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Chapter1 EconomicQuestionsandData1.1 MultipleChoice

1) AnalyzingthebehaviorofunemploymentratesacrossU.S.statesinMarchof2006isanexampleofusingA) timeseriesdata.B) paneldata.C) cross-sectionaldata.D) experimentaldata.

Answer: C

2) StudyinginflationintheUnitedStatesfrom1970to2006isanexampleofusingA) randomizedcontrolledexperiments.B) timeseriesdata.C) paneldata.D) cross-sectionaldata.

Answer: B

3) Analyzingtheeffectofminimumwagechangesonteenageemploymentacrossthe48contiguousU.S.statesfrom1980to2004isanexampleofusing

A) timeseriesdata.B) paneldata.C) havingatreatmentgroupvs.acontrolgroup,sinceonlyteenagersreceiveminimumwages.D) cross-sectionaldata.

Answer: B

4) PaneldataA) isalsocalledlongitudinaldata.B) isthesameastimeseriesdata.C) studiesagroupofpeopleatapointintime.D) typicallyusescontrolandtreatmentgroups.

Answer: A

5) EconometricscanbedefinedasfollowswiththeexceptionofA) thescienceoftestingeconomictheory.B) fittingmathematicaleconomicmodelstoreal-worlddata.C) asetoftoolsusedforforecastingfuturevaluesofeconomicvariables.D) measuringtheheightofeconomists.

Answer: D

6) ToprovidequantitativeanswerstopolicyquestionsA) itistypicallysufficienttousecommonsense.B) youshouldinterviewthepolicymakersinvolved.C) youshouldexamineempiricalevidence.D) istypicallyimpossiblesincepolicyquestionsarenotquantifiable.

Answer: C

7) AnexampleofarandomizedcontrolledexperimentiswhenA) householdsreceiveataxrebateinoneyearbutnottheother.B) oneU.S.stateincreasesminimumwagesandanadjacentstatedoesnot,andemploymentdifferencesare

observed.C) randomvariablesarecontrolledforbyholdingconstantotherfactors.D) some5thgradersinaspecificelementaryschoolareallowedtousecomputersatschoolwhileothersare

not,andtheirend-of-yearperformanceiscomparedholdingconstantotherfactors.Answer: D

Stock/Watson2e--CVC28/23/06-- Page1

8) IdealrandomizedcontrolledexperimentsineconomicsareA) oftenperformedinpractice.B) oftenusedbytheFederalReservetostudytheeffectsofmonetarypolicy.C) usefulbecausetheygiveadefinitionofacausaleffect.D) sometimesusedbyuniversitiestodeterminewhograduatesinfouryearsratherthanfive.

Answer: C

9) MosteconomicdataareobtainedA) throughrandomizedcontrolledexperiments.B) bycalibrationmethods.C) throughtextbookexamplestypicallyinvolvingtenobservationpoints.D) byobservingreal-worldbehavior.

Answer: D

10) Oneoftheprimaryadvantagesofusingeconometricsovertypicalresultsfromeconomictheory,isthatA) itpotentiallyprovidesyouwithquantitativeanswersforapolicyproblemratherthansimplysuggesting

thedirection(positive/negative)oftheresponse.B) teachingyouhowtousestatisticalpackagesC) learninghowtoinverta4by4matrix.D) alloftheabove.

Answer: A

11) InarandomizedcontrolledexperimentA) thereisacontrolgroupandatreatmentgroup.B) youcontrolfortheeffectthatrandomnumbersarenottrulyrandomlygeneratedC) youcontrolforrandomanswersD) thecontrolgroupreceivestreatmentonevendaysonly.

Answer: A

12) Thereasonwhyeconomistsdonotuseexperimentaldatamorefrequentlyisforallofthefollowingreasonsexceptthatreal-worldexperiments

A) cannotbeexecutedineconomics.B) withhumansaredifficulttoadminister.C) areoftenunethical.D) haveflawsrelativetoidealrandomizedcontrolledexperiments.

Answer: A

13) Themostfrequentlyusedexperimentalorobservationaldataineconometricsareofthefollowingtype:A) cross-sectionaldata.B) randomlygenerateddata.C) timeseriesdata.D) paneldata.

Answer: A


14) Inthegraphbelow,theverticalaxisrepresentsaveragerealGDPgrowthfor65countriesovertheperiod1960-1995,andthehorizontalaxisshowstheaveragetradesharewithinthesecountries.

ThisisanexampleofA) cross-sectionaldata.B) experimentaldata.C) atimeseries.D) longitudinaldata.

Answer: A


15) Theaccompanyinggraph

Isanexampleof

A) cross-sectionaldata.B) experimentaldata.C) atimeseries.D) longitudinaldata.

Answer: A


16) Theaccompanyinggraph

isanexampleofA) experimentaldata.B) cross-sectionaldata.C) atimeseries.D) longitudinaldata.

Answer: C

1.2 Essays1) Giveatleastthreeexamplesfromeconomicswhereeachofthefollowingtypeofdatacanbeused:

cross-sectionaldata,timeseriesdata,andpaneldata.Answer: Answerswillvarybystudent.Atthislevelofeconomics,studentsmostlikelyhaveheardofthe

followinguseofcross-sectionaldata:earningsfunctions,growthequations,theeffectofclasssizereductiononstudentperformance(inthischapter),demandfunctions(inthischapter:cigaretteconsumption);timeseries:thePhillipscurve(inthischapter),consumptionfunctions,Okunslaw;paneldata:variousU.S.statepanelstudiesonroadfatalities(inthisbook),unemploymentrateandunemploymentbenefitsvariations,growthregressions(acrossstatesandcountries),andcrimeandabortion(Freakonomics).


Chapter2 ReviewofProbability2.1 MultipleChoice

1) TheprobabilityofanoutcomeA) isthenumberoftimesthattheoutcomeoccursinthelongrun.B) equalsMN,whereMisthenumberofoccurrencesandN isthepopulationsize.C) istheproportionoftimesthattheoutcomeoccursinthelongrun.D) equalsthesamplemeandividedbythesamplestandarddeviation.

Answer: C

2) TheprobabilityofaneventAorB(Pr(A orB))tooccurequalsA) Pr(A)Pr(B).B) Pr(A)+Pr(B)ifAandBaremutuallyexclusive.

C) Pr(A)Pr(B)

.

D) Pr(A)+Pr(B)evenifAandBarenotmutuallyexclusive.Answer: B

3) ThecumulativeprobabilitydistributionshowstheprobabilityA) thatarandomvariableislessthanorequaltoaparticularvalue.B) oftwoormoreeventsoccurringatonce.C) ofallpossibleeventsoccurring.D) thatarandomvariabletakesonaparticularvaluegiventhatanothereventhashappened.

Answer: A

4) TheexpectedvalueofadiscreterandomvariableA) istheoutcomethatismostlikelytooccur.B) canbefoundbydeterminingthe50%valueinthec.d.f.C) equalsthepopulationmedian.D) iscomputedasaweightedaverageofthepossibleoutcomeofthatrandomvariable,wheretheweights

aretheprobabilitiesofthatoutcome.Answer: D

5) LetYbearandomvariable.Thenvar(Y)equals

A) E[Y-Y)2].

B) E (Y-Y) .

C) E (Y-Y)2 .

D) E (Y-Y) .

Answer: C


6) TheskewnessofthedistributionofarandomvariableY isdefinedasfollows:

A)E (Y3-Y)

2Y

B) E (Y-Y)3

C)E Y3- 3Y

3Y

D)E (Y-Y)

3

3Y

Answer: D

7) Theskewnessismostlikelypositiveforoneofthefollowingdistributions:A) Thegradedistributionatyourcollegeoruniversity.B) TheU.S.incomedistribution.C) SATscoresinEnglish.D) Theheightof18yearoldfemalesintheU.S.

Answer: B

8) Thekurtosisofadistributionisdefinedasfollows:

A)E Y-Y

4

4Y

B)E Y4- 4Y

2Y

C) skewnessvar(Y)

D) E[(Y-Y)4)

Answer: A

9) Foranormaldistribution,theskewness andkurtosismeasuresareasfollows:A) 1.96and4B) 0and0C) 0and3D) 1and2

Answer: C


10) TheconditionaldistributionofYgivenX = x,Pr(Y = y X=x),is

A) Pr(Y=y)Pr(X=x)

.

B)l

i=1Pr(X=xi,Y=y).

C) Pr(X=x,Y=y)Pr(Y=y)

D) Pr(X=x,Y=y)Pr(X=x)

.

Answer: D

11) TheconditionalexpectationofYgivenX,E(Y X= x),iscalculatedasfollows:

A)k

i=1YiPr(X=xi Y=y)

B) E E(Y X)]

C)k

i=1yiPr(Y=yi X=x)

D)l

i=1E(Y X=xi) Pr(X=xi)

Answer: C

12) TworandomvariablesXandYareindependentlydistributedifallofthefollowingconditionshold,withtheexceptionof

A) Pr(Y=y X=x)=Pr(Y=y).B) knowingthevalueofoneofthevariablesprovidesnoinformationabouttheother.C) iftheconditionaldistributionofY givenX equalsthemarginaldistributionofY.D) E(Y)=E[E(Y X)].

Answer: D

13) ThecorrelationbetweenXandYA) cannotbenegativesincevariancesarealwayspositive.B) isthecovariancesquared.C) canbecalculatedbydividingthecovariancebetweenX andY bytheproductofthetwostandard

deviations.

D) isgivenbycorr(X,Y)= cov(X,Y)var(X)var(Y)

.

Answer: C

14) Twovariablesareuncorrelatedinallofthecasesbelow,withtheexceptionofA) beingindependent.B) havingazerocovariance.

C) XY 2X

2Y .

D) E(Y X)=0.Answer: C


15) var(aX+bY)=

A) a2 2X+b2 2Y .

B) a2 2X+2abXY+b2 2Y .

C) XY+XY.

D) a 2X +b2Y .

Answer: B

16) TostandardizeavariableyouA) subtractitsmeananddividebyitsstandarddeviation.B) integratetheareabelowtwopointsunderthenormaldistribution.C) addandsubtract1.96timesthestandarddeviationtothevariable.D) divideitbyitsstandarddeviation,aslongasitsmeanis1.

Answer: A

17) AssumethatYisnormallydistributedN(,2).Movingfromthemean()1.96standarddeviationstotheleftand1.96standarddeviationstotheright,thentheareaunderthenormalp.d.f.is

A) 0.67B) 0.05C) 0.95D) 0.33

Answer: C

18) AssumethatYisnormallydistributedN(,2).TofindPr(c1Yc2),wherec1

21) Whentherearedegreesoffreedom,thet distribution

A) cannolongerbecalculated.B) equalsthestandardnormaldistribution.C) hasabellshapesimilartothatofthenormaldistribution,butwithfattertails.

D) equalsthe 2distribution.

Answer: B

22) ThesampleaverageisarandomvariableandA) isasinglenumberandasaresultcannothaveadistribution.B) hasaprobabilitydistributioncalleditssamplingdistribution.C) hasaprobabilitydistributioncalledthestandardnormaldistribution.D) hasaprobabilitydistributionthatisthesameasfortheY1,...,Yn i.i.d.variables.

Answer: B

23) Toinferthepoliticaltendenciesofthestudentsatyourcollege/university,yousample150ofthem.Onlyoneofthefollowingisasimplerandomsample:You

A) makesurethattheproportionofminoritiesarethesameinyoursampleasintheentirestudentbody.

B) calleveryfiftiethpersoninthestudentdirectoryat9a.m.Ifthepersondoesnotanswerthephone,youpickthenextnamelisted,andsoon.

C) gotothemaindininghalloncampusandinterviewstudentsrandomlythere.D) haveyourstatisticalpackagegenerate150randomnumbersintherangefrom1tothetotalnumberof

studentsinyouracademicinstitution,andthenchoosethecorrespondingnamesinthestudenttelephonedirectory.

Answer: D

24) ThevarianceofY, 2Y ,isgivenbythefollowingformula:

A) 2Y .

B)Yn.

C) 2Y

n.

D) 2Y

n.

Answer: C


25) ThemeanofthesampleaverageY,E(Y),is

A) 1nY.

B) Y.

C)Yn.

D)YY

forn>30.

Answer: B

26) Ineconometrics,wetypicallydonotrelyonexactorfinitesampledistributionsbecauseA) wehaveapproximatelyaninfinitenumberofobservations(thinkofre-sampling).B) variablestypicallyarenormallydistributed.C) thecovariancesofYi,Yjaretypicallynotzero.D) asymptoticdistributionscanbecountedontoprovidegoodapproximationstotheexactsampling

distribution(giventhenumberofobservationsavailableinmostcases).Answer: D

27) ConsistencyforthesampleaverageYcanbedefinedasfollows,withtheexceptionofA) YconvergesinprobabilitytoY.

B) Yhasthesmallestvarianceofallestimators.

C) YpY.

D) theprobabilityofYbeingintherangeYcbecomesarbitrarilyclosetooneasnincreasesforany

constantc>0.Answer: B

28) Thecentrallimittheoremstatesthat

A) thesamplingdistributionofY-YY

isapproximatelynormal.

B) YpY.

C) theprobabilitythatYisintherangeYcbecomesarbitrarilyclosetooneasnincreasesforanyconstant

c>0.D) thetdistributionconvergestotheF distributionforapproximatelyn > 30.

Answer: A

29) ThecentrallimittheoremA) statesconditionsunderwhichavariableinvolvingthesumof Y1,...,Yn i.i.d.variablesbecomesthe

standardnormaldistribution.B) postulatesthatthesamplemeanYisaconsistentestimatorofthepopulationmeanY.

C) onlyholdsinthepresenceofthelawoflargenumbers.D) statesconditionsunderwhichavariableinvolvingthesumofY1,...,Yni.i.d.variablesbecomesthe

Studenttdistribution.Answer: A


30) Thecovarianceinequalitystatesthat

A) 0 2XY

1.

B) 2XY

2X 2Y.

C) 2XY

- 2X 2

Y.

D) 2XY

2X

2Y

.

Answer: B

31)n

i=1(axi+byi+c)=

A) an

i=1

xi +bn

i=1

yi +nc

B) an

i=1

xi +bn

i=1

yi +c

C) ax+by+nc

D) an

i=1

xi +bn

i=1

yi

Answer: A

32) n

i=1(axi+b)

A) nax+ nbB) n(a+b)C)

D)Answer: A


33) Assumethatyouassignthefollowingsubjectiveprobabilitiesforyourfinalgradeinyoureconometricscourse(thestandardGPAscaleof4=Ato0=Fapplies):

Grade ProbabilityA 0.20B 0.50C 0.20D 0.08F 0.02

Theexpectedvalueis:

A) 3.0B) 3.5C) 2.78D) 3.25

Answer: C

34) ThemeanandvarianceofaBernoillerandomvariablearegivenasA) cannotbecalculatedB) npandnp(1-p)C) pand p(1-p)D) pand(1-p)

Answer: D

35) Considerthefollowinglineartransformationofarandomvariabley=x-xx

wherexisthemeanofxandx

isthestandarddeviation.ThentheexpectedvalueandthestandarddeviationofYaregivenasA) 0and1B) 1and1C) CannotbecomputedbecauseYisnotalinearfunctionofX

D) x

andx

Answer: A


2.2 EssaysandLongerQuestions1) ThinkofthesituationofrollingtwodiceandletM denotethesumofthenumberofdotsonthetwodice.(SoM

isanumberbetween1and12.)(a) Inatable,listallofthepossibleoutcomesfortherandomvariableMtogetherwithitsprobabilitydistributionandcumulativeprobabilitydistribution.Sketchbothdistributions.(b) CalculatetheexpectedvalueandthestandarddeviationforM.(c) Lookingatthesketchoftheprobabilitydistribution,younoticethatitresemblesanormaldistribution.Shouldyoubeabletousethestandardnormaldistributiontocalculateprobabilitiesofevents?Whyorwhynot?Answer: (a)

Outcome 2 3 4 5 6 7 8 9 10 11 12(sumofdots)Probability 0.0280.0560.0830.1110.1390.1670.1390.1110.0830.0560.028distributionCumulative0.0280.0830.1670.2780.4170.5830.7220.8330.9120.9721.000probabilitydistribution

(b)7.0;2.42.(c)Youcannotusethenormaldistribution(withoutcontinuitycorrection)tocalculateprobabilitiesofevents,sincetheprobabilityofanyeventequalszero.


2) Whatistheprobabilityofthefollowingoutcomes?(a) Pr(M=7)(b) Pr(M=2orM=10)(c) Pr(M=4orM4)(d) Pr(M=6andM=9)(e) Pr(M10)

Answer: (a) 0.167or 636

=16;

(b) 0.111or 439

=19;

(c) 1;(d) 0;(e) 0.583;

(f) 0.222or 836

=29.

3) Probabilitiesandrelativefrequenciesarerelatedinthattheprobabilityofanoutcomeistheproportionofthetimethattheoutcomeoccursinthelongrun.Henceconceptsofjoint,marginal,andconditionalprobabilitydistributionsstemfromrelatedconceptsoffrequencydistributions.

Youareinterestedininvestigatingtherelationshipbetweentheageofheadsofhouseholdsandweeklyearningsofhouseholds.Theaccompanyingdatagivesthenumberofoccurrencesgroupedbyageandincome.Youcollectdatafrom1,744individualsandthinkoftheseindividualsasapopulationthatyouwanttodescribe,ratherthanasamplefromwhichyouwanttoinferbehaviorofalargerpopulation.Aftersortingthedata,yougeneratetheaccompanyingtable:

JointAbsoluteFrequenciesofAgeandIncome,1,744Households

Ageofheadofhousehold X1 X2 X3 X4 X5HouseholdIncome 16-under20 20-under25 25-under45 45-under65 65and>Y1$0-under$200 80 76 130 86 24

Y2$200-under$400 13 90 346 140 8

Y3$400-under$600 0 19 251 101 6

Y4$600-under$800 1 11 110 55 1

Y5$800and> 1 1 108 84 2

Themedianoftheincomegroupof$800andaboveis$1,050.

(a)Calculatethejointrelativefrequenciesandthemarginalrelativefrequencies.Interpretoneofeachofthese.Sketchthecumulativeincomedistribution.(b)Calculatetheconditionalrelativeincomefrequenciesforthetwoagecategories16-under20,and45-under65.Calculatethemeanhouseholdincomeforbothagecategories.(c)Ifhouseholdincomeandageofheadofhouseholdwereindependentlydistributed,whatwouldyouexpectthesetwoconditionalrelativeincomedistributionstolooklike?Aretheysimilarhere?(d)Yourtextbookhasgivenyouaprimarydefinitionofindependencethatdoesnotinvolveconditionalrelativefrequencydistributions.Whatisthatdefinition?Doyouthinkthatageandincomeareindependenthere,usingthisdefinition?


Answer: (a) Thejointrelativefrequenciesandmarginalrelativefrequenciesaregivenintheaccompanyingtable.5.2percentoftheindividualsarebetweentheageof20and24,andmakebetween$200andunder$400.21.6percentoftheindividualsearnbetween$400andunder$600.

JointRelativeandMarginalFrequenciesofAgeandIncome,1,744Households

AgeofheadofhouseholdX1 X2 X3 X4 X5

HouseholdIncome 16-under2020-under2525-under4545-under6565and>TotalY1$0-under$2000.046 0.044 0.075 0.049 0.014 0.227Y2$200-under$4000.007 0.052 0.198 0.080 0.005 0.342Y3$400-under$6000.000 0.011 0.144 0.058 0.003 0.216Y4$600-under$8000.001 0.006 0.063 0.032 0.001 0.102Y5$800and>0.001 0.001 0.062 0.048 0.001 0.112

(b) Themeanhouseholdincomeforthe16-under20agecategoryisroughly$144.Itisapproximately$489forthe45-under65agecategory.

ConditionalRelativeFrequenciesofIncomeandAge16-under20,and45-under65,1,744Households

AgeofheadofhouseholdX1 X4

HouseholdIncome 16-under2045-under65Y1$0-under$2000.842 0.185Y2$200-under$4000.137 0.300


Y3$400-under$6000.000 0.217Y4$600-under$8000.001 0.118Y5$800and>0.001 0.180

(c)Theywouldhavetobeidentical,whichtheyclearlyarenot.(d)Pr(Y=y,X=x)=Pr(Y=y)Pr(X=x).Wecancheckthisbymultiplyingtwomarginalprobabilitiestoseeifthisresultsinthejointprobability.Forexample,Pr(Y=Y3)=0.216andPr(X=X3)=0.542,resultinginaproductof0.117,whichdoesnotequalthejointprobabilityof0.144.Giventhatwearelookingatthedataasapopulation,notasample,wedonothavetotesthowclose0.117isto0.144.

4) MathandverbalSATscoresareeachdistributednormallywithN(500,10000).(a)Whatfractionofstudentsscoresabove750?Above600?Between420and530?Below480?Above530?(b)Ifthemathandverbalscoreswereindependentlydistributed,whichisnotthecase,thenwhatwouldbethedistributionoftheoverallSATscore?Finditsmeanandvariance.(c)Next,assumethatthecorrelationcoefficientbetweenthemathandverbalscoresis0.75.Findthemeanandvarianceoftheresultingdistribution.(d)Finally,assumethatyouhadchosen25studentsatrandomwhohadtakentheSATexam.DerivethedistributionfortheiraveragemathSATscore.Whatistheprobabilitythatthisaverageisabove530?Whyisthissomuchsmallerthanyouranswerin(a)?Answer: (a)Pr(Y>750)=0.0062;Pr(Y>600)= 0.1587;Pr(420

Answer: (a)TestPositive(Y=1) TestNegative(Y=0) Total

HIV(X=1) 10,000NoHIV(X=0) 9,990,000Total 10,000,000

(b)TestPositive(Y=1) TestNegative(Y=0) Total

HIV(X=1) 9,500 500 10,000NoHIV(X=0) 499,500 9,490,500 9,990,000Total 10,000,000

(c)TestPositive(Y=1) TestNegative(Y=0) Total

HIV(X=1) 9,500 500 10,000NoHIV(X=0) 499,500 9,490,500 9,990000Total 509,000 9,491,000 10,000,000

Pr(X=1 Y=1)=0.0187.Althoughthetestisquiteaccurate,thereareveryfewpeoplewhohaveHIV(10,000),andmanywhodonothaveHIV(9,999,000).Asmallpercentageofthatlargenumber(499,500/9,990,000)islargewhencomparedtothehigherpercentageofthesmallernumber(9,500/10,000).d.Answerswillvarybystudent.Perhapsaniceillustrationistheprobabilitytobeamalegiventhatyouplayonthecollege/universitymensvarsityteam,versustheprobabilitytoplayonthecollege/universitymensvarsityteamgiventhatyouareamalestudent.


6) Youhavereadabouttheso-calledcatch-uptheorybyeconomichistorians,wherebynationsthatarefurtherbehindinpercapitaincomegrowfastersubsequently.Ifthisistruesystematically,theneventuallylaggardswillreachtheleader.Toputthetheorytothetest,youcollectdataonrelative(totheUnitedStates)percapitaincomefortwoyears,1960and1990,for24OECDcountries.Youthinkofthesecountriesasapopulationyouwanttodescribe,ratherthanasamplefromwhichyouwanttoinferbehaviorofalargerpopulation.Therelevantdataforthisquestionisasfollows:

Y X1 X2 YX1 Y2 X21 X

22

0.023 0.770 1.030 0.018 0.00053 0.593 1.06090.014 1.000 1.000 0.014 0.00020 1.000 1.0000. . . . . . .0.041 0.200 0.450 0.008 0.00168 0.040 0.20250.033 0.130 0.230 0.004 0.00109 0.017 0.05290.625 13.220 17.800 0.294 0.01877 8.529 13.9164

whereX1andX2arepercapitaincomerelativetotheUnitedStatesin1960and1990respectively,andYistheaverageannualgrowthrateinXoverthe1960-1990period.Numbersinthelastrowrepresentsumsofthecolumnsabove.(a)CalculatethevarianceandstandarddeviationofX1andX2.Foracatch-upeffecttobepresent,whatrelationshipmustthetwostandarddeviationsshow?Isthisthecasehere?(b)CalculatethecorrelationbetweenYand.Whatsignmustthecorrelationcoefficienthavefortheretobeevidenceofacatch-upeffect?Explain.Answer: (a)ThevariancesofX1andX2 are0.0520and0.0298respectively,withstandarddeviationsof0.2279

and0.1726.Forthecatch-upeffecttobepresent,thestandarddeviationwouldhavetoshrinkovertime.Thisisthecasehere.(b)Thecorrelationcoefficientis0.88.Ithastobenegativefortheretobeevidenceofacatch-upeffect.Ifcountriesthatwererelativelyaheadintheinitialperiodandintermsofpercapitaincomegrowbyrelativelylessovertime,theneventuallythelaggardswillcatch-up.

7) FollowingAlfredNobelswill,therearefiveNobelPrizesawardedeachyear.TheseareforoutstandingachievementsinChemistry,Physics,PhysiologyorMedicine,Literature,andPeace.In1968,theBankofSwedenaddedaprizeinEconomicSciencesinmemoryofAlfredNobel.Youthinkofthedataasdescribingapopulation,ratherthanasamplefromwhichyouwanttoinferbehaviorofalargerpopulation.Theaccompanyingtableliststhejointprobabilitydistributionbetweenrecipientsineconomicsandtheotherfiveprizes,andthecitizenshipoftherecipients,basedonthe1969-2001period.

JointDistributionofNobelPrizeWinnersinEconomicsandNon-EconomicsDisciplines,andCitizenship,1969-2001

U.S.Citizen(Y=0)

Non=U.S.Citizen(Y=1)

Total

EconomicsNobelPrize(X=0)

0.118 0.049 0.167

Physics,Chemistry,Medicine,Literature,andPeaceNobelPrize(X=1)

0.345 0.488 0.833

Total 0.463 0.537 1.00

(a)ComputeE(Y)andinterprettheresultingnumber.(b)CalculateandinterpretE(Y X=1)andE(Y X=0).


(c)ArandomlyselectedNobelPrizewinnerreportsthatheisanon-U.S.citizen.WhatistheprobabilitythatthisgeniushaswontheEconomicsNobelPrize?ANobelPrizeintheotherfivedisciplines?(d)Showwhatthejointdistributionwouldlooklikeifthetwocategorieswereindependent.Answer: (a)E(Y)=0.53.7.53.7percentofNobelPrizewinnerswerenon-U.S.citizens.

(b)E(Y X=1)=0.586.58.6percentofNobelPrizewinnersinnon-economicsdisciplineswerenon-U.S.citizens.E(Y X=0)=0.293.29.3percentoftheEconomicsNobelPrizewinnerswerenon-U.S.citizens.(c)Thereisa9.1percentchancethathehaswontheEconomicsNobelPrize,anda90.9percentchancethathehaswonaNobelPrizeinoneoftheotherfivedisciplines.(d)JointDistributionofNobelPrizeWinnersinEconomicsandNon-EconomicsDisciplines,

andCitizenship,1969-2001,underassumptionofindependence

U.S.Citizen(Y=0)

Non=U.S.Citizen(Y=1)

Total

EconomicsNobelPrize(X=0)

0.077 0.090 0.167

Physics,Chemistry,Medicine,Literature,andPeaceNobelPrize(X=1)

0.386 0.447 0.833

Total 0.463 0.537 1.00

8) AfewyearsagothenewsmagazineTheEconomist listedsomeofthestrangerexplanationsusedinthepasttopredictpresidentialelectionoutcomes.Theseincludedwhetherornotthehemlinesofwomensskirtswentupordown,stockmarketperformances,baseballWorldSerieswinsbyanAmericanLeagueteam,etc.Thinkingaboutthisproblemmoreseriously,youdecidetoanalyzewhetherornotthepresidentialcandidateforacertainpartydidbetterifhispartycontrolledthehouse.Accordinglyyoucollectdataforthelast34presidentialelections.Youthinkofthisdataascomprisingapopulationwhichyouwanttodescribe,ratherthanasamplefromwhichyouwanttoinferbehaviorofalargerpopulation.Yougeneratetheaccompanyingtable:

JointDistributionofPresidentialPartyAffiliationandPartyControlofHouseofRepresentatives,1860-1996

DemocraticControlofHouse(Y=0)

RepublicanControlofHouse(Y=1)

Total

DemocraticPresident(X=0)

0.412 0.030 0.441

RepublicanPresident(X=1)

0.176 0.382 0.559

Total 0.588 0.412 1.00

(a)Interpretoneofthejointprobabilitiesandoneofthemarginalprobabilities.(b)ComputeE(X).HowdoesthisdifferfromE(XY=0)?Explain.(c)IfyoupickedoneoftheRepublicanpresidentsatrandom,whatistheprobabilitythatduringhistermtheDemocratshadcontroloftheHouse?(d)Whatwouldthejointdistributionlooklikeunderindependence?Checkyourresultsbycalculatingthetwoconditionaldistributionsandcomparethesetothemarginaldistribution.


Answer: (a)38.2percentofthepresidentswereRepublicansandwereintheWhiteHousewhileRepublicanscontrolledtheHouseofRepresentatives.44.1percentofallpresidentswereDemocrats.(b)E(X)=0.559.E(XY=0)=0.701.E(X)givesyoutheunconditionalexpectedvalue,whileE(XY=0)istheconditionalexpectedvalue.(c)E(X)=0.559.55.9percentofthepresidentswereRepublicans.E(XY=0)=0.299.29.9percentofthosepresidentswhowereinofficewhileDemocratshadcontroloftheHouseofRepresentativeswereRepublicans.ThesecondconditionsonthoseperiodsduringwhichDemocratshadcontroloftheHouseofRepresentatives,andignorestheotherperiods.(d)JointDistributionofPresidentialPartyAffiliationandPartyControlofHouseof

Representatives,1860-1996,undertheAssumptionofIndependence

DemocraticControlofHouse(Y=0)

RepublicanControlofHouse(Y=1)

Total

DemocraticPresident(X=0)

0.259 0.182 0.441

RepublicanPresident(X=1)

0.329 0.230 0.559

Total 0.588 0.412 1.00

Pr(X=0 Y=0)=0.2590.588

=0.440(thereisasmallroundingerror).

Pr(Y=1 X=1)=0.2300.559

=0.411(thereisasmallroundingerror).

9) TheexpectationsaugmentedPhillipscurvepostulates

p=f(uu),

wherepistheactualinflationrate,istheexpectedinflationrate,anduistheunemploymentrate,withindicatingequilibrium(theNAIRUNon-AcceleratingInflationRateofUnemployment).Undertheassumptionofstaticexpectations(=p1),i.e.,thatyouexpectthisperiodsinflationratetoholdforthenextperiod(thesunshinestoday,itwillshinetomorrow),thenthepredictionisthatinflationwillaccelerateiftheunemploymentrateisbelowitsequilibriumlevel.Theaccompanyingtablebelowdisplaysinformationonacceleratingannualinflationandunemploymentratedifferencesfromtheequilibriumrate(cyclicalunemployment),wherethelatterisapproximatedbyafive-yearmovingaverage.Youthinkofthisdataasapopulationwhichyouwanttodescribe,ratherthanasamplefromwhichyouwanttoinferbehaviorofalargerpopulation.ThedataiscollectedfromUnitedStatesquarterlydatafortheperiod1964:1to1995:4.

JointDistributionofAcceleratingInflationandCyclicalUnemployment,1964:1-1995:4

(uu)>0(Y=0)

(uu)0(Y=1)

Total

pp1>0(X=0)

0.156 0.383 0.539

pp10(X=1)

0.297 0.164 0.461

Total 0.453 0.547 1.00

(a)ComputeE(Y)andE(X),andinterpretbothnumbers.(b)CalculateE(Y X=1)andE(Y X=0).Iftherewasindependencebetweencyclicalunemploymentandaccelerationintheinflationrate,whatwouldyouexpecttherelationshipbetweenthetwoexpectedvaluesto


be?Giventhatthetwomeansaredifferent,isthissufficienttoassumethatthetwovariablesareindependent?(c)Whatistheprobabilityofinflationtoincreaseifthereispositivecyclicalunemployment?Negativecyclicalunemployment?(d)Yourandomlyselectoneofthe59quarterswhentherewaspositivecyclicalunemployment((uu)>0).Whatistheprobabilitytherewasdeceleratinginflationduringthatquarter?Answer: (a)E(Y)=0.547.54.7percentofthequarterssawcyclicalunemployment.

E(Y)=0.461.46.1percentofthequarterssawdecreasinginflationrates.(b)E(Y X=1)=0.356;E(Y X=0)=0.711.Youwouldexpectthetwoconditionalexpectationstobethesame.Ingeneral,independenceinmeansdoesnotimplystatisticalindependence,althoughthereverseistrue.(c)Thereisa34.4percentprobabilityofinflationtoincreaseifthereispositivecyclicalunemployment.Thereisa70percentprobabilityofinflationtoincreaseifthereisnegativecyclicalunemployment.(d)Thereisa65.6percentprobabilityofinflationtodeceleratewhenthereispositivecyclicalunemployment.


10) Theaccompanyingtableshowsthejointdistributionbetweenthechangeoftheunemploymentrateinanelectionyearandtheshareofthecandidateoftheincumbentpartysince1928.Youthinkofthisdataasapopulationwhichyouwanttodescribe,ratherthanasamplefromwhichyouwanttoinferbehaviorofalargerpopulation.

JointDistributionofUnemploymentRateChangeandIncumbentPartysVoteShareinTotalVoteCastfortheTwoMajor-PartyCandidates,

1928-2000

(Incumbent-50%)>0(Y=0)

(Incumbent-50%)0(Y=1)

Total

u>0(X=0) 0.053 0.211 0.264u0(X=1) 0.579 0.157 0.736

Total 0.632 0.368 1.00

(a)ComputeandinterpretE(Y)andE(X).(b)CalculateE(Y X=1)andE(Y X=0).Didyouexpectthesetobeverydifferent?(c)Whatistheprobabilitythattheunemploymentratedecreasesinanelectionyear?(d)Conditionalontheunemploymentratedecreasing,whatistheprobabilitythatanincumbentwilllosetheelection?(e)Whatwouldthejointdistributionlooklikeunderindependence?Answer: (a)E(Y)=0.368;E(X)=0.736.Theprobabilityofanincumbenttohavelessthan50%oftheshareofvotes

castforthetwomajor-partycandidatesis0.368.Theprobabilityofobservingfallingunemploymentratesduringtheelectionyearis73.6percent.(b)E(Y X=1)=0.213;E(Y X=0)=0.799.Astudentwhobelievesthatincumbentswillattempttomanipulatetheeconomytowinelectionswillansweraffirmativelyhere.(c)Pr(X=1)=0.736.(d)Pr(Y=1 X=1)=0.213.(e)

JointDistributionofUnemploymentRateChangeandIncumbentPartysVoteShareinTotalVoteCastfortheTwoMajor-PartyCandidates,1928-2000underAssumptionofStatisticalIndependence



Total

u>0(X=0) 0.167 0.097 0.264u0(X=1) 0.465 0.271 0.736

Total 0.632 0.368 1.00


11) ThetableaccompanyingliststhejointdistributionofunemploymentintheUnitedStatesin2001bydemographiccharacteristics(raceandgender).

JointDistributionofUnemploymentbyDemographicCharacteristics,UnitedStates,2001

White(Y=0)

BlackandOther(Y=1)

Total

Age16-19(X=0)

0.13 0.05 0.18

Age20andabove(X=1)

0.60 0.22 0.82

Total 0.73 0.27 1.00

(a)Whatisthepercentageofunemployedwhiteteenagers?(b)Calculatetheconditionaldistributionforthecategorieswhiteandblackandother.(c)Givenyouranswerinthepreviousquestion,howdoyoureconcilethisfactwiththeprobabilitytobe60%offindinganunemployedadultwhiteperson,andonly22%forthecategoryblackandother.Answer: (a)Pr(Y=0,X=0)=0.13.

(b)ConditionalDistributionofUnemploymentbyDemographic

Characteristics,UnitedStates,2001

White(Y=0)

BlackandOther(Y=1)

Age16-19(X=0)

0.18 0.19

Age20andabove(X=1)

0.82 0.81

Total 1.00 1.00

(c)Theoriginaltableshowedthejointprobabilitydistribution,whilethetablein(b)presentedtheconditionalprobabilitydistribution.

12) FromtheStockandWatson(http://www.pearsonhighered.com/stock_watson )websitethechapter8CPSdataset(ch8_cps.xls)intoaspreadsheetprogramsuchasExcel.Fortheexercise,usethefirst500observationsonly.Usingdataforaveragehourlyearningsonly(ahe),describetheearningsdistribution.Usesummarystatistics,suchasthemean,meadian,variance,andskewness.Produceafrequencydistribution(histogram)usingreasonableearningsclasssizes.Answer: ahe

Mean 19.79StandardError 0.51Median 16.83Mode 19.23StandardDeviation 11.49SampleVariance 131.98Kurtosis 0.23Skewness 0.96Range 58.44Minimum 2.14


Maximum 60.58Sum 9897.45Count 500.0

Themeanis$19.79.Themedian($16.83)islowerthantheaverage,suggestingthatthemeanisbeingpulledupbyindividualswithfairlyhighaveragehourlyearnings.Thisisconfirmedbytheskewnessmeasure,whichispositive,andthereforesuggestsadistributionwithalongtailtotheright.Thevarianceis$2131.96,whilethestandarddeviationis$11.49.

TogeneratethefrequencydistributioninExcel,youfirsthavetosettleonthenumberofclassintervals.Onceyouhavedecidedonthese,thentheminimumandmaximuminthedatasuggeststheclasswidth.InExcel,youthendefinebins(theupperlimitsoftheclassintervals).Sturgessformulacanbeusedtosuggestthenumberofclassintervals(1+3.31log(n)),whichwouldsuggestabout9intervalshere.InsteadIsettledfor8intervalswithaclasswidthof$8minimumwagesinCaliforniaarecurrently$8andapproximatelythesameinotherU.S.states.

Thetableproducestheabsolutefrequencies,andrelativefrequenciescanbecalculatedinastraightforwardway.

bins Frequency rel.freq.8 50 0.116 187 0.37424 115 0.2332 68 0.13640 38 0.07648 33 0.06656 8 0.01666 1 0.002More 0

Substitutionoftherelativefrequenciesintothehistogramtablethenproducesthefollowinggraph(aftereliminatingthegapsbetweenthebars).


2.3 MathematicalandGraphicalProblems1) Thinkofanexampleinvolvingfivepossiblequantitativeoutcomesofadiscreterandomvariableandattacha

probabilitytoeachoneoftheseoutcomes.Displaytheoutcomes,probabilitydistribution,andcumulativeprobabilitydistributioninatable.Sketchboththeprobabilitydistributionandthecumulativeprobabilitydistribution.Answer: Answerswillvarybystudent.ThegeneratedtableshouldbesimilartoTable2.1inthetext,andfigures

shouldresembleFigures2.1and2.2inthetext.

2) Theheightofmalestudentsatyourcollege/universityisnormallydistributedwithameanof70inchesandastandarddeviationof3.5inches.Ifyouhadalistoftelephonenumbersformalestudentsforthepurposeofconductingasurvey,whatwouldbetheprobabilityofrandomlycallingoneofthesestudentswhoseheightis(a)tallerthan60?(b)between53and65?(c)shorterthan57,themeanheightoffemalestudents?(d)shorterthan50?(e)tallerthanShaqONeal,thecenteroftheMiamiHeat,whois71tall?Comparethistotheprobabilityofawomanbeingpregnantfor10months(300days),wheredaysofpregnancyisnormallydistributedwithameanof266daysandastandarddeviationof16days.Answer: (a)Pr(Z>0.5714)=0.2839;

(b)Pr(21.645)(g)Pr(1.96

4) UsingthefactthatthestandardizedvariableZ isalineartransformationofthenormallydistributedrandomvariableY,derivetheexpectedvalueandvarianceofZ.

Answer: Z=Y-YY

=-YY

+ 1Y

Y=a+bY,witha=-YY

andb= 1Y

.Given(2.29)and(2.30)inthetext,E(Z)=

-YY

+ 1Y

Y=0,andZ=1

2Z

2Z =1.

5) ShowinascatterplotwhattherelationshipbetweentwovariablesXandYwouldlooklikeiftherewas(a)astrongnegativecorrelation.(b)astrongpositivecorrelation.(c)nocorrelation.Answer: (a)

(b)

(c)


6) WhatwouldthecorrelationcoefficientbeifallobservationsforthetwovariableswereonacurvedescribedbyY=X2?Answer: Thecorrelationcoefficientwouldbezerointhiscase,sincetherelationshipisnon-linear.

7) Findthefollowingprobabilities:

(a)Yisdistributed 24 .FindPr(Y>9.49).

(b)Yisdistributedt.FindPr(Y>0.5).

(c)YisdistributedF4,.FindPr(Y696orY

8) Inconsideringthepurchaseofacertainstock,youattachthefollowingprobabilitiestopossiblechangesinthestockpriceoverthenextyear.

StockPriceChangeDuringNextTwelveMonths(%)

Probability

+15 0.2+5 0.30 0.45 0.0515 0.05

Whatistheexpectedvalue,thevariance,andthestandarddeviation?Whichisthemostlikelyoutcome?Sketchthecumulativedistributionfunction.

Answer: E(Y)=3.5; 2Y =8.49;Y=2.91;mostlikely:0.

9) YouconsidervisitingMontrealduringthebreakbetweentermsinJanuary.YougototherelevantWebsiteoftheofficialtouristofficetofigureoutthetypeofclothesyoushouldtakeonthetrip.ThesiteliststhattheaveragehighduringJanuaryis7C,withastandarddeviationof4C.UnfortunatelyyouaremorefamiliarwithFahrenheitthanwithCelsius,butfindthatthetwoarerelatedbythefollowinglinearfunction:

C= 59(F32).

FindthemeanandstandarddeviationfortheJanuarytemperatureinMontrealinFahrenheit.Answer: Usingequations(2.29)and(2.30)fromthetextbook,theresultis19.4and7.2.


10) Tworandomvariablesareindependentlydistributediftheirjointdistributionistheproductoftheirmarginaldistributions.ItisintuitivelyeasiertounderstandthattworandomvariablesareindependentlydistributedifallconditionaldistributionsofYgivenXareequal.Deriveoneofthetwoconditionsfromtheother.Answer: IfallconditionaldistributionsofY givenX areequal,then

Pr(Y=y X=1)=Pr(Y=y X=2)=...=Pr(Y=y X=l).

Butifallconditionaldistributionsareequal,thentheymustalsoequalthemarginaldistribution,i.e.,

Pr(Y=y X=x)=Pr(Y-y).

GiventhedefinitionoftheconditionaldistributionofYgivenX=x,youthenget

Pr(Y=y X=x)=Pr(Y=y,X=x)Pr(X=x)

=Pr(Y=y),

whichgivesyouthecondition

Pr(Y=y,X=x)=Pr(Y=y)Pr(X=x).

11) TherearefrequentlysituationswhereyouhaveinformationontheconditionaldistributionofY givenX,but

areinterestedintheconditionaldistributionofXgivenY.RecallingPr(Y=y X=x)=Pr(X=x,Y=y)Pr(X=x)

,derivea

relationshipbetweenPr(X=x Y=y)andPr(Y=y X=x).ThisiscalledBayestheorem.

Answer: GivenPr(Y=y X=x)=Pr(X= x Y = y)Pr(X=x)

,

Pr(Y=y X=x)Pr(X=x)=Pr(X=x,Y=y);

similarlyPr(X=x Y=y)=Pr(X=x Y=y)Pr(Y=y)

and

Pr(X=x Y=y)Pr(Y=y)=Pr(X=x,Y=y).EquatingthetwoandsolvingforPr(X=x Y=y)thenresultsin

Pr(X=x Y=y)=Pr(Y=y X=x)Pr(X=x)Pr(Y=y)

.

12) Youareatacollegeofroughly1,000studentsandobtaindatafromtheentirefreshmanclass(250students)onheightandweightduringorientation.Youconsiderthistobeapopulationthatyouwanttodescribe,ratherthanasamplefromwhichyouwanttoinfergeneralrelationshipsinalargerpopulation.Weight(Y)ismeasuredinpoundsandheight(X)ismeasuredininches.Youcalculatethefollowingsums:

n

i=1y 2i =94,228.8,

n

i=1x 2i =1,248.9,

n

i=1xiyi =7,625.9

(smalllettersrefertodeviationsfrommeansasinzi=ZiZ).

(a)Givenyourgeneralknowledgeabouthumanheightandweightofagivenage,whatcanyousayabouttheshapeofthetwodistributions?(b)Whatisthecorrelationcoefficientbetweenheightandweighthere?Answer: (a)Bothdistributionsareboundtobenormal.

(b)0.703.


13) Usethedefinitionfortheconditionaldistributionof Y givenX = x andthemarginaldistributionofX toderivetheformulaforPr(X=x,Y=y).Thisiscalledthemultiplicationrule.Useittoderivetheprobabilityfordrawingtwoacesrandomlyfromadeckofcards(nojoker),whereyoudonotreplacethecardafterthefirstdraw.Next,generalizingthemultiplicationruleandassumingindependence,findtheprobabilityofhavingfourgirlsinafamilywithfourchildren.

Answer: 452

351

=0.0045;0.0625or 12

4= 1

16.

14) Thesystolicbloodpressureoffemalesintheir20sisnormallydistributedwithameanof120withastandarddeviationof9.Whatistheprobabilityoffindingafemalewithabloodpressureoflessthan100?Morethan135?Between105and123?Youvisitthewomenssoccerteamoncampus,andfindthattheaveragebloodpressureofthe25membersis114.Isitlikelythatthisgroupofwomencamefromthesamepopulation?

Answer: Pr(Y135)=0.0478;Pr(105

17) TheEconomicReportofthePresidentgivesthefollowingagedistributionoftheUnitedStatespopulationfortheyear2000:

UnitedStatesPopulationByAgeGroup,2000

Outcome(agecategory

Under5 5-15 16-19 20-24 25-44 45-64 65andover

Percentage 0.06 0.16 0.06 0.07 0.30 0.22 0.13

Imaginethateverypersonwasassignedauniquenumberbetween1and275,372,000(thetotalpopulationin2000).Ifyougeneratedarandomnumber,whatwouldbetheprobabilitythatyouhaddrawnsomeoneolderthan65orunder16?Treatingthepercentagesasprobabilities,writedownthecumulativeprobabilitydistribution.Whatistheprobabilityofdrawingsomeonewhois24yearsoryounger?Answer: Pr(Y65)=0.35;

Outcome(agecategory

Under5 5-15 16-19 20-24 25-44 45-64 65andover

Cumulativeprobabilitydistribution

0.06 0.22 0.28 0.35 0.65 0.87 1.00

Pr(Y24)=0.35.

18) Theaccompanyingtablegivestheoutcomesandprobabilitydistributionofthenumberoftimesastudentcheckshere-maildaily:

ProbabilityofCheckingE-Mail

Outcome(numberofe-mailchecks)

0 1 2 3 4 5 6

Probabilitydistribution

0.05 0.15 0.30 0.25 0.15 0.08 0.02

Sketchtheprobabilitydistribution.Next,calculatethec.d.f.fortheabovetable.Whatistheprobabilityofhercheckinghere-mailbetween1and3timesaday?Ofcheckingitmorethan3timesaday?Answer: Outcome

(numberofe-mailchecks)

0 1 2 3 4 5 6


0.05 0.20 0.50 0.75 0.90 0.98 1.00

Pr(1Y3)0.70;Pr(Y>0.25).


19) Theaccompanyingtableliststheoutcomesandthecumulativeprobabilitydistributionforastudentrentingvideosduringtheweekwhileoncampus.

VideoRentalsperWeekduringSemester

Outcome(numberofweeklyvideorentals)

0 1 2 3 4 5 6

Probabilitydistribution 0.05 0.55 0.25 0.05 0.07 0.02 0.01

Sketchtheprobabilitydistribution.Next,calculatethecumulativeprobabilitydistributionfortheabovetable.Whatistheprobabilityofthestudentrentingbetween2and4aweek?Oflessthan3aweek?Answer: Thecumulativeprobabilitydistributionisgivenbelow.Theprobabilityofrentingbetweentwoandfour

videosaweekis0.37.Theprobabilityofrentinglessthanthreeaweekis0.85.

Outcome(numberofweeklyvideorentals)

0 1 2 3 4 5 6


0.05 0.60 0.85 0.90 0.97 0.99 1.00

20) ThetextbookmentionedthatthemeanofY,E(Y)iscalledthefirstmomentofY,andthattheexpectedvalueofthesquareofY,E(Y2)iscalledthesecondmomentofY,andsoon.Thesearealsoreferredtoasmomentsabouttheorigin.Arelatedconceptismomentsaboutthemean,whicharedefinedasE[(YY)r].Whatdoyoucallthesecondmomentaboutthemean?Whatdoyouthinkthethirdmoment,referredtoasskewness,measures?Doyoubelievethatitwouldbepositiveornegativeforanearningsdistribution?Whatmeasureofthethirdmomentaroundthemeandoyougetforanormaldistribution?Answer: Thesecondmomentaboutthemeanisthevariance.Skewnessmeasuresthedeparturefromsymmetry.

Forthetypicalearningsdistribution,itwillbepositive.Forthenormaldistribution,itwillbezero.

21) Explainwhythetwoprobabilitiesareidenticalforthestandardnormaldistribution:Pr(1.96X 1.96)andPr(1.96

22) SATscoresinMathematicsarenormallydistributedwithameanof500andastandarddeviationof100.The

formulaforthenormaldistributionisf(Y)= 1

2 2Y

e-12(Y-YY

)2Usethescatterplotoptioninastandard

spreadsheetprogram,suchasExcel,toplottheMathematicsSATdistributionusingthisformula.Startbyentering300asthefirstSATscoreinthefirstcolumn(thelowestscoreyoucangetinthemathematicssectionaslongasyoufillinyournamecorrectly),andthenincrementthescoresby10untilyoureach800.Inthesecondcolumn,usetheformulaforthenormaldistributionandcalculatef(Y).Thenusethescatterplotoption,whereyoueventuallyremovemarkersandsubstitutethesewiththesolidlineoption.

Answer:

23) Useastandardspreadsheetprogram,suchasExcel,tofindthefollowingprobabilitiesfromvariousdistributionsanalyzedinthecurrentchapter:

a.IfYisdistributedN(1,4),findPr(Y3)b.IfYisdistributedN(3,9),findPr(Y>0)c.IfYisdistributedN(50,25),findPr(40Y52)d.IfYisdistributedN(5,2),findPr(6Y8)Answer: TheanswersherearegiventogetherwiththerelevantExcelcommands.

a. =NORMDIST(3,1,2,TRUE)=0.8413b. =1-NORMDIST(0,3,3,TRUE)=0.8413c. =NORMDIST(52,50,5,TRUE)-NORMDIST(40,50,5,TRUE)=0.6326d. =NORMDIST(8,5,SQRT(2),TRUE)-NORMDIST(6,5,SQRT(2),TRUE)=0.2229


24) LookingatalargeCPSdatasetwithover60,000observationsfortheUnitedStatesandtheyear2004,youfindthattheaveragenumberofyearsofeducationisapproximately13.6.However,asurprisinglargenumberofindividuals(approximately800)havequitealowvalueforthisvariable,namely6yearsorless.Youdecidetodroptheseobservations,sincenoneofyourrelativesorfriendshavethatfewyearsofeducation.Inaddition,youareconcernedthatiftheseindividualscannotreporttheyearsofeducationcorrectly,thentheobservationsonothervariables,suchasaveragehourlyearnings,canalsonotbetrusted.Asamatteroffactyouhavefoundseveralofthesetobebelowminimumwagesinyourstate.Discussifdroppingtheobservationsisreasonable.Answer: Whileitisalwaysagoodideatocheckthedatacarefullybeforeconductingaquantitativeanalysis,you

shouldneverdropdatabeforecarefullythinkingabouttheproblemathand.WhileitisnotplausibletofindmanyindividualsintheU.S.whowereraisedherewiththatfewyearsofeducation,therewillbeimmigrantsinthesurvey.Averageyearsofeducationcanbequitelowinothercountries.Forexample,Brazilsaverageyearsofschoolingislessthan6years.Thepointoftheexerciseistothinkhardwhetherornotobservationsareoutliersgeneratedbyfaultydataentryorifthereisareasonforobservingvalueswhichmayappearstrangeatfirst.

25) Useastandardspreadsheetprogram,suchasExcel,tofindthefollowingprobabilitiesfromvariousdistributionsanalyzedinthecurrentchapter:

a. IfYisdistributed 24 ,findPr(Y7.78)

b. IfYisdistributed 210 ,findPr(Y>18.31)

c. IfYisdistributedF10,,findPr(Y>1.83)d. IfYisdistributedt15,findPr(Y>1.75)e. IfYisdistributedt90,findPr(-1.99Y1.99)f. IfYisdistributedN(0,1),findPr(-1.99Y1.99)g. IfYisdistributedF7,4,findPr(Y>4.12)h. IfYisdistributedF7,120,,findPr(Y>2.79)

Answer: TheanswersherearegiventogetherwiththerelevantExcelcommands.a. =1-CHIDIST(7.78,4)=0.90b. =CHIDIST(18.31,10)=0.05c. =FDIST(1.83,10,1000000)=0.05d. =TDIST(1.75,15,1)=0.05e. =1-TDIST(1.99,90,2)=0.95f. =NORMDIST(1.99,0,1,1)-NORMDIST(-1.99,0,1,1)=0.953g. =FDIST(4.12,7,4)=0.10h. =FDIST(2.79,7,120)=0.01


Chapter3 ReviewofStatistics3.1 MultipleChoice

1) AnestimatorisA) anestimate.B) aformulathatgivesanefficientguessofthetruepopulationvalue.C) arandomvariable.D) anonrandomnumber.

Answer: C

2) AnestimateisA) efficientifithasthesmallestvariancepossible.B) anonrandomnumber.C) unbiasedifitsexpectedvalueequalsthepopulationvalue.D) anotherwordforestimator.

Answer: B

3) Anestimator^YofthepopulationvalueYisunbiasedif

A) ^Y=Y.

B) Yhasthesmallestvarianceofallestimators.

C) Yp

Y.

D) E(^Y)=Y.

Answer: D

4) Anestimator^YofthepopulationvalueYisconsistentif

A) ^Yp

Y.B) itsmeansquareerroristhesmallestpossible.C) Yisnormallydistributed.

D) Yp

0.Answer: A

5) Anestimator^YofthepopulationvalueYismoreefficientwhencomparedtoanotherestimator

~Y,if

A) E(^Y)>E(

~Y).

B) ithasasmallervariance.C) itsc.d.f.isflatterthanthatoftheotherestimator.

D) bothestimatorsareunbiased,andvar(^Y)

7) ThestandarderrorofY,SE(Y)=^Yisgivenbythefollowingformula:

A) 1n

n

i=1(Yi Y)2.

B)S 2Y

n.

C) SY.

D)SYn.

Answer: D

8) Thecriticalvalueofatwo-sidedt-testcomputedfromalargesampleA) is1.64ifthesignificancelevelofthetestis5%.B) cannotbecalculatedunlessyouknowthedegreesoffreedom.C) is1.96ifthesignificancelevelofthetestis5%.D) isthesameasthep-value.

Answer: C

9) AtypeIerrorisA) alwaysthesameas(1-typeII)error.B) theerroryoumakewhenrejectingthenullhypothesiswhenitistrue.C) theerroryoumakewhenrejectingthealternativehypothesiswhenitistrue.D) always5%.

Answer: B

10) AtypeIIerrorA) istypicallysmallerthanthetypeIerror.B) istheerroryoumakewhenchoosingtypeIIortypeI.C) istheerroryoumakewhennotrejectingthenullhypothesiswhenitisfalse.D) cannotbecalculatedwhenthealternativehypothesiscontainsan=.

Answer: C

11) ThesizeofthetestA) istheprobabilityofcommittingatypeIerror.B) isthesameasthesamplesize.C) isalwaysequalto(1-thepoweroftest).D) canbegreaterthan1inextremeexamples.

Answer: A

12) ThepowerofthetestisA) dependentonwhetheryoucalculateatorat2statistic.B) oneminustheprobabilityofcommittingatypeIerror.C) asubjectiveviewtakenbytheeconometriciandependentonthesituation.D) oneminustheprobabilityofcommittingatypeIIerror.

Answer: D


13) Whenyouaretestingahypothesisagainstatwo-sidedalternative,thenthealternativeiswrittenasA) E(Y)>Y,0.

B) E(Y)=Y,0.

C) YY,0.

D) E(Y)Y,0.

Answer: D

14) AscatterplotA) showshowYandXarerelatedwhentheirrelationshipisscatteredallovertheplace.B) relatesthecovarianceofXandY tothecorrelationcoefficient.C) isaplotofnobservationsonXiandYi,whereeachobservationisrepresentedbythepoint(Xi,Yi).D) showsnobservationsofYovertime.

Answer: C

15) Thefollowingtypesofstatisticalinferenceareusedthroughouteconometrics,withtheexceptionofA) confidenceintervals.B) hypothesistesting.C) calibration.D) estimation.

Answer: C

16) AmongallunbiasedestimatorsthatareweightedaveragesofY1,...,YnY,isA) theonlyconsistentestimatorofY.

B) themostefficientestimatorofY.

C) anumberwhich,bydefinition,cannothaveavariance.D) themostunbiasedestimatorofY.

Answer: B

17) ToderivetheleastsquaresestimatorY,youfindtheestimatormwhichminimizes

A)n

i=1(Yim)2 .

B)n

i=1(Yim) .

C)n

i=1mY 2i .

D)n

i=1(Yim) .

Answer: A

18) IfthenullhypothesisstatesH0:E(Y)= Y,0,thenatwo-sidedalternativehypothesisis

A) H1:E(Y)Y,0.

B) H1:E(Y)Y,0.

C) H1:YY,0.

Answer: A


19) Thep-valueisdefinedasfollows:A) p=0.05.B) PrH0[ YY,0 > Y

actY,0 ].

C) Pr(z>1.96).D) PrH0[ YY,0

23) Thet-statisticisdefinedasfollows:

A) t=YY,0

2Y

n

.

B) t=YY,0SE(Y)

.

C) t=(YY,0)

2

SE(Y).

D) 1.96.Answer: A

24) ThepowerofthetestA) istheprobabilitythatthetestactuallyincorrectlyrejectsthenullhypothesiswhenthenullistrue.B) dependsonwhetheryouuseYorY2forthet-statistic.C) isoneminusthesizeofthetest.D) istheprobabilitythatthetestcorrectlyrejectsthenullwhenthealternativeistrue.

Answer: D

25) Thesamplecovariancecanbecalculatedinanyofthefollowingways,withtheexceptionof:

A) 1n1

n

i=1

(XiX)(Yi Y).

B) 1n1

n

i=1

XiYi n

n1 XY.

C) 1n

n

i=1

(XiX)(Yi Y).

D) rXYSYSY,whererXYisthecorrelationcoefficient.

Answer: C

26) Whenthesamplesizenislarge,the90%confidenceintervalforY is

A) Y1.96SE(Y).B) Y1.64SE(Y).C) Y1.64Y.

D) Y1.96.Answer: B


27) ThestandarderrorforthedifferenceinmeansiftworandomvariablesM andW ,whenthetwopopulationvariancesaredifferent,is

A)S 2M+S

2W

nM+nW.

B)SMnM

+SW

nW.

C) 12(S 2M

nM+S 2W

nW).

D)S 2M

nM+S 2W

nW.

Answer: D

28) Thet-statistichasthefollowingdistribution:A) standardnormaldistributionforn < 15B) Studenttdistributionwithn1degreesoffreedomregardlessofthedistributionoftheY.C) Studenttdistributionwithn1degreesoffreedomiftheY isnormallydistributed.D) astandardnormaldistributionifthesamplestandarddeviationgoestozero.

Answer: C

29) Thefollowingstatementaboutthesamplecorrelationcoefficientistrue.A) 1rXY1.

B) r 2XYp

corr(Xi,Yi).

C) rXY

31) Whentestingfordifferencesofmeans,thet-statistict=Ym-Yw

SE(Ym-Yw),whereSE(Ym-Yw)=

s 2m

nm+

s 2w

nwhas

A) astudenttdistributionifthepopulationdistributionofY isnotnormalB) astudenttdistributionifthepopulationdistributionofYisnormalC) anormaldistributioneveninsmallsamplesD) cannotbecomputedunlessnw=nm

Answer: B

32) Whentestingfordifferencesofmeans,youcanbasestatisticalinferenceontheA) StudenttdistributioningeneralB) normaldistributionregardlessofsamplesizeC) StudenttdistributioniftheunderlyingpopulationdistributionofY isnormal,thetwogroupshavethe

samevariances,andyouusethepooledstandarderrorformulaD) Chi-squareddistributionwith(nw + nm - 2)degreesoffreedom

Answer: C

33) Assumethatyouhave125observationsontheheight(H)andweight(W)ofyourpeersincollege.LetsHW=68,sH=3.5,sW=29.Thesamplecorrelationcoefficientis

A) 1.22B) 0.50C) 0.67D) Cannotbecomputedsincemalesandfemaleshavenotbeenseparatedout.

Answer: C

34) Youhavecollecteddataontheaverageweeklyamountofstudyingtime(T)andgrades(G)fromthepeersatyourcollege.Changingthemeasurementfromminutesintohourshasthefollowingeffectonthecorrelationcoefficient:

A) decreasestherTGbydividingtheoriginalcorrelationcoefficientby60B) resultsinahigherrTGC) cannotbecomputedsincesomestudentsstudylessthananhourperweekD) doesnotchangetherTG

Answer: A,D

35) AlowcorrelationcoefficientimpliesthatA) thelinealwayshasaflatslopeB) inthescatterplot,thepointsfallquitefarawayfromthelineC) thetwovariablesareunrelatedD) youshoulduseatighterscaleoftheverticalandhorizontalaxistobringtheobservationsclosertotheline

Answer: B

3.2 EssaysandLongerQuestions1) Thinkofatleastnineexamples,threeofeach,thatdisplayapositive,negative,ornocorrelationbetweentwo

economicvariables.Ineachofthepositiveandnegativeexamples,indicatewhetherornotyouexpectthecorrelationtobestrongorweak.Answer: Answerswillvarybystudent.Studentsfrequentlybringupthefollowingcorrelations.Positive

correlations:earningsandeducation(hopefullystrong),consumptionandpersonaldisposableincome(strong),percapitaincomeandinvestment-outputratioorsavingrate(strong);negativecorrelation:OkunsLaw(strong),incomevelocityandinterestrates(strong),thePhillipscurve(strong);nocorrelation:productivitygrowthandinitiallevelofpercapitaincomeforallcountriesoftheworld(beta-convergenceregressions),consumptionandthe(real)interestrate,employmentandrealwages.


2) Adultmalesaretaller,onaverage,thanadultfemales.VisitingtworecentAmericanYouthSoccerOrganization(AYSO)under12yearold(U12)soccermatchesonaSaturday,youdonotobserveanobviousdifferenceintheheightofboysandgirlsofthatage.Yousuggesttoyourlittlesisterthatshecollectdataonheightandgenderofchildrenin4thto6thgradeaspartofherscienceproject.Theaccompanyingtableshowsherfindings.

HeightofYoungBoysandGirls,Grades4-6,ininches

Boys Girls

YBoys SBoys nBoys YGirls SGirls nGirls57.8 3.9 55 58.4 4.2 57

(a)Letyournullhypothesisbethatthereisnodifferenceintheheightoffemalesandmalesatthisagelevel.Specifythealternativehypothesis.(b)Findthedifferenceinheightandthestandarderrorofthedifference.(c)Generatea95%confidenceintervalforthedifferenceinheight.(d)Calculatethet-statisticforcomparingthetwomeans.Isthedifferencestatisticallysignificantatthe1%level?Whichcriticalvaluedidyouuse?Whywouldthisnumberbesmallerifyouhadassumedaone-sidedalternativehypothesis?Whatistheintuitionbehindthis?Answer: (a)H0:Boys-Girls=0vs.H1:Boys - Girls 0

(b)YBoys-YGirls=-0.6,SE(YBoys-YGirls)=3.9255

+4.22

57=0.77.

(c)-0.61.960.77=(-2.11,0.91).(d)t=-0.78,so t

3) MathSATscores(Y)arenormallydistributedwithameanof500andastandarddeviationof100.Aneveningschooladvertisesthatitcanimprovestudentsscoresbyroughlyathirdofastandarddeviation,or30points,iftheyattendacoursewhichrunsoverseveralweeks.(AsimilarclaimismadeforattendingaverbalSATcourse.)Thestatisticianforaconsumerprotectionagencysuspectsthatthecoursesarenoteffective.Sheviewsthesituationasfollows:H0:Y=500vs.H1:Y=530.(a)Sketchthetwodistributionsunderthenullhypothesisandthealternativehypothesis.(b)Theconsumerprotectionagencywantstoevaluatethisclaimbysending50studentstoattendclasses.Oneofthestudentsbecomessickduringthecourseanddropsout.Whatisthedistributionoftheaveragescoreoftheremaining49studentsunderthenull,andunderthealternativehypothesis?(c)Assumethataftergraduatingfromthecourse,the49participantstaketheSATtestandscoreanaverageof520.Isthisconvincingevidencethattheschoolhasfallenshortofitsclaim?Whatisthe p-valueforsuchascoreunderthenullhypothesis?(d)Whatwouldbethecriticalvalueunderthenullhypothesisifthesizeofyourtestwere5%?(e)Giventhiscriticalvalue,whatisthepowerofthetest?Whatoptionsdoesthestatisticianhaveforincreasingthepowerinthissituation?Answer: (a)

(b)Yofthe49participantsisnormallydistributed,withameanof500andastandarddeviationof14.286underthenullhypothesis.Underthealternativehypothesis,itisnormallydistributedwithameanof530andastandarddeviationof14.286.(c)Itispossiblethattheconsumerprotectionagencyhadchosenagroupof49studentswhoseaveragescorewouldhavebeen490withoutattendingthecourse.Thecrucialquestionishowlikelyitisthat49students,chosenrandomlyfromapopulationwithameanof500andastandarddeviationof100,willscoreanaverageof520.Thep-valueforthisscoreis0.081,meaningthatiftheagencyrejectedthenullhypothesisbasedonthisevidence,itwouldmakeamistake,onaverage,roughly1outof12times.Hencetheaveragescoreof520wouldallowrejectionofthenullhypothesisthattheschoolhashadnoeffectontheSATscoreofstudentsatthe10%level.(d)Thecriticalvaluewouldbe523.(e)Pr(Y

errorofYaccordingly.(c)Foreachofthetwentyobservationsin(c)a95%confidenceintervalisconstructed.Drawtheseconfidenceintervals,usingthesamegraphasin(c).Howmanyofthese20confidenceintervalswouldyouexpecttoweigh5poundsunderthenullhypothesis?Answer: (a)Onaverage,thereshouldbeonebagineverysampleof20whichweighslessthan4.9poundsor

morethan5.1pounds.

(b)Theaverageweightof25bagswillbenormallydistributed,withameanof5poundsandastandarddeviationof0.01pounds.(Samegraphasin(a),butwiththefollowinglowerandupperbounds.)

(c)Youwouldexpect19ofthe20confidenceintervalstocontain5pounds.


5) Assumethattwopresidentialcandidates,callthemBushandGore,receive50%ofthevotesinthepopulation.YoucanmodelthissituationasaBernoullitrial,whereYisarandomvariablewithsuccessprobabilityPr(Y=

1)=p,andwhereY=1ifapersonvotesforBushandY=0otherwise.Furthermore,letp^bethefractionof

successes(1s)inasample,whichisdistributedN(p,p(1-p)n

)inreasonablylargesamples,sayforn40.

(a)Givenyourknowledgeaboutthepopulation,findtheprobabilitythatinarandomsampleof40,Bushwouldreceiveashareof40%orless.(b)Howwouldthissituationchangewitharandomsampleof100?(c)Givenyouranswersin(a)and(b),wouldyoubecomfortabletopredictwhatthevotingintentionsforthe

entirepopulationareifyoudidnotknowpbuthadpolled10,000individualsatrandomandcalculatedp^?

Explain.(d)Thisresultseemstoholdwhetheryoupoll10,000peopleatrandomintheNetherlandsortheUnitedStates,wheretheformerhasapopulationoflessthan20millionpeople,whiletheUnitedStatesis15timesaspopulous.Whydoesthepopulationsizenotcomeintoplay?

Answer: (a)Pr(p^

6) Youhavecollectedweeklyearningsandagedatafromasub-sampleof1,744individualsusingtheCurrentPopulationSurveyinagivenyear.(a)Giventheoverallmeanof$434.49andastandarddeviationof$294.67,constructa99%confidenceintervalforaverageearningsintheentirepopulation.Statethemeaningofthisintervalinwords,ratherthanjustinnumbers.Ifyouconstructeda90%confidenceintervalinstead,woulditbesmallerorlarger?Whatistheintuition?(b)Whendividingyoursampleintopeople45yearsandolder,andyoungerthan45,theinformationshowninthetableisfound.

AgeCategory AverageEarningsY

StandardDeviationSY

N

Age45 $488.87 $328.64 507Age7)= Pr(Z>1)=0.1587.

(b)62.58 250

=60.73=(5.27,6.73).

(c) 12(2.58 2

50)=2.581

2 2

50=2.58 2

450,orn=200.


8) U.S.NewsandWorldReportrankscollegesanduniversitiesannually.Yourandomlysample100ofthenationaluniversitiesandliberalartscollegesfromtheyear2000issue.Theaveragecost,whichincludestuition,fees,androomandboard,is$23,571.49withastandarddeviationof$7,015.52.(a)Basedonthissample,constructa95%confidenceintervaloftheaveragecostofattendingauniversity/collegeintheUnitedStates.(b)Costvariesbyquiteabit.Oneofthereasonsmaybethatsomeuniversities/collegeshaveabetterreputationthanothers.U.S.NewsandWorldReportstriestomeasurethisfactorbyaskinguniversitypresidentsandchiefacademicofficersaboutthereputationofinstitutions.Therankingisfrom1(marginal)to5(distinguished).Youdecidetosplitthesampleaccordingtowhethertheacademicinstitutionhasareputationofgreaterthan3.5ornot.Forcomparison,in2000,Caltechhadareputationrankingof4.7,SmithCollegehad4.5,andAuburnUniversityhad3.1.Thisgivesyouthestatisticsshownintheaccompanyingtable.

ReputationCategory

AverageCostY

StandarddeviationofCost(SY)

N

Ranking>3.5 $29,311.31 $5,649.21 29Ranking3.5 $21,227.06 $6,133.38 71

Testthehypothesisthattheaveragecostforalluniversities/collegesisthesameindependentofthereputation.Whatalternativehypothesisdidyouuse?(c)Whatotherfactorsshouldyouconsiderbeforemakingadecisionbasedonthedatain(b)?

Answer: (a)23,571.491.967,015.52100

=23,571.49701.55=(22,869.94,24,273.04).

(b)Assumingunequalpopulationvariances,t= (29311.31-21,227.06)

5,649.21229

+6,133.382

71

=6.33,whichisstatistically

significantwhetherornotyouuseaone-sidedortwo-sidedhypothesistest.Yourpriorexpectationisthatacademicinstitutionswithahigherreputationwillchargemoreforattending,andhenceaone-sidedalternativewouldhavebeenappropriatehere.(c)Theremaybeothervariableswhichpotentiallyhaveaneffectonthecostofattendingtheacademicinstitution.Someofthesefactorsmightbewhetherornotthecollege/universityisprivateorpublic,itssize,whetherornotithasareligiousaffiliation,etc.Itisonlyaftercontrollingforthesefactorsthatthepurerelationshipbetweenreputationandcostcanbeidentified.


9) ThedevelopmentofficeandtheregistrarhaveprovidedyouwithanonymousmatchesofstartingsalariesandGPAsfor108graduatingeconomicsmajors.Yoursamplecontainsavarietyofjobs,fromchurchpastortostockbroker.(a)Theaveragestartingsalaryforthe108studentswas$38,644.86withastandarddeviationof$7,541.40.Constructa95%confidenceintervalforthestartingsalaryofalleconomicsmajorsatyouruniversity/college.(b)Asimilarsampleforpsychologymajorsindicatesasignificantlylowerstartingsalary.Giventhatthesestudentshadthesamenumberofyearsofeducation,doesthisindicatediscriminationinthejobmarketagainstpsychologymajors?(c)Youwonderifitpays(nopunintended)togetgoodgradesbycalculatingtheaveragesalaryforeconomicsmajorswhograduatedwithacumulativeGPAofB+orbetter,andthosewhohadaBorworse.Thedataisasshownintheaccompanyingtable.

CumulativeGPA AverageEarningsY

StandarddeviationSY

n

B+orbetter $39,915.25 $8,330.21 59Borworse $37,083.33 $6,174.86 49

Conductat-testforthehypothesisthatthetwostartingsalariesarethesameinthepopulation.Giventhatthisdatawascollectedin1999,doyouthinkthatyourresultswillholdforotheryears,suchas2002?

Answer: (a)38,644.861.967,541.40108

=38,644.861,422.32=(37,222.54,40,067.18).

(b)Itsuggeststhatthemarketvaluescertainqualificationsmorehighlythanothers.Comparingmeansandidentifyingthatoneissignificantlylowerthanothersdoesnotindicatediscrimination.

(c)Assumingunequalpopulationvariances,t= (39,915.25-37,083.33)

8,33.21259

+6,174.862

49

=2.03.Thecriticalvaluefora

one-sidedtestis1.64,foratwo-sidedtest1.96,bothatthe5%level.Henceyoucanrejectthenullhypothesisthatthetwostartingsalariesareequal.Presumablyyouwouldhavechosenasanalternativethatbetterstudentsreceivebetterstartingsalaries,sothatthisbecomesyournewworkinghypothesis.1999wasaboomyear.Ifbetterstudentsreceivebetterstartingoffersduringaboomyear,whenthelabormarketforgraduatesistight,thenitisverylikelythattheyreceiveabetterofferduringarecessionyear,assumingthattheyreceiveanofferatall.


10) Duringthelastfewdaysbeforeapresidentialelection,thereisafrenzyofvotingintentionsurveys.Onagivenday,quiteoftenthereareconflictingresultsfromthreemajorpolls.(a)Thinkofeachofthesepollsasreportingthefractionofsuccesses(1s)ofaBernoullirandomvariableY,

wheretheprobabilityofsuccessisPr(Y=1)=p.Letp^bethefractionofsuccessesinthesampleandassumethat

thisestimatorisnormallydistributedwithameanofpandavarianceof p(1-p)n

.Whyaretheresultsforall

pollsdifferent,eventhoughtheyaretakenonthesameday?

(b)Giventheestimatorofthevarianceofp^, p

^(1-p

^)

n,constructa95%confidenceintervalforp

^.Forwhichvalue

ofp^isthestandarddeviationthelargest?Whatvaluedoesittakeinthecaseofamaximum p

^?

(c)Whentheresultsfromthepollsarereported,youaretold,typicallyinthesmallprint,thatthemarginoferrorisplusorminustwopercentagepoints.Usingtheapproximationof1.962,andassuming,conservatively,themaximumstandarddeviationderivedin(b),whatsamplesizeisrequiredtoaddandsubtract(marginoferror)twopercentagepointsfromthepointestimate?(d)Whatsamplesizewouldyouneedtohalvethemarginoferror?

Answer: (a)Sinceallpollsareonlysamples,thereisrandomsamplingerror.Asaresult,p^willdifferfromsample

tosample,andmostlikelyalsofromp.

(b)p^1.96 p

^(1-p

^)

n.Abitofthoughtorcalculuswillshowthatthestandarddeviationwillbelargest

forp^=0.5,inwhichcaseitbecomes 0.5

n.

(c)n=2,500.(d)n=10,000.

11) AttheStockandWatson(http://www.pearsonhighered.com/stock_watson )websitegotoStudentResourcesandselecttheoptionDatasetsforReplicatingEmpiricalResults.ThenselecttheCPSDataUsedinChapter8(ch8_cps.xls)andopenitinExcel.Thisisaratherlargedatasettoworkwith,sojustcopythefirst500observationsintoanewWorksheet(thesearerows1to501).

InthenewlycreatedWorksheet,markA1toA501,thenselecttheDatatabandclickonsort.Adialogboxwillopen.FirstselectAddlevelfromoneoftheoptionsontheleft.ThenselectsortbyandchooseNortheastandLargesttoSmallest.RepeatthesamefortheSouthasasecondoption.Finallypressok.

Thisshouldgiveyou209observationsforaveragehourlyearningsfortheNortheastregion,followedby205observationsfortheSouth.

a. Foreachofthe209averagehourlyearningsobservationsfortheNortheastregionandseparatelyfortheSouthregion,calculatethemeanandsamplestandarddeviation.

b UsetheappropriatetesttodeterminewhetherornotaveragehourlyearningsintheNortheastregionthesameasintheSouthregion.

c Findthe1%,5%,and10%confidenceintervalforthedifferencesbetweenthetwopopulatioonmeans.Isyourconclusionconsistentwiththetestinpart(b)?

d Inallthreecasesofusingtheconfidenceintervalin(c),thepowerofthetestisquitelow(5%).Whatcanyoudotoincreasethepowerofthetestwithoutreducingthesizeofthetest?


Answer: a.YNortheast=$21.12;YSouth=$18.18;sNortheast=$11.86;sSouth=$11.18

b.t= 21.12-18.80

11.862209

+11.182

205

=2.05Youcannotrejectthenullhypothesisofequalaverageearningsinthetwo

regionsatthe1%level,butyouareabletorejectitatthe10%and5%significancelevel.

c.Forthe10%significancelevel,theconfidenceintervalis($0.46,$4.18).Forthe5%significancelevel,theintervalbecomeslargerandis($0.10,$4.54).Ineitheroneofthecasesyoucanrejectthenullhypothesis,since$0isnotcontainedintheconfidenceinterval.Itisonlyforthe1%significancelevelthatthenullhypothesiscannotberejected.Inthatcase,theconfidenceintervalis($-0.60,$5.24).

d.Youwouldhavetoincreasethesamplesize,sincethatwouldshrinkthestandarderror(assumingthatthesamplemeanandvariancewillnotchange).

3.3 MathematicalandGraphicalProblems1) YourtextbookdefinedthecovariancebetweenX andY asfollows:

1n1

n

i=1(XiX)(YiY)

Provethatthisisidenticaltothefollowingalternativespecification:

1n-1

n

i=1XiYi-

nn-1

XY

Answer: 1n-1

n

i=1(Xi-X)(Yi-Y) =

1n-1

n

i=1(XiYi-XYi-YXi+YX)

= 1n-1

(n

i=1XiYi-X

n

i=1Yi-Y

n

i=1Xi+nYX) =

1n-1

(n

i=1XiYi-nXY-nYX+nYX)

= 1n-1

n

i=1XiYi-

nn-1

XY.


2) Foreachoftheaccompanyingscatterplotsforseveralpairsofvariables,indicatewhetheryouexpectapositiveornegativecorrelationcoefficientbetweenthetwovariables,andthelikelymagnitudeofit(youcanuseasmallrange).

(a)

(b)

(c)


(d)

Answer: (a) Positivecorrelation.Theactualcorrelationcoefficientis0.46.(b)Norelationship.Theactualcorrelationcoefficientis0.00007.(c) Negativerelationship.Theactualcorrelationcoefficientis0.70.(d) Nonlinear(invertedU)relationship.Theactualcorrelationcoefficientis0.23.


3) Yourtextbookdefinesthecorrelationcoefficientasfollows:

r=

1n-1

n

i=1(YiY)2(XiX)2

1n-1

n

i=1(YiY)2

1n-1

n

i=1(Xi-X)2

Anothertextbookgivesanalternativeformula:

r=

nn

i=1YiXi- (

n

i=1Yi)(

n

i=1Xi)

nn

i=1Y 2i -(

n

i=1Yi)2 n

n

i=1X 2i -(

n

i=1Xi)2

Provethatthetwoarethesame.

Answer: r=

1n-1

n

i=1(Yi-Y)2(Xi-X)2

1n-1

n

i=1(Yi-Y)2

1n-1

n

i=1(Xi-X)2

=

1n-1

n

i=1

(YiXi-YXi-XYi +YX)

1n-1

n

i=1(Y 2i -2YYi+Y2)

n

i=1(X 2i-2XXi+X2)

=

n

i=1YiXi-nYX

n

i=1Y 2i-nY2

n

i=1X 2i-nX2

=

nn

i=1YiXi-nYnX

nn

i=1Y 2i -nY2

n

i=1X 2i-X2

=

nn

i=1YiXi- (

n

i=1Yi) (

n

i=1Xi)

nn

i=1Y 2i-(

n

i=1Yi)2 n

n

i=1X 2i-(

n

i=1Xi)2

.

4) IQsofindividualsarenormallydistributedwithameanof100andastandarddeviationof16.Ifyousampledstudentsatyourcollegeandassumed,asthenullhypothesis,thattheyhadthesameIQasthepopulation,theninarandomsampleofsize(a)n=25,findPr(Y97).(c)n=144,findPr(101

5) Considerthefollowingalternativeestimatorforthepopulationmean:

Y~=1n( 14Y1+

74Y2+

14Y3+

74Y4+...+

14Yn1+

74Yn)

ProvethatY~isunbiasedandconsistent,butnotefficientwhencomparedtoY.

Answer: E(Y~)=1n( 14E(Y1)+

74E(Y2)+

14E(Y3)+

74E(Y4)+...+

14E(Yn-1)+

74E(Yn))

=1nY(2+2+...+

14+7

4)=nnY=Y.HenceY

~isunbiased.

var(Y~)=E(Y

~)-Y)

2=E[ 1n( 14Y1+

74Y2+

14Y3+

74Y4+...+

14Yn-1+

74Yn)-Y]

2

=1

n2E[ 1

4(Y1-Y)+

74(Y2-Y)+...+

14(Yn-1-Y)+

74(Yn-Y)]

2

=1

n2[ 116E(Y1-Y)

2+4916E(Y2-Y)

2+...+ 116E(Yn-1-Y)

2+4916E(Yn-Y)

2]

=1

n2[ 116

2Y +4916

2Y +...+116

2Y +4916

2Y ]= 2Y

n2[n2( 116

+496)]=1.5625

2Y

n.

Sincevar(Y~)0asn,Y

~isconsistent.Y

~hasalargervariancethanYandisthereforenotas

efficient.

6) Imaginethatyouhadsampled1,000,000femalesand1,000,000malestotestwhetherornotfemaleshaveahigherIQthanmales.IQsarenormallydistributedwithameanof100andastandarddeviationof16.YouareexcitedtofindthatfemaleshaveanaverageIQof101inyoursample,whilemaleshaveanIQof99.Doesthisdifferenceseemimportant?Doyoureallyneedtocarryoutat-testfordifferencesinmeanstodeterminewhetherornotthisdifferenceisstatisticallysignificant?Whatdoesthisresulttellyouabouttestinghypotheseswhensamplesizesareverylarge?Answer: Thedifferenceseemsverysmall,bothintermsofabsolutevaluesand,moreimportantly,intermsof

standarddeviations.Withasamplesizeaslargeasn=1,000,000,thestandarderrorbecomesextremelysmall.Thisimpliesthatthedistributionofmeans,ordifferencesinmeans,hasalmostturnedintoaspike.Inessence,youare(verycloseto)observingthepopulation.Itisthereforeunnecessarytotestwhetherornotthedifferenceisstatisticallysignificant.Afterall,ifinthepopulation,themaleIQwere99.99andthefemaleIQwere100.01,theywouldbedifferent.Ingeneral,whensamplesizesbecomeverylarge,itisveryeasytorejectnullhypothesesaboutpopulationmeans,whichinvolvesamplemeansasanestimator,evenifhypothesizeddifferencesareverysmall.Thisistheresultofthedistributionofsamplemeanscollapsingfairlyrapidlyassamplesizesincrease.


7) LetYbeaBernoullirandomvariablewithsuccessprobabilityPr(Y = 1)= p,andletY1,...,Ynbei.i.d.draws

fromthisdistribution.Letp^bethefractionofsuccesses(1s)inthissample.Inlargesamples,thedistributionof

p^willbeapproximatelynormal,i.e.,p

^isapproximatelydistributedN(p,p(1-p)

n).NowletXbethenumberof

successesandnthesamplesize.Inasampleof10voters(n=10),iftherearesixwhovoteforcandidateA,thenX

=6.RelateX,thenumberofsuccess,top^,thesuccessproportion,orfractionofsuccesses.Next,usingyour

knowledgeoflineartransformations,derivethedistributionofX.

Answer: X=np^.Henceifp

^isdistributedN(p,p(1- p)

n),then,giventhatXisalineartransformationofp

^,Xis

distributedN(np,np(1-p)).

8) Whenyouperformhypothesistests,youarefacedwithfourpossibleoutcomesdescribedintheaccompanyingtable.

Decisionbasedon Truth(Population)sample H0istrue H1istrueRejectH0 I DonnotrejectH0 II

indicatesacorrectdecision,andIandIIindicatethatanerrorhasbeenmade.Inprobabilityterms,statethemistakesthathavebeenmadeinsituationIandII,andrelatethesetotheSizeofthetestandthePowerofthetest(ortransformationsofthese).Answer: I:Pr(rejectH0 H0iscorrect)= Sizeofthetest.

II:Pr(rejectH1 H1iscorrect)=(1-Powerofthetest).

9) Assumethatunderthenullhypothesis,Yhasanexpectedvalueof500andastandarddeviationof20.Underthealternativehypothesis,theexpectedvalueis550.Sketchtheprobabilitydensityfunctionforthenullandthealternativehypothesisinthesamefigure.Pickacriticalvaluesuchthatthep-valueisapproximately5%.Marktheareas,whichshowthesizeandthepowerofthetest.Whathappenstothepowerofthetestifthealternativehypothesismovesclosertothenullhypothesis,i.e.,,Y=540,530,520,etc.?

Answer: Foragivensizeofthetest,thepowerofthetestislower.


10) Thenetweightofabagofflourisguaranteedtobe5poundswithastandarddeviationof0.05pounds.Youareconcernedthattheactualweightisless.Totestforthis,yousample25bags.Carefullystatethenullandalternativehypothesisinthissituation.Determineacriticalvaluesuchthatthesizeofthetestdoesnotexceed5%.Findingtheaverageweightofthe25bagstobe4.7pounds,canyourejectthenullhypothesis?Whatisthepowerofthetesthere?Whyisitsolow?Answer: LetYbethenetweightofthebagofflour.ThenH0 :E(Y)= 5andH1 :E(Y)

13) Yourtextbookstatesthatwhenyoutestfordifferencesinmeansandyouassumethatthetwopopulationvariancesareequal,thenanestimatorofthepopulationvarianceisthefollowingpooledestimator:

S 2pooled =1

nm+nw-2

nm

i=1(Yi-Ym)2 +

nw

i=1(Yi-Yw)2

Explainwhythispooledestimatorcanbelookedatastheweightedaverageofthetwovariances.

Answer: S 2pooled =1

nm+nw-2

nm

i=1(Yi-Ym)2 +

nw

i=1(Yi-Yw)2

= 1nm+nw-2

(nm-1) s2m+(nw-1) s

2w

=(nm-1)nm+nw-2

S 2m+(nw-1)

nm+nw-2S 2w .

14) Yourtextbooksuggestsusingthefirstobservationfromasampleofn asanestimatorofthepopulationmean.

Itisshownthatthisestimatorisunbiasedbuthasavarianceof 2Y ,whichmakesitlessefficientthanthe

samplemean.Explainwhythisestimatorisnotconsistent.Youdevelopanotherestimator,whichisthesimpleaverageofthefirstandlastobservationinyoursample.Showthatthisestimatorisalsounbiasedandshowthatitismoreefficientthantheestimatorwhichonlyusesthefirstobservation.Isthisestimatorconsistent?Answer: Theestimatorisnotconsistentbecauseitsvariancedoesnotvanishasngoestoinfinity,i.e.,var(Y1) 0

asndoesnothold.

Y~=12(Y1+Yn).E(Y

~)=1

2(E(Y1)+E(Yn))=

12(Y+Y)=Y.HenceY

~isunbiased.var(Y

~)=E(Y

~-Y)

2=

E[( 12Y1+

12Yn)-Y]

2

=E[( 12(Y1-Y)+

12(Yn-Y)]

2= 14[E(Y1+Y]

2+E(Yn-Y)2]=1

4[ 2Y +

2Y ]

= 2Y

2.

Sincevar(Y~)0asn,doesnothold,Y

~isnotconsistent.

var(Y~)

15) LetpbethesuccessprobabilityofaBernoullirandomvariableY,i.e.,p=Pr(Y=1).Itcanbeshownthatp^,the

fractionofsuccessesinasample,isasymptoticallydistributedN(p,p(1p)n

.Usingtheestimatorofthevariance

ofp^, p

^(1-p

^)

n,constructa95%confidenceintervalforp.Showthatthemarginforsamplingerrorsimplifiesto

1/ nifyouused2insteadof1.96assuming,conservatively,thatthestandarderrorisatitsmaximum.Constructatableindicatingthesamplesizeneededtogenerateamarginofsamplingerrorof1%,2%,5%and10%.Whatdoyounoticeabouttheincreaseinsamplesizeneededtohalvethemarginoferror?(Themarginof

samplingerroris1.96SE(p^).)

Answer: The95%confidenceintervalforpisp^1.96 p

^(1-p

^)

n. p

^(1-p

^)

nisatamaximumforp

^=0.5,inwhich

casetheconfidenceintervalreducestop^1.96 0.25

np

^ 1

n,andthemarginofsamplingerroris

1n.

1n

n

0.01 10,0000.02 2,5000.05 4000.10 100

Tohalvethemarginoferror,thesamplesizehastoincreasefourfold.

16) LetYbeaBernoullirandomvariablewithsuccessprobabilityPr(Y = 1)= p,andletY1,...,Ynbei.i.d.draws

fromthisdistribution.Letp^bethefractionofsuccesses(1s)inthissample.Giventhefollowingstatement

Pr(-1.96

17) Yourtextbookmentionsthatdividingthesamplevariancebyn 1insteadofn iscalledadegreesoffreedomcorrection.Themeaningofthetermstemsfromthefactthatonedegreeoffreedomisusedupwhenthemeanisestimated.Hencedegreesoffreedomcanbeviewedasthenumberofindependentobservationsremainingafterestimatingthesamplemean.

Consideranexamplewhereinitiallyyouhave20independentobservationsontheheightofstudents.Aftercalculatingtheaverageheight,yourinstructorclaimsthatyoucanfigureouttheheightofthe20thstudentifsheprovidesyouwiththeheightoftheother19studentsandthesamplemean.Henceyouhavelostonedegreeoffreedom,orthereareonly19independentbitsofinformation.Explainhowyoucanfindtheheightofthe20thstudent.

Answer: SinceY= 120

20

i=1Yi, 20Y=

20

i=1

Yi =Y20+19

i=1

Yi .Henceknowledgeofthesamplemeanandthe

heightoftheother19studentsissufficientforfindingtheheightofthe20thstudent.

18) Theaccompanyingtableliststheheight(STUDHGHT)ininchesandweight(WEIGHT)inpoundsoffivecollegestudents.Calculatethecorrelationcoefficient.

STUDHGHTWEIGHT

74 165 73 165 72 145 68 155 66 140

Answer: r=0.72.

19) (Requirescalculus.)LetYbeaBernoullirandomvariablewithsuccessprobabilityPr(Y=1)=p.Itcanbe

shownthatthevarianceofthesuccessprobabilitypis p(1p)n

.Usecalculustoshowthatthisvarianceis

maximizedforp=0.5.

Answer:p(1-p)np

=1-pn

-pn=0.Hence1-2p=0orp=1

2.


20) Considertwoestimators:onewhichisbiasedandhasasmallervariance,theotherwhichisunbiasedandhasalargervariance.Sketchthesamplingdistributionsandthelocationofthepopulationparameterforthissituation.Discussconditionsunderwhichyoumayprefertousethefirstestimatoroverthesecondone.Answer: Thebiasindicateshowfaraway,onaverage,theestimatorisfromthepopulationvalue.Althoughthis

averageiszeroforanunbiasedestimator,theremaybequitesomevariationaroundthepopulationmean.Inasingledraw,thereisthereforeahighprobabilityofbeingsomedistanceawayfromthepopulationmean.Ontheotherhand,ifthevarianceisverysmallandtheestimatorisbiasedbyasmallamount,thentheprobabilityofbeingclosertothepopulationvaluemaybehigher.(Thebiasedestimatormayhaveasmallermeansquareerrorthantheunbiasedestimator.)


21) AttheStockandWatson(http://www.pearsonhighered.com/stock_watson )websitegotoStudentResourcesandselecttheoptionDatasetsforReplicatingEmpiricalResults.Thenselectthechapter8CPSdataset(ch8_cps.xls)intoaspreadsheetprogramsuchasExcel.Fortheexercise,usethefirst500observationsonly.Usingdataforaveragehourlyearningsonly(ahe)andyearsofeducation(yrseduc),produceascatterplotwithearningsontheverticalaxisandeducationlevelonthehorizontalaxis.Whatkindofrelationshipdoesthescatterplotsuggest?Confirmyourimpressionbyaddingalineartrendline.Findthecorrelationcoefficientbetweenthetwoandinterpretit.

Answer:

Withoutthetrendlineadded,theredoesnotseemtobemuchofalinearrelationshipbetweenaveragehourlyearningsandyearsofeducation.Perhapsalinearrelationshipisnotplausiblesinceitwouldimplythatthereturnstoeducationwouldbecomesmallerasfurtheryearsofeducationareadded.However,andregardlessofthelinearityissues,thereisapositiverelationshipinthedatabetweenthetwovariables,whichbecomesvisiblewhenthetrendlineisadded.Thecorrelationcoefficientispositiveandhasavalueof46.9%,whichisreasonablyhigh(thecorrelationbetweenheightandweightforcollegestudentsisapproximately50%bycomparison).

22) IQscoresarenormallydistributedwithanaverageof100andastandarddeviationof16.Someresearchsuggeststhatleft-handedindividualshaveahigherIQscorethanright-handedindividuals.Totestthishypothesis,aresearcherrandomlyselects132individualsandfindsthattheiraverageIQis103.2withasamplestandarddeviationof14.6.Usingtheresultsfromthesample,canyourejectthenullhypothesisthatleft-handedpeoplehaveanIQof100vs.thealternativethattheyhaveahigherIQ?Whatcriticalvalueshouldyouchooseifthesizeofthetestis5%?

Answer: ThehypothesisisH0:=100versusthealternativeH1:>100.Theteststatisticist=103.2-100

14.6132

=2.52.

Sincethecriticalvaluefortheone-sidedalternativeis1.645atthe5%significancelevel,theresearchershouldrejectthenullhypothesisthatleft-handedindividualshaveanIQof100.


23) AttheStockandWatson(http://www.pearsonhighered.com/stock_watson )websitegotoStudentResourcesandselecttheoptionDatasetsforReplicatingEmpiricalResults.ThenselecttheTestScoredatasetusedinChapters4-9(caschool.xls)andopentheExceldataset.Nextproduceascatterplotoftheaveragereadingscore(horizontalaxis)andtheaveragemathematicsscore(verticalaxis).Whatdoesthescatterplotsuggest?Calculatethecorrelationcoefficientbetweenthetwoseriesandgiveaninterpretation.

Answer:

Thescatterplotsuggeststhat,onaverage,schoolswhichperformhighlyonthereadingscorewillalsoperformhighlyonthemathematicsscore.Thesamplecorrelationbetweenthetwoseriesis92.3%,suggestingahighpositivecorrelationbetweenthetwovariables.

24) In2007,astudyofcloseto250,00018-19year-oldNorwegianmalesfoundthatfirst-bornshaveanIQthatis2.3pointshigherthanthosewhoaresecond-born.Toseeifyoucanfindasimilarevidenceatyouruniversity,youcollectdatafrom250students,ofwhich140arefirst-borns.AftersubjectingeachoftheseindividualstoanIQtest,youfindthatthefirst-bornsscore108.3withastandarddeviationof13.2,whilethesecondbornsachieve107.1withastandarddeviationof11.6.Youhypothesizethatfirst-bornsandsecond-bornsinauniversitypopulationhaveidenticalIQsagainsttheone-sidedalternativehypothesisthatfirstbornshavehigherIQs.Usingasizeofthetestof5%,whatisyourconclusion?

Answer: GiventhatyournullhypothesisstatesH0:first=second,yourteststatisticist=108.3- 107.1

13.22140

+11.62

110

=

0.76.Sincethecriticalvaluefortheone-sidedalternativetestis1.64,youcannotrejectthenullhypothesis.


Chapter4 LinearRegressionwithOneRegressor4.1 MultipleChoice

1) Whentheestimatedslopecoefficientinthesimpleregressionmodel,^1,iszero,then

A) R2=Y.B) 0TSSD) R2=1-(ESS/TSS)

Answer: A

5) BinaryvariablesA) aregenerallyusedtocontrolforoutliersinyoursample.B) cantakeonmorethantwovalues.C) excludecertainindividualsfromyoursample.D) cantakeononlytwovalues.

Answer: D


6) Thefollowingareallleastsquaresassumptionswiththeexceptionof:A) Theconditionaldistributionofui givenXi hasameanofzero.B) Theexplanatoryvariableinregressionmodelisnormallydistributed.C) (Xi,Yi),i=1,...,nareindependentlyandidenticallydistributed.D) Largeoutliersareunlikely.

Answer: B

7) ThereasonwhyestimatorshaveasamplingdistributionisthatA) economicsisnotaprecisescience.B) individualsresponddifferentlytoincentives.C) inreallifeyoutypicallygettosamplemanytimes.D) thevaluesoftheexplanatoryvariableandtheerrortermdifferacrosssamples.

Answer: D

8) Inthesimplelinearregressionmodel,theregressionslopeA) indicatesbyhowmanypercentY increases,givenaonepercentincreaseinX.B) whenmultipliedwiththeexplanatoryvariablewillgiveyouthepredictedY.C) indicatesbyhowmanyunitsYincreases,givenaoneunitincreaseinX.D) representstheelasticityofYonX.

Answer: C

9) TheOLSestimatorisderivedbyA) connectingtheYicorrespondingtothelowestXi observationwiththeYi correspondingtothehighestXi

observation.B) makingsurethatthestandarderroroftheregressionequalsthestandarderroroftheslopeestimator.C) minimizingthesumofabsoluteresiduals.D) minimizingthesumofsquaredresiduals.

Answer: D

10) InterpretingtheinterceptinasampleregressionfunctionisA) notreasonablebecauseyouneverobservevaluesoftheexplanatoryvariablesaroundtheorigin.B) reasonablebecauseundercertainconditionstheestimatorisBLUE.C) reasonableifyoursamplecontainsvaluesofXi aroundtheorigin.D) notreasonablebecauseeconomistsareinterestedintheeffectofachangeinXonthechangeinY.

Answer: C

11) ThevarianceofYiisgivenby

A) 20 +21 var(Xi)+var(ui).

B) thevarianceofui.

C) 21 var(Xi)+var(ui).

D) thevarianceoftheresiduals.Answer: C

12) (RequiresAppendix)ThesampleaverageoftheOLSresidualsisA) somepositivenumbersinceOLSusessquares.B) zero.C) unobservablesincethepopulationregressionfunctionisunknown.D) dependentonwhethertheexplanatoryvariableismostlypositiveornegative.

Answer: B


13) TheOLSresiduals,uî,aredefinedasfollows:

A) Yî-

^0-

^1Xi

B) Yi-0-1Xi

C) Yi-Yî

D) (Yi-Y)2

Answer: C

14) Theslopeestimator,1,hasasmallerstandarderror,otherthingsequal,ifA) thereismorevariationintheexplanatoryvariable,X.B) thereisalargevarianceoftheerrorterm,u.C) thesamplesizeissmaller.D) theintercept,0,issmall.

Answer: A

15) TheregressionR2isameasureofA) whetherornotXcausesY.B) thegoodnessoffitofyourregressionline.C) whetherornotESS>TSS.D) thesquareofthedeterminantofR.

Answer: B

16) (RequiresAppendix)ThesampleregressionlineestimatedbyOLSA) willalwayshaveaslopesmallerthantheintercept.B) isexactlythesameasthepopulationregressionline.C) cannothaveaslopeofzero.D) willalwaysrunthroughthepoint(X,Y).

Answer: D

17) TheOLSresidualsA) canbecalculatedusingtheerrorsfromtheregressionfunction.B) canbecalculatedbysubtractingthefittedvaluesfromtheactualvalues.C) areunknownsincewedonotknowthepopulationregressionfunction.D) shouldnotbeusedinpracticesincetheyindicatethatyourregressiondoesnotrunthroughallyour

observations.Answer: B

18) Thenormalapproximationtothesamplingdistributionof^1ispowerfulbecause

A) manyexplanatoryvariablesinreallifearenormallydistributed.B) itallowseconometricianstodevelopmethodsforstatisticalinference.C) manyotherdistributionsarenotsymmetric.D) isimpliesthatOLSistheBLUEestimatorfor1.

Answer: B


19) Ifthethreeleastsquaresassumptionshold,thenthelargesamplenormaldistributionof^1is

A) N(0,1nvar[Xi-X)ui]

[var(Xi)]2).

B) N(1,1nvar(ui)]2

[var(Xi)]2).

C) N(1, 2u

n

i=1(Xi-X)2

.

D) N(1,1nvar(ui)]

[var(Xi)]2).

Answer: B

20) InthesimplelinearregressionmodelYi = 0 + 1Xi+ ui,A) theinterceptistypicallysmallandunimportant.B) 0+1Xirepresentsthepopulationregressionfunction.C) theabsolutevalueoftheslopeistypicallybetween0and1.D) 0+1Xirepresentsthesampleregressionfunction.

Answer: B

21) Toobtaintheslopeestimatorusingtheleastsquaresprinciple,youdividetheA) samplevarianceofXbythesamplevarianceofY.B) samplecovarianceofXandYbythesamplevarianceofY.C) samplecovarianceofXandYbythesamplevarianceofX.D) samplevarianceofXbythesamplecovarianceofX andY.

Answer: C

22) Todecidewhetherornottheslopecoefficientislargeorsmall,A) youshouldanalyzetheeconomicimportanceofagivenincreaseinX.B) theslopecoefficientmustbelargerthanone.C) theslopecoefficientmustbestatisticallysignificant.D) youshouldchangethescaleoftheX variableifthecoefficientappearstobetoosmall.

Answer: A

23) E(ui Xi)=0saysthatA) dividingtheerrorbytheexplanatoryvariableresultsinazero(onaverage).B) thesampleregressionfunctionresidualsareunrelatedtotheexplanatoryvariable.C) thesamplemeanoftheXsismuchlargerthanthesamplemeanoftheerrors.D) theconditionaldistributionoftheerrorgiventheexplanatoryvariablehasazeromean.

Answer: D

24) Inthelinearregressionmodel,Yi=0+ 1Xi + ui,0 + 1XiisreferredtoasA) thepopulationregressionfunction.B) thesampleregressionfunction.C) exogenousvariation.D) theright-handvariableorregressor.

Answer: A


25) Multiplyingthedependentvariableby100andtheexplanatoryvariableby100,000leavestheA) OLSestimateoftheslopethesame.B) OLSestimateoftheinterceptthesame.C) regressionR2thesame.D) varianceoftheOLSestimatorsthesame.

Answer: C

26) Assumethatyouhavecollectedasampleofobservationsfromover100householdsandtheirconsumptionandincomepatterns.Usingtheseobservations,youestimatethefollowingregressionCi=0+1Yi+uiwhereCisconsumptionandYisdisposableincome.Theestimateof1willtellyou

A) IncomeConsumption

B) Theamountyouneedtoconsumetosurvive

C) IncomeConsumption

D) ConsumptionIncome

Answer: D

27) Inwhichofthefollowingrelationshipsdoestheintercepthaveareal-worldinterpretation?A) therelationshipbetweenthechangeintheunemploymentrateandthegrowthrateofrealGDP

(OkunsLaw)B) thedemandforcoffeeanditspriceC) testscoresandclass-sizeD) weightandheightofindividuals

Answer: A

28