Upload
vlabrague6426
View
400
Download
40
Embed Size (px)
Citation preview
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
Stock/Watson2e--CVC28/23/06-- Page2
14) Inthegraphbelow,theverticalaxisrepresentsaveragerealGDPgrowthfor65countriesovertheperiod1960-1995,andthehorizontalaxisshowstheaveragetradesharewithinthesecountries.
ThisisanexampleofA) cross-sectionaldata.B) experimentaldata.C) atimeseries.D) longitudinaldata.
Answer: A
Stock/Watson2e--CVC28/23/06-- Page3
15) Theaccompanyinggraph
Isanexampleof
A) cross-sectionaldata.B) experimentaldata.C) atimeseries.D) longitudinaldata.
Answer: A
Stock/Watson2e--CVC28/23/06-- Page4
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).
Stock/Watson2e--CVC28/23/06-- Page5
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
Stock/Watson2e--CVC28/23/06-- Page6
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
Stock/Watson2e--CVC28/23/06-- Page7
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
Stock/Watson2e--CVC28/23/06-- Page8
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
Stock/Watson2e--CVC28/23/06-- Page10
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
Stock/Watson2e--CVC28/23/06-- Page11
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
Stock/Watson2e--CVC28/23/06-- Page12
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
Stock/Watson2e--CVC28/23/06-- Page13
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.
Stock/Watson2e--CVC28/23/06-- Page14
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?
Stock/Watson2e--CVC28/23/06-- Page15
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
Stock/Watson2e--CVC28/23/06-- Page16
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.
Stock/Watson2e--CVC28/23/06-- Page18
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).
Stock/Watson2e--CVC28/23/06-- Page19
(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.
Stock/Watson2e--CVC28/23/06-- Page20
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
Stock/Watson2e--CVC28/23/06-- Page21
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.
Stock/Watson2e--CVC28/23/06-- Page22
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
(Incumbent-50%)>0(Y=0)
(Incumbent-50%)>0(Y=1)
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
Stock/Watson2e--CVC28/23/06-- Page23
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
Stock/Watson2e--CVC28/23/06-- Page24
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).
Stock/Watson2e--CVC28/23/06-- Page25
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)
Stock/Watson2e--CVC28/23/06-- Page27
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.
Stock/Watson2e--CVC28/23/06-- Page29
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.
Stock/Watson2e--CVC28/23/06-- Page30
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
Cumulativeprobabilitydistribution
0.05 0.20 0.50 0.75 0.90 0.98 1.00
Pr(1Y3)0.70;Pr(Y>0.25).
Stock/Watson2e--CVC28/23/06-- Page32
Stock/Watson2e--CVC28/23/06-- Page33
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
Cumulativeprobabilitydistribution
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
Stock/Watson2e--CVC28/23/06-- Page35
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
Stock/Watson2e--CVC28/23/06-- Page36
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
Stock/Watson2e--CVC28/23/06-- Page38
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
Stock/Watson2e--CVC28/23/06-- Page39
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
Stock/Watson2e--CVC28/23/06-- Page41
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.
Stock/Watson2e--CVC28/23/06-- Page43
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.
Stock/Watson2e--CVC28/23/06-- Page46
Stock/Watson2e--CVC28/23/06-- Page47
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.
Stock/Watson2e--CVC28/23/06-- Page49
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.
Stock/Watson2e--CVC28/23/06-- Page50
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.
Stock/Watson2e--CVC28/23/06-- Page51
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?
Stock/Watson2e--CVC28/23/06-- Page52
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.
Stock/Watson2e--CVC28/23/06-- Page53
2) Foreachoftheaccompanyingscatterplotsforseveralpairsofvariables,indicatewhetheryouexpectapositiveornegativecorrelationcoefficientbetweenthetwovariables,andthelikelymagnitudeofit(youcanuseasmallrange).
(a)
(b)
(c)
Stock/Watson2e--CVC28/23/06-- Page54
(d)
Answer: (a) Positivecorrelation.Theactualcorrelationcoefficientis0.46.(b)Norelationship.Theactualcorrelationcoefficientis0.00007.(c) Negativerelationship.Theactualcorrelationcoefficientis0.70.(d) Nonlinear(invertedU)relationship.Theactualcorrelationcoefficientis0.23.
Stock/Watson2e--CVC28/23/06-- Page55
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.
Stock/Watson2e--CVC28/23/06-- Page57
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.
Stock/Watson2e--CVC28/23/06-- Page58
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.
Stock/Watson2e--CVC28/23/06-- Page62
20) Considertwoestimators:onewhichisbiasedandhasasmallervariance,theotherwhichisunbiasedandhasalargervariance.Sketchthesamplingdistributionsandthelocationofthepopulationparameterforthissituation.Discussconditionsunderwhichyoumayprefertousethefirstestimatoroverthesecondone.Answer: Thebiasindicateshowfaraway,onaverage,theestimatorisfromthepopulationvalue.Althoughthis
averageiszeroforanunbiasedestimator,theremaybequitesomevariationaroundthepopulationmean.Inasingledraw,thereisthereforeahighprobabilityofbeingsomedistanceawayfromthepopulationmean.Ontheotherhand,ifthevarianceisverysmallandtheestimatorisbiasedbyasmallamount,thentheprobabilityofbeingclosertothepopulationvaluemaybehigher.(Thebiasedestimatormayhaveasmallermeansquareerrorthantheunbiasedestimator.)
Stock/Watson2e--CVC28/23/06-- Page63
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.
Stock/Watson2e--CVC28/23/06-- Page64
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.
Stock/Watson2e--CVC28/23/06-- Page65
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
Stock/Watson2e--CVC28/23/06-- Page66
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
Stock/Watson2e--CVC28/23/06-- Page67
13) TheOLSresiduals,u^i,aredefinedasfollows:
A) Y^i-
^0-
^1Xi
B) Yi-0-1Xi
C) Yi-Y^i
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
Stock/Watson2e--CVC28/23/06-- Page68
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
Stock/Watson2e--CVC28/23/06-- Page69
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