Economics Chapter 12 Multiple Choice Choose The One Alternative That

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Ch.12 InferenceonCategoricalData

12.1 Goodness–o

–FitTest

1 Performagoodness–of–fittest.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Provideanappropriateresponse.

1) Determinetheexpectedcountsforeachoutcome.

n=600

pi0.25 0.05 0.35 0.35

ExpectedCounts

pi0.25 0.05 0.35 0.35

ExpectedCounts 150 30 210 210

pi0.25 0.05 0.35 0.35

ExpectedCounts 25 5 35 35

pi0.25 0.05 0.35 0.35

ExpectedCounts 154 28 211 207

pi0.25 0.05 0.35 0.35

ExpectedCounts 1500 300 2100 2100

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

2) Amultinomialexperimentwithk=4cellsandn=300producedthedatashowninthefollowingtable.

Cell

1234

ni65 69 80 86

Dothesedataprovidesufficientevidencetocontradictthenullhypothesisthatp1=0.20,p2=0.20,

p3=0.30,andp4=0.30?Testusingα=0.05.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

3) Manytrackhurdlersbelievethattheyhaveabetterchanceofwinningiftheystartintheinsidelanethatis

closesttothefield.Forthedatabelow,thelaneclosesttothefieldisLane1,thenextlaneisLane2,andsoon

untiltheoutermostlane,Lane6.Thedataliststhenumberofwinsfortrackhurdlersinthedifferentstarting

positions.Calculatethechi–squareteststatisticχ2totesttheclaimthattheprobabilitiesofwinningarethe

sameinthedifferentpositions.Useα=0.05.Theresultsarebasedon240wins.

StartingPosition 123456

NumberofWins 33 50 44 32 36 45

A) 6.750 B) 9.326 C) 12.592 D) 15.541

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4) Manytrackhurdlersbelievethattheyhaveabetterchanceofwinningiftheystartintheinsidelanethatis

closesttothefield.Forthedatabelow,thelaneclosesttothefieldisLane1,thenextlaneisLane2,andsoon

untiltheoutermostlane,Lane6.Thedataliststhenumberofwinsfortrackhurdlersinthedifferentstarting

positions.Findthecriticalvalueχ 2

0totesttheclaimthattheprobabilitiesofwinningarethesameinthe

differentpositions.Useα=0.05.Theresultsarebasedon240wins.

StartingPosition 123456

NumberofWins 45 36 32 33 44 50

A) 11.070 B) 9.236 C) 15.086 D) 12.833

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

5) Manytrackhurdlersbelievethattheyhaveabetterchanceofwinningiftheystartintheinsidelanethatis

closesttothefield.Forthedatabelow,thelaneclosesttothefieldisLane1,thenextlaneisLane2,andsoon

untiltheoutermostlane,Lane6.Thedataliststhenumberofwinsfortrackhurdlersinthedifferentstarting

positions.Testtheclaimthattheprobabilitiesofwinningarethesameinthedifferentpositions.Useα=0.05.

Theresultsarebasedon240wins.

StartingPosition 123456

NumberofWins 32 44 36 33 45 50

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

6) Acompanywantstodetermineifitsemployeeshaveanypreferenceamong5differenthealthplanswhichit

offerstothem.Asampleof200employeesprovidedthedatabelow.Calculatethechi –squareteststatisticχ2to

testtheclaimthattheprobabilitiesshownopreference.Useα=0.01.

Plan 12345

Employees 65 32 18 55 30

A) 37.45 B) 45.91 C) 48.91 D) 55.63

7) Acompanywantstodetermineifitsemployeeshaveanypreferenceamong5differenthealthplanswhichit

offerstothem.Asampleof200employeesprovidedthedatabelow.Findthecriticalvalueχ2

0totesttheclaim

thattheprobabilitiesshownopreference.Useα=0.01.

Plan 12345

Employees 30 55 32 18 65

A) 13.277 B) 9.488 C) 11.143 D) 14.860

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

8) Acompanywantstodetermineifitsemployeeshaveanypreferenceamong5differenthealthplanswhichit

offerstothem.Asampleof200employeesprovidedthedatabelow.Testtheclaimthattheprobabilitiesshow

nopreference.Useα=0.01.

Plan 12345

Employees 32 18 55 65 30

Page313

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

9) Ateacherfiguresthatfinalgradesinthechemistrydepartmentaredistributedas:A,25%;B,25%; C,40%;

D,5%;F,5%.Attheendofarandomlyselectedsemester,thefollowingnumberofgradeswererecorded.

Calculatethechi–squareteststatisticχ2todetermineifthegradedistributionforthedepartmentisdifferent

thanexpected.Useα=0.01.

Grade ABCDF

Number 42 36 60 14 8

A) 5.25 B) 6.87 C) 0.6375 D) 4.82

10) Ateacherfiguresthatfinalgradesinthechemistrydepartmentaredistributedas:A,25%;B,25%; C,40%;

D,5%;F,5%.Attheendofarandomlyselectedsemester,thefollowingnumberofgradeswererecorded.Find

thecriticalvalueχ 2

0todetermineifthegradedistributionforthedepartmentisdifferentthanexpected.Useα

=0.01.

Grade ABCDF

Number 36 42 60 814

A) 13.277 B) 15.086 C) 9.488 D) 7.779

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

11) Ateacherfiguresthatfinalgradesinthechemistrydepartmentaredistributedas:A,25%;B,25%; C,40%;

D,5%;F,5%.Attheendofarandomlyselectedsemester,thefollowingnumberofgradeswererecorded.

Determineifthegradedistributionforthedepartmentisdifferentthanexpected.Useα

=0.01.

Grade ABCDF

Number 36 42 60 14 8

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

12) Aspinnerismountedonapieceofcardboarddividedintosixareasofequalsize.Eachofthesixareasisa

differentcolor(blue,yellow,red,green,whiteandorange).Whenthespinnerisspun,eachcolorshouldbe

selectedbythespinnerapproximately1

6ofthetime.Astudentsuspectsthatacertainspinnerisdefective.The

suspectedspinnerisspun90times.Theresultsareshownbelow.Calculatethechi –squareteststatisticχ2to

testthestudentʹsclaim.Useα=0.10.

Color Blue Yellow Red Green White Orange

Frequency 15 12 11 17 19 16

A) 3.067 B) 2.143 C) 5.013 D) 4.312

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13) Aspinnerismountedonapieceofcardboarddividedintosixareasofequalsize.Eachofthesixareasisa

differentcolor(blue,yellow,red,green,whiteandorange).Whenthespinnerisspun,eachcolorshouldbe

selectedbythespinnerapproximately1

6ofthetime.Astudentsuspectsthatacertainspinnerisdefective.The

suspectedspinnerisspun90times.Theresultsareshownbelow.Findthecriticalvalueχ2

0totestthe

studentʹsclaim.Useα=0.10.

Color Blue Yellow Red Green White Orange

Frequency 12 11 16 19 17 15

A) 9.236 B) 1.610 C) 10.645 D) 11.071

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

14) Aspinnerismountedonapieceofcardboarddividedintosixareasofequalsize.Eachofthesixareasisa

differentcolor(blue,yellow,red,green,whiteandorange).Whenthespinnerisspun,eachcolorshouldbe

selectedbythespinnerapproximately1

6ofthetime.Astudentsuspectsthatacertainspinnerisdefective.The

suspectedspinnerisspun90times.Theresultsareshownbelow.Testthestudentʹsclaim.Useα

=0.10.

Color Blue Yellow Red Green White Orange

Frequency 12 11 17 16 19 15

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

15) Arandomsampleof160carpurchasesareselectedandcategorizedbyage.Theresultsarelistedbelow.The

agedistributionofdriversforthegivencategoriesis18%fortheunder26group,39%forthe26 –45group,31%

forthe45–65group,and12%forthegroupover65.Calculatethechi–squareteststatisticχ2totesttheclaim

thatallageshavepurchaseratesproportionaltotheirdrivingrates.Useα

=0.05.

Age Under26 26–45 46–65 Over65

Purchases 66 39 25 30

A) 75.101 B) 85.123 C) 101.324 D) 95.431

16) Arandomsampleof160carpurchasesareselectedandcategorizedbyage.Theresultsarelistedbelow.The

agedistributionofdriversforthegivencategoriesis18%fortheunder26group,39%forthe26–45group,31%

forthe45–65group,and12%forthegroupover65.Findthecriticalvalueχ2

0totesttheclaimthatallages

havepurchaseratesproportionaltotheirdrivingrates.Useα=0.05.

Age Under26 26–45 46–65 Over65

Purchases 66 39 25 30

A) 7.815 B) 6.251 C) 11.143 D) 9.348

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

17) Arandomsampleof160carpurchasesareselectedandcategorizedbyage.Theresultsarelistedbelow.The

agedistributionofdriversforthegivencategoriesis18%fortheunder26group,39%forthe26–45group,31%

forthe46–65group,and12%forthegroupover65.Testtheclaimthatallageshavepurchaserates

proportionaltotheirdrivingrates.Useα=0.05.

Age Under26 26–45 46–65 Over65

Purchases 66 39 25 30

Page315

18) Theresultsofarecentnationalsurveyreportedthat70%ofAmericansrecycleatleastsomeofthetime.Aspart

oftheirfinalprojectinstatisticsclass,NaylaandRobertosurvey5randomstudentsoncampusandaskthemif

theyrecycleatleastsomeofthetime.Theythenrepeatthisexperiment1000times.Theresultsoftheirresearch

areshownbelow.

X(numberwhorecycleoutof5) 012345

Frequency 227 139 305 358 169

IsthereevidencetosupportthebeliefthattherandomvariableX,thenumberofstudentsoutof5whorecycle

atleastsomeofthetime,isabinomialrandomvariablewithp=0.7attheα=0.05level?

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

19) Asthenumberofthedegreesoffreedomincreases,theχ2distribution

ecomesmoresymmetric.

ecomeslesssymmetric.

C) doesnotchangeshapeasthedegreesoffreedomchange.

ecomesexponential.

20) A__________________testisaninferentialprocedureusedtodeterminewhetherafrequencydistribution

followsadefineddistribution.

A) goodness–of–fit B) χ2C) F D) normality

21) Thedegreesoffreedomforaχ2goodness–of–fittestwhenthereare6categoriesandasampleofsize1200is

A) 5 B) 6 C) 1199 D) 1205

12.2 TestsforIndependenceandtheHomogeneityofProportions

1 Performatestforindependence.

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Provideanappropriateresponse.

1) Testthenullhypothesisofindependenceofthetwoclassifications,AandB,ofthe3×3contingencytable

shownbelow.Testusingα=0.10.

B

B1B2B3

A119 40 60

255 23 22

A331 42 47

Page316

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

2) Thecontingencytablebelowshowstheresultsofarandomsampleof200registeredvotersthatwasconducted

toseewhethertheiropinionsonabillarerelatedtotheirpartyaffiliation.Useα

=0.05.

Party Opinion

Republican

Democrat

Independent

Approve Disapprove NoOpinion

42 20 14

50 24 18

10 16 6

Findthecriticalvalueχ 2

0,totesttheclaimofindependence.

A) 9.488 B) 7.779 C) 11.143 D) 13.277

3) Thecontingencytablebelowshowstheresultsofarandomsampleof200registeredvotersthatwasconducted

toseewhethertheiropinionsonabillarerelatedtotheirpartyaffiliation.

Party Opinion

Republican

Democrat

Independent

Approve Disapprove NoOpinion

42 20 14

50 24 18

10 16 6

Findthechi–squareteststatistic,χ2,totesttheclaimofindependence.

A) 8.030 B) 11.765 C) 7.662 D) 9.483

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

4) Thecontingencytablebelowshowstheresultsofarandomsampleof200registeredvotersthatwasconducted

toseewhethertheiropinionsonabillarerelatedtotheirpartyaffiliation.

Party Opinion

Republican

Democrat

Independent

Approve Disapprove NoOpinion

42 20 14

50 24 18

10 16 6

Testtheclaimofindependence.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

5) Aresearcherwantstodetermineifthenumberofminutesspentwatchingtelevisionperdayisindependentof

gender.Arandomsampleof315adultswasselectedandtheresultsareshownbelow.Findthecriticalvalue

χ2

0todetermineifthereisenoughevidencetoconcludethatthenumberofminutesspentwatchingtelevision

perdayisrelatedtogender.Useα=0.05.

Gender MinutesspentwatchingTVperday

Male

Female

0–30 30–60 60–90 90–over

25 35 75 45

30 45 45 15

A) 7.815 B) 9.348 C) 11.345 D) 6.251

Page317

6) Aresearcherwantstodetermineifthenumberofminutesspentwatchingtelevisionperdayisindependentof

gender.Arandomsampleof315adultswasselectedandtheresultsareshownbelow.Calculatethechi –square

statisticχ2todetermineifthereisenoughevidencetoconcludethatthenumberofminutesspentwatching

televisionperdayisrelatedtogender.Useα=0.05.

Gender MinutesspentwatchingTVperday

Male

Female

0–30 30–60 60–90 90–over

25 35 75 45

30 45 45 15

A) 18.146 B) 19.874 C) 20.912 D) 21.231

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

7) Aresearcherwantstodetermineifthenumberofminutesspentwatchingtelevisionperdayisindependentof

gender.Arandomsampleof315adultswasselectedandtheresultsareshownbelow.Isthereenough

evidencetoconcludethatthenumberofminutesspentwatchingtelevisionperdayisrelatedtogender?Useα

=0.05.

Gender MinutesspentwatchingTVperday

Male

Female

0–30 30–60 60–90 90–over

25 35 75 45

30 45 45 15

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

8) Amedicalresearcherisinterestedindeterminingifthereisarelationshipbetweenadultsover50whoexercise

regularlyandlow,moderate,andhighbloodpressure.Arandomsampleof236adultsover50isselectedand

theresultsaregivenbelow.Findthecriticalvalueχ2

0totesttheclaimthatregularexerciseandlow,moderate,

andhighbloodpressureareindependent.Useα=0.01.

BloodPressure Low Moderate High

Reg.Exercise 35 62 25

NoReg.Exercise 21 65 28

A) 9.210 B) 9.348 C) 6.251 D) 5.991

9) Amedicalresearcherisinterestedindeterminingifthereisarelationshipbetweenadultsover50whoexercise

regularlyandlow,moderate,andhighbloodpressure.Arandomsampleof236adultsover50isselectedand

theresultsaregivenbelow.Calculatethechi–squareteststatisticχ2totesttheclaimthatregularexerciseand

low,moderate,andhighbloodpressureareindependent.Useα=0.01.

BloodPressure Low Moderate High

Reg.Exercise 35 62 25

NoReg.Exercise 21 65 28

A) 3.473 B) 16.183 C) 18.112 D) 6.003

Page318

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

10) Amedicalresearcherisinterestedindeterminingifthereisarelationshipbetweenadultsover50whoexercise

regularlyandlow,moderate,andhighbloodpressure.Arandomsampleof236adultsover50isselectedand

theresultsaregivenbelow.Testtheclaimthatregularexerciseandlow,moderate,andhighbloodpressureare

independent.Useα=0.01.

BloodPressure Low Moderate High

Reg.Exercise 35 62 25

NoReg.Exercise 21 65 28

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

11) Asportsstatisticianisinterestedindeterminingifthereisarelationshipbetweenthenumberofhometeam

andvisitingteamlossesanddifferentsports.Arandomsampleof526gamesisselectedandtheresultsare

givenbelow.Findthecriticalvalueχ 2

0totesttheclaimthatthenumberofhometeamandvisitingteamlosses

isindependentofthesport.Useα=0.01.

Football Basketball Soccer Baseball

Hometeamlosses 39 156 25 83

Visitingteamlosses 31 98 19 75

A) 11.345 B) 12.838 C) 7.815 D) 9.348

12) Asportsstatisticianisinterestedindeterminingifthereisarelationshipbetweenthenumberofhometeam

andvisitingteamlossesanddifferentsports.Arandomsampleof526gamesisselectedandtheresultsare

givenbelow.Calculatethechi–squareteststatisticχ2totesttheclaimthatthenumberofhometeamand

visitingteamlossesisindependentofthesport.Useα=0.01.

Football Basketball Soccer Baseball

Hometeamlosses 39 156 25 83

Visitingteamlosses 31 98 19 75

A) 3.290 B) 2.919 C) 5.391 D) 4.192

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

13) Asportsstatisticianisinterestedindeterminingifthereisarelationshipbetweenthenumberofhometeam

andvisitingteamlossesanddifferentsports.Arandomsampleof526gamesisselectedandtheresultsare

givenbelow.Testtheclaimthatthenumberofhometeamandvisitingteamlossesisindependentofthesport.

Useα=0.01.

Football Basketball Soccer Baseball

Hometeamlosses 39 156 25 83

Visitingteamlosses 31 98 19 75

14) Thedatabelowshowtheageandfavoritetypeofreadingof779randomlyselectedpeople.Testtheclaimthat

ageandpreferredreadingtypeareindependent.Useα=0.05.

Age CurrentEvents Mystery ScienceFiction History

15–21 21 45 90 33

21–30 68 55 42 48

30–40 65 47 31 57

40–50 60 39 25 53

Page319

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

15) Inaχ2testofindependence,thenullhypothesisisthat

A) thereisnotanassociation.

B) thereisanassociation.

C) eachelementofeachsethasthesameprobabilityofoccurrence.

D) therandomvariablesaredependent.

2 Performatestforhomogeneityofproportions.

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Provideanappropriateresponse.

1) Arandomsampleof400menand400womenwasrandomlyselectedandaskedwhethertheyplannedto

attendaconcertinthenextmonth.Theresultsarelistedbelow.Performahomogeneityofproportionstestto

testtheclaimthattheproportionofmenwhoplantoattendaconcertinthenextmonthisthesameasthe

proportionofwomenwhoplantoattendaconcertinthenextmonth.Useα

=0.05.

Men Women

Plantoattendconcert 230 255

Donʹtplantoattendconcert 170 145

2) Arandomsampleof100employeesfrom5differentcompanieswasrandomlyselected,andthenumberwho

takepublictransportationtoworkwasrecorded.Theresultsarelistedbelow.Performahomogeneityof

proportionstesttotesttheclaimthattheproportionwhotakepublictransportationtoworkisthesameinall5

companies.Useα=0.01.

Companies

12345

UsePublicTrans. 18 25 12 33 22

DonʹtUsePublicTrans. 82 75 88 67 78

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

3) Inachi–squaretestofhomogeneityofproportionswetesttheclaimsthat

A) differentpopulationshavethesameproportionsofindividualswiththesamecharacteristics.

B) acrossasinglesampletheproportionofindividualswiththesamecharacteristicisthesameasthe

population.

C) theproportionofindividualswithagivencharacteristicdoesnʹtchangeovertime.

D) theproportionofapopulationhavingagivencharacteristicisbasedonthehomogeneityofthe

population.

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Ch.12 InferenceonCategoricalData

AnswerKey

12.1 Goodness–o

–FitTest

1 Performagoodness–of–fittest.

12.2 TestsforIndependenceandtheHomogeneityofProportions

1 Performatestforindependence.

Page321

2 Performatestforhomogeneityofproportions.

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