《AP统计学讲义》PPT课件.ppt

统计学精讲精练 主讲老师 黄逸昆 2004APSTATISTICSFREE RESPONSEQUESTIONS FORMB STATISTICSSECTION PartAQuestions1 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 STATISTICSSECTION PartBQuestions6Spendabout25minutesonthispartoftheexam 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 2006APSTATISQUESTIONSSTATISTICSSECTION PartAQuestions1 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 RESPONSEQUESTIONSSTATISTICSSECTION PartBQuestions6Spendabout25minutesonthispartoftheexam 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 RESPONSEQUESTIONSSTATISTICSSECTION PartAQuestions1 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 X0 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 DoesnothaveahighschooldiplomaHasahighschooldiplomaThesurveywillbeconductedusingatelephoneinterview Random digitdialingwillbeusedtoselectthesample a Forthissurvey stateonepotentialsourceofbiasanddescribehowitmightaffecttheestimateoftheproportionofadultheadsofhouseholdsintheUnitedStateswhodonothaveahighschooldiploma b Apilotsurveyindicatedthatabout22percentofthepopulationofadultheadsofhouseholdsdonothaveahighschooldiploma Usingthisinformation howmanyrespondentsshouldbeobtainedifthegoalofthesurveyistoestimatetheproportionofthepopulationwhodonothaveahighschooldiplomatowithin0 03with95percentconfidence Justifyyouranswer c Sinceeducationislargelytheresponsibilityofeachstate theagencywantstobesurethatestimatesareavailableforeachstateaswellasforthenation Identifyasamplingmethodthatwillachievethisadditionalgoalandbrieflydescribeawaytoselectthesurveysampleusingthismethod 2005APSTATISTICSFREE RESPONSEQUESTIONSSTATISTICSSECTION PartBQuestions6Spendabout25minutesonthispartoftheexam 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 urban andtherelationshipbetweenthetwoontheamountofleadonthedominanthandofchildrenafterplayinthiscity Justifyyouranswer Thanks