统计学精讲精练主讲老师：黄逸昆2004APSTATISTICSFREE-RESPONSEQUESTIONS(FORMB)STATISTICSSECTIONⅡPartAQuestions1-5Spendabout65minutesonthispartoftheexam.PercentofSectionIIgrade—751TheEarth'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.3TrainscarrybauxiteorefromamineinCanadatoanaluminumprocessingplantinnorthernNewYorkstateinhoppercars.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)Anassistantprincipalreasonedthatamuchnarrowerconfidenceintervalcouldbeobtainedifthestudentswerepairedbasedo...
