.(b)Notethats=1.99821 in the computer output.Interpret this value in the context of this study.(c)Identifyandinterpretthestandarderroroftheslope.3.Thedepthfromt
28、hesurfaceofEarthtoarefractinglayerbeneaththesurfacecanbeestimatedusingmethodsdevelopedbyseismologists.Onemethodisbasedonthetimerequiredforvibrationstotravelfromadistantexplosiontoareceivingpoint.Thedepthmeasurement(M)isthesumofthetruedepth(D)andtherandommeasurementerror(E).That is,M-D+E.The measurem
29、ent error(E)is assumed to be normally distributed with mean 0 feet and standard deviation 1.5 feet.(a)Ifthetruedepthatacertainpointis2feet,whatistheprobabilitythatthedepthmeasurementwillbenegative?(b)Supposethreeindependentdepthmeasurementsaretakenatthepointwherethetruedepthis2feet.Whatistheprobabil
30、itythatatleastoneofthesemeasurementswillbenegative?(c)Whatistheprobabilitythatthemeanofthethreeindependentdepthmeasurementstakenatthepointwherethetruedepthis2feetwillbenegative?4.Patientswithheart-attacksymptomsarriveatanemergencyroomeitherbyambulanceorself-transportationprovidedbythemselves,family,
31、orfriends.Whenapatientarrivesattheemergencyroom,thetimeofarrivalisrecorded.Thetimewhenthepatientsdiagnostictreatmentbeginsisalsorecorded.Anadministratorofalargehospitalwantedtodeterminewhetherthemeanwaittime(timebetweenarrivalanddiagnostictreatment)forpatientswithheart-attacksymptomsdiffersaccording
32、tothemodeoftransportation.Arandomsampleof150patientswithheart-attacksymptomswhohadreportedtotheemergencyroomwasselected.Foreachpatient,themodeoftransportationandwaittimewererecorded.Summarystatisticsforeachmodeoftransportationareshowninthetablebelow.(a)Usea99percentconfidenceintervaltoestimatethedif
33、ferencebetweenthemeanwaittimesforambulance-transportedpatientsandself-transportedpatientsatthisemergencyroom.(b)Basedonlyonthisconfidenceinterval,doyouthinkthedifferenceinthemeanwaittimesisstatisticallysignificant?Justifyyouranswer.5.Abiologistisinterestedinstudyingtheeffectofgrowth-enhancingnutrien
34、tsanddifferentsalinity(salt)levelsinwateronthegrowthofshrimps.Thebiologisthasorderedalargeshipmentofyoungtigershrimpsfromasupplyhouseforuseinthestudy.Theexperimentistobeconductedinalaboratorywhere10tigershrimpsareplacedrandomlyintoeachof12similartanksinacontrolledenvironment.Thebiologistisplanningto
35、use3differentgrowth-enhancingnutrients(A,B,andC)andtwodifferentsalinitylevels(lowandhigh).(a)Listthetreatmentsthatthebiologistplanstouseinthisexperiment.(b)Usingthetreatmentslistedinpart(a),describeacompletelyrandomizeddesignthatwillallowthebiologisttocomparetheshrimpsgrowthafter3weeks.(c)Giveonesta
36、tisticaladvantagetohavingonlytigershrimpsintheexperiment.Explainwhythisisanadvantage.(d)Giveonestatisticaldisadvantagetohavingonlytigershrimpsintheexperiment.Explainwhythisisadisadvantage.2006 AP STATISTICS FREE-2006 AP STATISTICS FREE-RESPONSE QUESTIONSRESPONSE QUESTIONSSTATISTICS STATISTICS SECTIO
37、N SECTION Part B Part B Questions 6 Questions 6 Spend about 25 minutes on this part Spend about 25 minutes on this part of the exam.of the exam.Percent of Section II grade25Percent of Section II grade256.Amanufacturerofthermostatsisconcernedthatthereadingsofitsthermostatshavebecomelessreliable(morev
38、ariable).Inthepast,thevariancehasbeen1.52degreesFahrenheit(F)squared.Arandomsampleof10recentlymanufacturedthermostatswasselectedandplacedinaroomthatwasmaintainedat68F.Thereadingsforthose10thermostatsaregiveninthetablebelow.(a)Statethenullandalternativehypothesesthatthemanufacturerisinterestedintesti
39、ng.(b)Calculatethevalueof(n-1)S2/1.52forthesedata.(c)Assumethatthepopulationofthermostattemperaturesfollowsanormaldistribution.Usetheteststatistic(n-1)S2/1.52frompart(b)andthechi-squaredistributiontotestthehypothesesinpart(a).(d)Forthetestconductedinpart(c),whatisthesmallestvalueoftheteststatisticth
40、atwouldhaveledtotherejectionofthenullhypothesisatthe5percentsignificancelevel?Markthisvalueoftheteststatisticonthegraphofthechi-squaredistributionbelow.Indicatetheregionthatcontainsallofthevaluesthatwouldhaveledtotherejectionofthenullhypothesis.(e)Usingsimulation,1,000samples,eachofsize10,wererandom
41、lygeneratedfrom3populationswithdifferentvariances.Eachpopulationwasnormallydistributedwithmean68andvariancegreaterthan1.52.Thehistogramsbelowshowthesimulatedsamplingdistributionof(n-1)S2/1.52foreachpopulation.Marktheregionidentifiedinpart(d)oneachofthehistogramsbelow.2006 AP STATISTICS FREE-2006 AP
42、STATISTICS FREE-RESPONSE QUESTIONSRESPONSE QUESTIONS(f)Basedontheregionsthatyoumarkedinpart(e),identifythesimulatedsamplingdistributionthatcorrespondstothepopulationwiththelargestvariance.Thenidentifythesimulatedsamplingdistributionthatcorrespondstothepopulationwiththesmallestvariance.Justifyyourcho
43、ices.2005 AP STATISTICS FREE-2005 AP STATISTICS FREE-RESPONSE QUESTIONSRESPONSE QUESTIONSSTATISTICS STATISTICS SECTION SECTION Part A Part A Questions 1-5 Questions 1-5 Spend about 65 minutes on this part Spend about 65 minutes on this part of the exam.of the exam.Percent of Section II grade75Percen
44、t of Section II grade751.ThegoalofanutritionalstudywastocomparethecaloricintakeofadolescentsIrvinginruralareasoftheUnitedStateswiththecaloricintakeofadolescentslivinginurbanareasoftheUnitedStates.Arandomsampleofninth-gradestudentsfromonehighschoolinaruralareawasselected.Anotherrandomsampleofninthgra
45、dersfromonehighschoolinanurbanareawasalsoselected.Eachstudentineachsamplekeptrecordsofallthefoodheorsheconsumedinoneday.Theback-to-backstemplotbelowdisplaysthenumberofcaloriesoffoodconsumedperkilogramofbodyweightforeachstudentonthatday.(a)Writeafewsentencescomparingthedistributionofthedailycaloricin
46、takeofninth-gradestudentsintheruralhighschoolwiththedistributionofthedailycaloricintakeofninth-gradestudentsintheurbanhighschool.(b)Isitreasonabletogeneralizethefindingsofthisstudytoallruralandurbanninth-gradestudentsintheUnitedStates?Explain.Researcherswhowanttoconductasimilarstudyaredebatingwhicho
47、fthefollowingtwoplanstouse.PlanI:Haveeachstudentinthestudyrecordallthefoodheorsheconsumedinoneday.Thenresearcherswouldcomputethenumberofcaloriesoffoodconsumedperkilogramofbodyweightforeachstudentforthatday.PlanII:Haveeachstudentinthestudyrecordallthefoodheorsheconsumedoverthesame7-dayperiod.Thenrese
48、archerswouldcomputetheaveragedailynumberofcaloriesoffoodconsumedperkilogramofbodyweightforeachstudentduringthat7-dayperiod.Assumingthatthestudentskeepaccuraterecords,whichplan,IorII,wouldbettermeetthegoalofthestudy?Justifyyouranswer.2.LettherandomvariableX representthenumberoftelephonelinesinusebyth
49、etechnicalsupportcenterofasoftwaremanufactureratnooneachday.TheprobabilitydistributionofX isshowninthetablebelow.(a)Calculatetheexpectedvalue(themean)ofX.(b)Usingpastrecords,thestaffatthetechnicalsupportcenterrandomlyselected20daysandfoundthatanaverageof1.25telephonelineswereinuseatnoononthosedays.T
50、hestaffproposestoselectanotherrandomsampleof1,000daysandcomputetheaveragenumberoftelephonelinesthatwereinuseatnoononthosedays.Howdoyouexpecttheaveragefromthisnewsampletocomparetothatofthefirstsample?Justifyyourresponse.(c)Themedianofarandomvariableisdefinedasanyvaluex suchthatP(X 0.5andP(X x)0.5.For