AP Statistics – Chapter 3 Practice Quiz MODEL SOLUTIONS Multiple Choice: 1.Ciscorrect.Thisdescriptionaddressestheform(linear),direction(positive),andstrength(moderatelyweak)oftherelationship.Therearenoobservationsthatdonotfittheoverallpattern.Theotheroptionsarenotthebestchoicesbecauseadescriptionoftherelationshipbetweentwoquantitativevariablesshouldaddresstheform,direction,andstrengthoftherelationship,alongwithanyobservationthatdonotfittheoverallpattern.

2.Biscorrect.Theteacherwantstoknowiftextbookreadinghasanimpactongrades,sothereisaclearexplanatory-responserelationshipwheretextbookreadingistheexplanatoryvariableandAPStatsgradesaretheresponsevariable.

3.Discorrect.Choice(a)isincorrectbecausecorrelationdoesnotimplycausation.Theremaybeothervariablesinvolvedthatareassociatedwithtextbookreadingandgrades,suchasincreasedtimeforstudyingorahealthyhomeenvironmentthatisconducivetowardsstudying.Both(b)and(c)arecorrectbecause,whilethiscorrelationdoesnotestablishacausallinkbetweenthesetwovariables,itiscloseto+1,implyingastrongassociation.Thismaysuggestthatstudentsshouldconsiderincreasingtheamountoftimetheyspendreadingtheirtextbooks(butitdoesNOTmeanthatitWILLforsureincreasetheirgrades–otherfactorsareinvolved).

4.Aiscorrect.Theleast-squareslineisthelinethatmakesthesumofthesquaredresidualsassmallaspossible.Choice(b)isincorrectbecausetheleast-squareslineisbasedondistancesbetweenobservedvaluesandtheregressionline,butnottheperpendiculardistance(45degreestotheregressionline).(c)isincorrectbecauseitreferstothedistancesbetweentheexplanatoryvaluesinsteadoftheresponsevalues.

5.Biscorrect.Ifonelooksonlyattheo’scorrespondingtofemales,thereisacleardownwardtrendindicatinganegativecorrelationbetweenweightandtime.Thesameistrueforthe+’scorrespondingtomales.Choice(a)isincorrectbecausethecorrelationrmeasuresassociationbetweentwoquantitativevariables.Since“gender”isnotquantitative,risnotanappropriatemeasureofassociationbetweengenderandweight.Choice(c)isincorrectbecauseifthisstatementweretrue,thentheclusterof+’scorrespondingtomaleswouldliedistinctlybelowtheclusterofo’scorrespondingtofemales(thatis,the+’swouldtendtohavesmallery-coordinatesthantheo’s),clearlynotthecasehere.Choice(d)isincorrectbecausether–valueshouldbenegativesincethereisanegativeassociation(asweightincreases,timetoraisepulseto140bpmdecreases).Choice(e)isincorrectfortworeasons:1)wedon’thavealeast-squaresregressionlinegraphedorgiventous,and2)predictingthetimeittakesfora300lb.malewouldbeconsideredextrapolationsince300ismuchhigherthantheintervalof90poundsto180pounds.

Oops,thereisno#6.7.Biscorrect.Thenumberofsharksandbeachdeathsarebothquantitativevariables,andther=0.33isbetween-1and1anditisunit-less.Choice(a)isincorrectbecausezipcodesareactuallyacategoricalvariable,thereforewecannotdrawanassociationusingr.Choice(c)isincorrectbecausethecorrelationcoefficientshouldnothaveunits.Choice(d)isincorrectbecausewecannotconcludewhetheralinearmodelisappropriatewithoutlookingatthescatterplotandresidualplot,evenifther-valueappearstobeverycloseto1or-1.Choice(e)isincorrectbecause(d)isincorrect.

8.Ciscorrect.Choice(a)isincorrectbecauseyouhavemisinterpretedr2,theproportionofvariationintheresponsevariablethatcanbeexplainedbyregressionontheexplanatoryvariable,asr.Choice(b)isincorrectbecauseyouperformedthewrongoperationtotransformthegiveninformation(youshouldsquareroot0.81insteadofsquareit).Choice(d)isobviouslyincorrectforbeingnegativebecausetheassociationgivenintheproblemimpliesapositivecorrelationsinceincreasedstudyingisassociatedwithbetterscores.

9.Discorrect.Choice(IV)isthecorrectchoicebecausefootlengthiscorrectlyidentifiedastheexplanatoryvariableandheightiscorrectlyidentifiedasthepredictedresponsevariable,witha“hat”ontop.Additionally,theslopeiscorrectlyidentifiedas1.878andthey-interceptis117.99.Choice(I)isincorrectbecauseyshouldbe𝑦sincetheleast-squaresregressionlineisapredictionmodelandalsobecausethevariablesxand𝑦shouldbedefinedfortheequation.Choice(II)isincorrectforsimilarreasonsastowhy(I)isincorrect,aswellasthefactthattheslopeandy-interceptareswitched.Choice(III)isincorrectbecause,again,theslopeandy-interceptareswitched.Choice(V)isincorrectbecausetheexplanatoryandresponsevariablesareswitchedaswellastheslopeandy-intercept.

Free Response: 10.ThefollowingscatterplotwascreatedusingStapplet.com(twoquantitativevariablesapplet).11.Thereisalinear,negative,fairlystrongassociationbetweentimeandnumberofpeoplepayingattention.Therearenooutliersorunusualfeatures.

12a) Slope:𝑏 = 𝑟 ∙ !!

!!= −0.99 ∙ !".!"

!.!"= −9.39125 ≈ −9.39 𝑝𝑒𝑜𝑝𝑙𝑒 𝑝𝑒𝑟 ℎ𝑜𝑢𝑟

***UseyourL1andL2,LinReg,and2ndSTAT>MATH>StdDevoptionstofind𝑟,𝑠! ,and𝑠! . Y-intercept:𝑎 = 𝑦 − 𝑏𝑥 = 51.5714− −9.39125 3 = 79.74517857 ≈ 79.75LeastsquaresRegressionLine𝑦 = 𝑎 + 𝑏𝑥canbewritteneitherway:

• 𝑦 = 79.75− 9.39𝑥wherexrepresentstime(hours)and𝑦representspredictednumberofpeoplepayingattention(carefultonotforget“predicted”!)

• 𝑝𝑒𝑜𝑝𝑙𝑒 𝑝𝑎𝑦𝚤𝑛𝑔 𝑎𝑡𝑡𝑒𝑛𝑡𝚤𝑜𝑛 = 79.76− 9.39(𝑡𝑖𝑚𝑒)

12b)𝑦 = 79.75− 9.39𝑥isthesameaswhatwecalculatedbyhand.13)𝑦 = 79.75− 9.39 2.5 = 56.2678 ≈ 56people Thereareabout56peoplepredictedtobepayingattentionat2.5hours.14)𝑟 = −0.99 Thereisastrongandnegativeassociationbetweentimeandnumberofpeoplepayingattention.15)𝑟! = 0.98 98%ofthevariationinthenumberofpeoplepayingattentioncanbeaccountedforbytheleastsquaresregressionlinerelatingtimeandnumberofpeoplepayingattention.

16)slope=-9.39 Foreachadditionalhourthatpasses,thenumberofpeoplepayingattentionispredictedtodecreaseby9.39people.

17)y-intercept=79.75 Atthestartofthetraining(time=0hours),thepredictednumberofpeoplepayingattentionis79.75people.

18)No,youshouldnotmakeapredictionforthenumberofpeoplepayingattentionaftera13-hourconferencebecausethisisconsideredextrapolation.13hoursis7hoursabovetheintervalof0to6hoursusedtocalculatetheleastsquaresregressionline𝑦 = 79.75− 9.39𝑥,thereforeourpredictionwouldbeunreliablebecauseitcouldpossiblyoverestimateorunderestimatethepredictednumberofpeoplepayingattention.

19)TheMinitaboutputgivesusthesameinformationasabove. slope=-9.393 y-intercept=79.75 𝑦 = 79.75− 9.39𝑥wherexrepresentstime(hours)and𝑦representspredictednumberofpeople20)𝑆 = 3.143 Wewilltypically(oronaverage)beoffbyabout3.143peoplewhenweusetheleastsquaresregressionlinetopredictthenumberofpeoplepayingattentionfromthetimepassed.

21)𝑦 = 79.75− 9.39 4 = 42.178people***IusedY1(4)inmycalculator 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 = 𝑦 − 𝑦 = 48− 42.178 = 5.82 Theactualnumberofpeoplepayingattentionafter4hoursis5.81peoplehigherthanpredictedbytheleastsquaresregressionline.

22)Alinearmodelisappropriatetobecausethereisnoleftoverpattern(orrandomscatter)inthe

residualplot.

23) Yes,MichaelJordanwalkingintothetrainingwouldbeinfluentialbecausethepoint(7,82)liesoutsidetheoverallpatternofobservations.

Beforeweaddedthisnewpoint(7,82),theleastsquaresregressionlinewas𝑦 = 79.75− 9.39𝑥andhadanr-valueof-0.99.Afteraddingthepoint(7,82),thelinechangedsignificantlyto𝑦 = 68.42− 3.73𝑥andther–valueweakenedto-0.418.

Sincetheleastsquaresregressionlineisnonresistanttoinfluentialobservations,theslopedrasticallyincreasedfrom-9.39to-3.73,andthey-interceptdecreasedfrom79.75to68.42.

Thecorrelationcoefficientisalsononresistanttoinfluentialpoints.Thecorrelationbetweentimeandpeoplepayingattentionweakenedbecausethecorrelationcoefficientr=-0.418iscloserto0thanr=-0.99.Thishappenedbecausethepoint(7,82)doesnotfollowthelineartrend.