Chapter 11 A random sample of 20 days yields the data below

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Ch.11 NonparametricTests

11.1 TheSignTest

1 PerformaSignTest

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

Provideanappropriateresponse.

1) Aconveniencestoreownerbelievesthatthemediannumberoflotteryticketssoldperdayis49.Arandom

sampleof20daysyieldsthedatabelow.Findthecriticalvaluetotesttheownerʹsclaim.

Useα=0.05.

32 48 59 64 31 55 70 27 33 38

47 54 54 44 44 49 49 59 54 38

A) 4 B) 2 C) 3 D) 5

2) Aconveniencestoreownerbelievesthatthemediannumberoflotteryticketssoldperdayis73.Arandom

sampleof20daysyieldsthedatabelow.Findtheteststatisticxtotesttheownerʹsclaim.

56 72 83 88 55 79 94 51 57 62

71 78 78 68 68 73 73 83 78 62

A) 8 B) 10 C) 2 D) 18

3) Arealestateagentsurmisesthatthemedianrentforaone–

edroomapartmentinabeachcommunityin

southernCaliforniaisatleast$1300permonth.Therentsforarandomsampleof15one–bedroomapartments

arelistedbelow.Findthecriticalvaluetotesttheagentʹsclaim.Useα=0.01.

$1600 $1550 $1000 $1175 $1035

$2050 $1475 $970 $1690 $2300

$1295 $1300 $1375 $1300 $1080

A) 1 B) 2 C) 3 D) 4

4) Arealestateagentsurmisesthatthemedianrentforaone–

edroomapartmentinabeachcommunityin

southernCaliforniaisatleast$1700permonth.Therentsforarandomsampleof15one–bedroomapartments

arelistedbelow.Findtheteststatisticxtotesttheagentʹsclaim.

$2000 $1950 $1400 $1575 $1435

$2450 $1875 $1370 $2090 $2700

$1695 $1700 $1775 $1700 $1480

A) 6 B) 7 C) 13 D) 1

5) Aclubprofessionalatamajorgolfcourseclaimsthatthecourseissotoughthatevenprofessionalgolfers

rarelybreakparof68.Thescoresfromarandomsampleof20professionalgolfersarelistedbelow.Findthe

criticalvaluetotesttheclubprofessionalʹsclaim.Useα=0.05.

67 65 68 68 71 70 62 74 68 73

65 67 69 69 76 74 68 70 71 61

A) 4 B) 3 C) 2 D) 1

Page261

6) Aclubprofessionalatamajorgolfcourseclaimsthatthecourseissotoughthatevenprofessionalgolfers

rarelybreakparof73.Thescoresfromarandomsampleof20professionalgolfersarelistedbelow.Findthe

teststatisticxtotesttheclubprofessionalʹsclaim.

72 70 73 73 76 75 67 79 73 78

70 72 74 74 81 79 73 75 76 66

A) 6 B) 4 C) 14 D) 10

7) Agovernmentagencyclaimsthatthemedianhourlywagesforworkersatfastfoodrestaurantsinthewestern

U.S.is$6.65.Inarandomsampleof100workers,68werepaidlessthan$ 6.65,10werepaid$6.65,andtherest

morethan$6.65.Findthecriticalvaluestotestthegovernmentʹsclaim.Useα=0.05.

A) –1.96 B) 1.96 C) –1.645 D) 2.575

8) Agovernmentagencyclaimsthatthemedianhourlywagesforworkersatfastfoodrestaurantsinthewestern

U.S.is$6.55.Inarandomsampleof100workers,68werepaidlessthan$6.55,10werepaid

$6.55,andtherestmorethan$6.55.Findtheteststatisticztotestthegovernmentʹsclaim.

A) –4.743 B) –3.912 C) –3.187 D) –2.386

9) Acollegeresearcherclaimsthatthemedianhoursworkedbyfulltimestudentsisatleast10hoursperweek.In

arandomsampleof100students,65workedmorethan10hours,10workedexactly10hoursandtherest

workedlessthan10hours.Findthecriticalvaluetotesttheresearcherʹsclaim.Useα=0.05.

A) –1.645 B) –1.96 C) –2.33 D) –2.575

10) Acollegeresearcherclaimsthatthemedianhoursworkedbyfulltimestudentsisatleast13hoursperweek.In

arandomsampleof100students,58workedmorethan13hours,10workedexactly13hoursandtherest

workedlessthan13hours.Findtheteststatistictotesttheresearcherʹsclaim.

A) -2.635 B) -2.741 C) -3.5 D) -3.6

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

Performtheindicatedsigntest.Besuretodothefollowing:Identifytheclaimmathematicallyandstatethenulland

alternativehypotheses.Determinethecriticalvalueandfindtheteststatistic.Decidewhethertorejectorfailtoreject

thenullhypothesisandinterpretthedecisioninthecontextoftheoriginalclaim.

11) Aconveniencestoreownerbelievesthatthemediannumberoflotteryticketssoldperdayis44.Arandom

sampleof20daysyieldsthedatabelow.Testtheownerʹsclaim.Useα=0.05.

27 43 54 59 26 50 65 22 28 33

42 49 49 39 39 44 44 54 49 33

12) Arealestateagentsurmisesthatthemedianrentforaone–

edroomapartmentinabeachcommunityin

southernCaliforniaisatleast$1500permonth.Therentsforarandomsampleof15one–bedroomapartments

arelistedbelow.Testtheagentʹsclaim.Useα=0.01.

$1800 $1750 $1200 $1375 $1235

$2250 $1675 $1170 $1890 $2500

$1495 $1500 $1575 $1500 $1280

13) Aclubprofessionalatamajorgolfcourseclaimsthatthecourseissotoughthatthemedianscoreisgreater

than75.Thescoresfromarandomsampleof20professionalgolfersarelistedbelow.Testtheclub

professionalʹsclaim.Useα=0.05.

74 72 75 75 78 77 69 81 75 80

72 74 76 76 83 81 75 77 78 68

Page262

14) Acardealerclaimsthatitsnewmodelcarstillgetsatleast21 milespergallonofgas.Tencarsaretested.The

resultsaregivenbelow.Testthedealerʹsclaim.Useα=0.05.

15.8 13.6 19.8 14.9 18 20.2 23.3 17.9 12.7 19

15) Agovernmentagencyclaimsthatthemedianhourlywagesforworkersatfastfoodrestaurantsinthewestern

U.S.is$6.35.Inarandomsampleof100workers,68werepaidlessthan$6.35,10werepaid

$6.35,andtherestmorethan$6.35.Testthegovernmentʹsclaim.Useα=0.05.

16) Testthehypothesisthatthemedianageofstatisticsteachersis51 years.Arandomsampleof60statistics

teachersfound25above51yearsand35below51years.Useα=0.01.

17) NinestudentstooktheSATtest.Theirscoresarelistedbelow.Lateron,theytookatestpreparationcourseand

retooktheSAT.Theirnewscoresarelistedbelow.Usethesigntesttotesttheclaimthatthetestpreparation

hasnoeffectontheirscores.Useα=0.05.

Student 123456789

BeforeScore 860 820 910 990 1000 930 870 1180 920

AfterScore 880 820 900 1030 1030 940 860 1220 940

18) Aweight–liftingcoachclaimsthatweight–lifterscanincreasestrengthbytakingvitaminE.Totestthetheory,

thecoachrandomlyselects9athletesandgivesthemastrengthtestusingabenchpress.Thirtydayslater,after

regulartrainingsupplementedbyvitaminE,theyaretestedagain.Theresultsarelistedbelow.Usethesign

testtotesttheclaimthatthevitaminEsupplementiseffectiveinincreasingtheathletesʹstrength.Useα



=0.05.

Athlete 123456789

Before 217 264 273 202 180 237 242 268 274

After 227 269 273 200 187 252 247 263 279

19) Apharmaceuticalcompanywishestotestanewdrugwiththeexpectationofloweringcholesterollevels.Ten

subjectsarerandomlyselectedandtheircholesterollevelsarerecorded.Theresultsarelistedbelow.The

subjectswereplacedonthedrugforaperiodof6months,afterwhichtheircholesterollevelsweretested

again.Theresultsarelistedbelow.(Allunitsaremilligramsperdeciliter.)Usethesigntesttotestthe

companyʹsclaimthatthedruglowerscholesterollevels.Useα=0.01.

Subject 12345678910

Before 220 231 230 215 241 254 179 199 171186

After 205 226 238 205 236 254 149 181 169171

20) Inastudyoftheeffectivenessofphysicalexerciseinweightloss,20peoplewererandomlyselectedto

participateinaprogramfor30days.Usethesigntesttotesttheclaimthatexercisehasnoeffectonweightloss.

Useα=0.02.

WeightBeforeProgram

(inPounds) 178 210 156 188 193 225 190 165 168200

WeightAfterProgram

(inPounds) 182 205 156 190 183 220 195 155 165200

WeightBeforeProgram

(inPounds) 186 172 166 184 225 145 208 214 148174

WeightAfterProgram

(inPounds) 180 173 165 186 240 138 203 203 142170

Page263

21) Alocalschooldistrictisconcernedaboutthenumberofschooldaysmissedbyitsteachersduetoillness.A

randomsampleof10teachersisselected.Thenumbersofabsencesinoneyeararelistedbelow.Anincentive

programisofferedinanattempttodecreasethenumberofdaysabsent.Thenumbersofabsencesintheyear

aftertheincentiveprogramarealsolisted.Usethesigntesttotesttheclaimthattheincentiveprogramreduces

thenumberofdaysmissedbyteachers.Useα=0.05.

Teacher 12345678910

DaysAbsent

BeforeIncentive 47010 860375

DaysAbsent

AfterIncentive 3608741255

22) Aphysicianʹsgroupclaimsthatapersonʹsdiastolicbloodpressurecanbeloweredbylisteningtoarelaxation

tapeeachevening.Tensubjectsarerandomlyselectedandtheirbloodpressuresaremeasured.The10patients

aregiventhetapesandtoldtolistentothemeacheveningforonemonth.Attheendofthemonth,theirblood

pressuresaremeasuredagain.ThebloodpressuresinmmHgarelistedbelow.Usethesigntesttotestthe

physicianʹsclaim.Useα=0.01.

Patient 12345678910

Before 85 96 92 83 80 91 79 98 9396

After 82 90 92 75 74 80 82 88 8980

23) Acollegeresearcherclaimsthatthemediantimeworkedbyfulltimestudentsisatleast10hoursperweek.In

arandomsampleof100students,59workedmorethan10hours,10workedexactly10hoursandtherest

workedlessthan10hours.Testtheresearcherʹsclaim.Useα=0.05.

24) Testthehypothesisthatthemedianageofstatisticsteachersislessthan45years.Arandomsampleof60

statisticsteachersfound25above45years,33below45years,andtherestexactly45years.Useα

=0.01.

25) Acompanyclaimsthatthemedianmonthlyearningsofitsfarmworkersisgreaterthan $919.Totesttheclaim,

100workersarerandomlyselectedandaskedtoprovidetheirmonthlyearnings.Thedataisshownbelow.Test

thecompanyʹsclaim.Useα=0.05.

WeeklyEarnings NumberofWorkers

Lessthan$919 44

$919 5

Morethan$919 51

26) Onehundredpeoplegoonaspecialdietwiththeintentionoflosingweight.Attheendof6weeks,59lost

weight,27gainedweightandtherestremainedthesame.Testthehypothesisthatthedietiseffectivein

reducingweight.Useα=0.05.(Note:Thedietwillbeeffectiveifatleast50%loseweight.)

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11.2 TheWilcoxonTests

1 PerformaWilcoxonTest

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

Provideanappropriateresponse.

1) NinestudentstooktheSATtest.Theirscoresarelistedbelow.Lateron,theytookatestpreparationcourseand

retooktheSAT.Theirnewscoresarelistedbelow.UsetheWilcoxonsigned–ranktesttofindtheteststatistic

wstotesttheclaimthatthetestpreparationhadnoeffectontheirscores.Useα

=0.05.

Student 123456789

BeforeScore 820 920 810 1180 1060 1100 990 850 1150

AfterScore 840 920 800 1220 1090 1110 980 890 1170

A) 4 B) –4C)41D)

–41

2) Aweight–liftingcoachclaimsthataweight–liftercanincreasestrengthbytakingvitaminE.Totestthetheory,

thecoachrandomlyselects9athletesandgivesthemastrengthtestusingabenchpress.Theresultsarelisted

below.Thirtydayslater,afterregulartrainingsupplementedbyvitaminE,theyaretestedagain.Thenew

resultsarelistedbelow.FindthecriticalvalueforaWilcoxonsigned–ranktesttotesttheclaimthatthevitamin

Esupplementiseffectiveinincreasingtheathletesʹstrength.Useα=0.05.

Athlete 123456789

Befor 230 233 247 202 201 193 232 239 198

After 240 238 247 200 208 208 237 234 203

A) 6 B) 4 C) 2 D) 3

3) Apharmaceuticalcompanywishestotestanewdrugwiththeexpectationofloweringcholesterollevels.Ten

subjectsarerandomlyselectedandtheircholesterollevelsarerecorded.Theresultsarelistedbelow.The

subjectswereplacedonthedrugforaperiodof6months,afterwhichtheircholesterollevelsweretested

again.Theresultsarelistedbelow.(Allunitsaremilligramsperdeciliter.)UsetheWilcoxonsigned –ranktestto

findtheteststatisticwstotestthecompanyʹsclaimthatthedruglowerscholesterollevels.

Subject 12345678910

Before 219 245 253 200 244 243 238 197 227224

After 204 240 261 190 239 243 208 179 225209

A) 4 B) 7.5 C) –2 D) 5.5

4) Aphysicianclaimsthatapersonʹsdiastolicbloodpressurecanbelowered,if,insteadoftakingadrug,the

personlistenstoarelaxationtapeeachevening.Tensubjectsarerandomlyselected.Theirbloodpressures,

measuredinmillimetersofmercury,arelistedbelow.The10patientsaregiventhetapesandtoldtolistento

themeacheveningforonemonth.Attheendofthemonth,theirbloodpressuresaretakenagain.Thedataare

listedbelow.FindthecriticalvalueforaWilcoxonsigned–ranktesttotestthephysicianʹsclaim.Useα=0.05.

Patient 12345678910

Before 84 98 87 92 82 80 97 85 9190

After 81 92 87 84 76 69 100 75 8774

A) 8 B) 6 C) 2 D) 4

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5) VerbalSATscoresforstudentsrandomlyselectedfromtwodifferentschoolsarelistedbelow.Usethe

WilcoxonranksumtesttofindRtotesttheclaimthatthereisnodifferenceinthescoresfromeachschool.

School1School2

580 550 800

510 780 560

610 810 640

620 760 780

520 470 710

460 740 620

720 580 560

660 670 570

A) 128.5 B) 171.5 C) 75.5 D) 38.5

6) Aresearcherwantstoknowifthetimespentinprisonforaparticulartypeofcrimewasthesameformenand

women.Arandomsampleofmenandwomenwereeachaskedtogivethelengthofsentencereceived.The

data,inyears,arelistedbelow.Whatistheappropriatetesttotesttheclaimthatthereisnodifferenceinthe

sentencereceivedbyeachsex?

Men 12 24 18 20 21 28

Women 11 14 11 16 28 14

Men 16 24 14 21 25 26

Women 36 10 12 15 19 29

A) Wilcoxonranksumtest B) Wilcoxonsigned–ranktest

C) Signtest D) t–test

7) Aresearcherwantstoknowifthetimespentinprisonforaparticulartypeofcrimewasthesameformenand

women.Arandomsampleofmenandwomenwereeachaskedtogivethelengthofsentencereceived.The

data,inyears,arelistedbelow.UsetheWilcoxonranksumtesttofindR,thesumoftheranksforthesmaller

sample,totesttheclaimthatthereisnodifferenceinthesentencereceivedbyeachgender.

Men 11 23 17 19 20 27

Women 10 13 10 15 27 13

Men 15 23 13 20 24 25

Women 35 911 14 18 28

A) 125.5 B) 115.5 C) 173.5 D) 155.5

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

PerformtheindicatedWilcoxontest.Besuretodothefollowing:Identifytheclaimmathematicallyandstatethenull

andalternativehypotheses.Determinethecriticalvalueandfindtheteststatistic.Decidewhethertorejectorfailto

rejectthenullhypothesisandinterpretthedecisioninthecontextoftheoriginalclaim.

8) ArandomsampleofninestudentstooktheSATtest.andlateronretookthetestaftertakingatestpreparation

course.Theirscoresarelistedbelow.UsetheWilcoxonsigned–ranktesttotesttheclaimthatthetest

preparationhasnoeffectontheirscores.Useα=0.05.

Student 123456789

BeforeScore 820 1180 840 1080 850 1010 1160 940 980

AfterScore 840 1180 830 1120 880 1020 1150 980 1000

Page266

9) Aweight–liftingcoachclaimsthatweight–lifterscanincreasetheirstrengthbytakingvitaminE.Totestthe

theory,thecoachrandomlyselects9athletesandgivesthemastrengthtestusingabenchpress.Thirtydays

later,afterregulartrainingsupplementedbyvitaminE,theyaretestedagain.Theresultsarelistedbelow.Use

theWilcoxonsigned–ranktesttotesttheclaimthatthevitaminEsupplementiseffectiveinincreasingathletesʹ

strength.Useα=0.05.

Athlete 123456789

Before 185 241 251 187 216 210 204 219 183

After 195 246 251 185 223 225 209 214 188

10) Apharmaceuticalcompanywishestotestanewdrugwiththeexpectationofloweringcholesterollevels.Ten

subjectsarerandomlyselectedandtheircholesterollevelsarerecorded.Thesubjectswerethenplacedonthe

drugforaperiodof6months,afterwhichtheircholesterollevelsweretestedagain.Theresults(inmgper

deciliter)arelistedbelow.UsetheWilcoxonsigned–ranktesttotestthecompanyʹsclaimthatthedruglowers

cholesterollevels.Useα=0.05.

Subject 12345678910

Before 216 239 228 244 251 227 231 260 241270

After 201 234 236 234 246 227 201 242 239255

11) Aphysicianclaimsthatapersonʹsdiastolicbloodpressurecanbeloweredbylisteningtoarelaxationtapeeach

evening.Tensubjectsarerandomlyselectedandtheirbloodpressuresaremeasured.The10patientsthenlisten

tothetapeseacheveningforonemonth.Attheendofthemonth,theirbloodpressuresaremeasuredagain.

Thedata(inmmHg)arelistedbelow.UsetheWilcoxonsigned–ranktesttotestthephysicianʹsclaim.Useα=

0.05.

Patient 12345678910

Before 83 82 99 86 90 80 91 94 9281

After 80 76 99 78 84 69 94 84 8865

12) Inastudyoftheeffectivenessofphysicalexerciseonweightloss,20peoplewererandomlyselectedto

participateinaprogramfor30days.UsetheWilcoxonsigned–ranktesttotesttheclaimthatexercisehasno

effectonweightloss.Useα=0.02.

WeightBeforeProgram

(inPounds) 178 210 156 188 193 225 190 165 168200

WeightAfterProgram

(inPounds) 182 205 156 190 183 220 195 155 165200

WeightBeforeProgram

(inPounds) 186 172 166 184 225 145 208 214 148174

WeightAfterProgram

(inPounds) 180 173 165 186 240 138 203 203 142170

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13) Alocalschooldistrictisconcernedaboutthenumberofschooldaysmissedbyitsteachersduetoillness.A

randomsampleof10teachersisselected.Thenumbersofabsencesinoneyeararerecorded.Anincentive

programisthenofferedinanattempttoreducetheabsences.Thenumbersofabsencesintheyearafterthe

incentiveprogramarerecorded.Theresultsarelistedbelow.UsetheWilcoxonsigned –ranktesttotestthe

claimthattheincentiveprogramiseffectiveinreducingthenumberofdaysmissedbyteachers.Useα



=0.05.

Teacher 12345678910

DaysAbsent

BeforeIncentive 847310 52837

DaysAbsent

AfterIncentive 6371930917

14) VerbalSATscoresforstudentsrandomlyselectedfromtwodifferentschoolsarelistedbelow.Usethe

Wilcoxonranksumtesttotesttheclaimthatthereisnodifferenceinthescoresfromthetwoschools.

Useα=0.05.

School1School2

540 510 760

470 740 520

570 770 600

580 720 740

480 430 670

420 700 580

680 540 520

620 630 530

15) Aresearcherwantstoknowifthelengthofsentencesreceivedforaparticulartypeofcrimewasthesamefor

menandwomen.Thelengthofsentencereceivedwasrecordedforarandomsampleofmenandwomen.The

data,inyears,arelistedbelow.UsetheWilcoxonranksumtesttotesttheclaimthatthereisnodifference

betweenthesentencesreceivedbymenandthesentencesreceivedbywomen.Useα



=0.05.

Men 618 12 14 15 22

Women 58510 22 8

Men 10 18 815 19 20

Women 30 46913 23

Page268

2 PerformaWilcoxonSigned–RankTestforn>30

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

PerformtheindicatedWilcoxontest.Besuretodothefollowing:Identifytheclaimmathematicallyandstatethenull

andalternativehypotheses.Determinethecriticalvalueandfindtheteststatistic.Decidewhethertorejectorfailto

rejectthenullhypothesisandinterpretthedecisioninthecontextoftheoriginalclaim.

1) Astatisticsteacherwantedtoseewhethertherewasasignificantdifferenceinagebetweendaystudentsand

nightstudents.Arandomsampleof35studentsfromeachgroupwasselected.Thedataaregivenbelow.Use

theWilcoxonranksumtesttotesttheclaimthatthereisnodifferenceinagebetweenthetwogroups.Useα

0.05.

DayStudents

2224242319192322182121

1818252924232222212020

2027171918212023263024

2125

EveningStudents

1823252321212324273134

2020231925242723202021

2524232820192324202721

2930

11.3 TheKruskal–WallisTest

1 PerformaKruskal–WallisTest

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

Provideanappropriateresponse.

1) Amedicalresearcherwishestotrythreedifferenttechniquestolowerbloodpressureofpatientswithhigh

bloodpressure.Thesubjectsarerandomlyselectedandassignedtooneofthreegroups.Group1isgiven

medication,Group2isgivenanexerciseprogram,andGroup3isassignedadietprogram.Attheendofsix

weeks,thereductionineachsubjectʹsbloodpressureisrecorded.UsetheKruskal–Wallistesttofindthecritical

valuetotesttheclaimthatthereisnodifferenceinthedistributionofthepopulations.Useα



=0.05.

Group1Group2Group3

13 10 8

14 7 14

11 4 6

17 5 10

15 6 11

10 2 6

A) 5.991 B) 4.153 C) 3.195 D) 1.960

Page269

2) Amedicalresearcherwishestotrythreedifferenttechniquestolowerbloodpressureofpatientswithhigh

bloodpressure.Thesubjectsarerandomlyselectedandassignedtooneofthreegroups.Group1isgiven

medication,Group2isgivenanexerciseprogram,andGroup3isassignedadietprogram.Attheendofsix

weeks,thereductionineachsubjectʹsbloodpressureisrecorded.UsetheKruskal–Wallistesttofindthetest

statisticHtotesttheclaimthatthereisnodifferenceinthedistributionofthepopulations.

Group1Group2Group3

13 10 8

14 7 14

11 4 6

17 5 10

15 6 11

10 2 6

A) 10.187 B) 8.312 C) 6.813 D) 5.321

3) Fourdifferenttypesoffertilizersareusedonraspberryplants.Thenumberofraspberriesoneachrandomly

selectedplantisgivenbelow.UsetheKruskal–WallistesttofindtheteststatisticHtotesttheclaimthatthere

isnodifferenceinthedistributionofthepopulations.

Fertilizer1 Fertilizer2 Fertilizer3 Fertilizer4

11 10 11 8

10 13 8 10

11 10 9 8

12 10 8 9

12 10 7 10

11 11 8 9

A) 12.833 B) 10.922 C) 15.364 D) 14.818

4) Fourdifferenttypesoffertilizersareusedonraspberryplants.Thenumberofraspberriesoneachrandomly

selectedplantisgivenbelow.UsetheKruskal–WallistesttofindthecriticalvalueHtotesttheclaimthatthere

isnodifferenceinthedistributionofthepopulations.Useα=0.05.

Fertilizer1Fertilizer2 Fertilizer3 Fertilizer4

11 10 11 8

10 13 8 10

11 10 9 8

12 10 8 9

12 10 7 10

11 11 8 9

A) 7.815 B) 7.352 C) 6.531 D) 5.198

Page270

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

PerformtheindicatedKruskall–Wallistest.Besuretodothefollowing:Identifytheclaimmathematicallyandstatethe

nullandalternativehypotheses.Determinethecriticalvalueandfindtheteststatistic.Decidewhethertorejectorfail

torejectthenullhypothesisandinterpretthedecisioninthecontextoftheoriginalclaim.

5) Amedicalresearcherwishestotrythreedifferenttechniquestolowerbloodpressureofpatientswithhigh

bloodpressure.Thesubjectsarerandomlyselectedandrandomlyassignedtooneofthreegroups.Group1is

givenmedication,Group2isgivenanexerciseprogram,andGroup3isassignedaspecialdiet.Attheendof

sixweeks,thereductionineachsubjectʹsbloodpressureisrecorded.UsetheKruskal–Wallistesttotestthe

claimthatthereisnodifferenceinthedistributionsofthebloodpressuresofthethreepopulations.Use

α=0.05.

Group1Group2Group3

13 10 8

14 7 14

11 4 6

17 5 10

15 6 11

10 2 6

6) Fourdifferenttypesoffertilizersareusedonraspberryplants.Thenumberofraspberriesoneachrandomly

selectedplantisgivenbelow.UsetheKruskal–Wallistesttotesttheclaimthatthereisnodifferenceinthe

distributionsofthenumberofraspberriesforthefourdifferenttypesoffertilizer.Use

α=0.05.

Fertilizer1Fertilizer2 Fertilizer3 Fertilizer4

9896

81168

9876

10 8 6 7

10 8 5 8

9967

7) Aresearcherwishestotesttheclaimthatthereisadifferenceinthedistributionofagesofelementaryschool,

highschool,andcommunitycollegeteachers.Teachersarerandomlyselectedfromeachgroup.Theiragesare

recordedbelow.Canyouconcludethatthedistributionsofteachersʹagesatthesedifferentlevelsofeducation

aredifferent?Useα=0.05.

ElementarySchool

Teachers

HighSchool

Teachers

CommunityCollege

Teachers

28 41 44

33 46 50

32 43 41

57 52 66

42 47 50

30 36 40

Page271

8) Thegradepointaveragesofstudentsparticipatingindifferentsportsatacollegearetobecompared.Thedata

arelistedbelow.UsetheKruskal–Wallistesttotesttheclaimthatthereisadifferenceinthedistributionofthe

GPAsofthethreedifferentsportsgroups.Useα=0.05.

Tennis Golf Swimming

3.6 2.2 3.1

3 2.5 3.4

2.9 3.7 3.2

3.9 2.3 2.9

3.5 2.7 2.9

2.5 2.4 2.8

9) Aresearcherwishestocomparetimesittakesworkerstoassembleacertaincomputercomponentusing

differentmachines.Workersarerandomlyselectedandrandomlyassignedtooneofthreedifferentmachines.

Thetime(inminutes)ittakeseachworkertoassemblethecomponentisrecorded.UsetheKruskal–Wallistest

totesttheclaimthatthereisadifferenceinthedistributionsoftheassemblytimesforthethreemachines.Use

α=0.05.

Machine1Machine2Machine3

43 51 39

42 40 36

43 49 40

41 44 42

44 46 41

42 43 38

43 47 48

10) Arealtorwishestocomparethesquarefootageofhousesofsimilarpricesin4differentcities.Thedataare

listedbelow.UsetheKruskal–Wallistesttotesttheclaimthatthereisadifferenceinthedistributionsofthe

squarefootagesinthe4differentcities.Useα=0.05.

City1 City2 City3 City4

2150 1780 1530 2400

1980 1540 1670 2350

2000 1690 1580 2600

2210 1650 1600 2150

1900 1700 1500 2000

2050 1750 2200

1650 2350

2250

Page272

2 ComparetheResultsofaKruskal–WallisTestandaOne–WayANOVA

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

Provideanappropriateresponse.

1) Fourdifferenttypesoffertilizersareusedonraspberryplants.Thenumberofraspberriesoneachrandomly

selectedplantisgivenbelow.Testtheclaimusing(a)aKurskal–Wallistestand(b)aone–wayANOVAtest

thatthetypeoffertilizermakesnodifferenceinthemeannumberofraspberriesperplant.Comparetheresults.

Useα=0.01.

Fertilizer1 Fertilizer2 Fertilizer3 Fertilizer4

6563

5835

6543

7534

7525

6634

2) Aresearcherclaimsthatthelifetimes(inhours)offluorescentlightbulbsarethesameregardlessof

manufacturer.Randomsamplesareselectedfrom3differentmanufacturers.Thedataarelistedbelow.Testthe

claimthatthesamplescomefromidenticalpopulationsbyusing(a)aone–wayANOVAtestand

(b)aKruskal–Wallistest.Comparetheresults.Useα=0.05.

Manufacturer1Manufacturer2Manufacturer3

190 220

235 215

225 230

215 220

180 170

200 175

175 180

190 185

200 210

195 205

200 205

205 205

11.4 RankCorrelation

1 TestaClaimAboutaCorrelation

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

Provideanappropriateresponse.

1) ThetablebelowliststheverbalandmathSATscoresof10studentsselectedatrandom.Findthecriticalvalue

totesttheclaimofnocorrelationbetweenverbalandmathSATscores.Useα



=0.05.

Verbal 495 580 585 490 570

Math 580 650 675 610 660

Verbal 600 500 550 620 510

Math 625 710 630 500 510

A) 0.648 B) 0.564 C) 0.794 D) –0.648

Page273

2) ThetablebelowliststheverbalandmathSATscoresof10studentsselectedatrandom.Findtheteststatisticrs

totesttheclaimofnocorrelationbetweenverbalandmathSATscores.

Verbal 395 480 485 390 470

Math 480 550 575 510 560

Verbal 500 400 450 520 410

Math 525 610 530 400 410

A) –0.006 B) –0.0192 C) –0.218 D) –0.326

3) Thedramadepartmentatacollegeaskedprofessorsandstudentsinthedramadepartmenttorank8actors

accordingtotheirperformance.Thedataarelistedbelow.A10isthehighestrankinganda1thelowest

ranking.Findthecriticalvaluetotesttheclaimofnocorrelationbetweentherankings.

Useα=0.05.

Actor 1 2 3 4 5678

Professors 236108154

Students 4 3 1 4 5796

A) 0.738 B) 0.643 C) 0.881 D) 0.833

4) Thedramadepartmentatacollegeaskedprofessorsandstudentsinthedramadepartmenttorank8actors

accordingtotheirperformance.Thedataarelistedbelow.A10isthehighestrankinganda1thelowest

ranking.Findtheteststatistictotesttheclaimofnocorrelationbetweentherankings.

Actor 1 234 5678

Professors 236108154

Students 4 3 1 4 5796

A) –0.208 B) –0.198 C) –0.354 D) –0.278

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

UsetheSpearmanrankcorrelationcoefficienttodeterminewhetherthecorrelationbetweenthegivenvariablesis

significant.Besuretodothefollowing:Identifytheclaimmathematicallyandstatethenullandalternativehypotheses.

Determinethecriticalvalueandfindtheteststatistic.Decidewhethertorejectorfailtorejectthenullhypothesisand

interpretthedecisioninthecontextoftheoriginalclaim.

5) ThetablebelowliststheverbalandmathSATscoresof10studentsselectedatrandom.Testtheclaimthat

thereisnocorrelationbetweenverbalandmathSATscores.Useα=0.05.

Verbal 295 380 385 290 370

Math 380 450 475 410 460

Verbal 400 300 350 420 310

Math 425 510 430 300 310

6) Thedramadepartmentatacollegeaskedprofessorsandstudentsinthedramadepartmenttorank8actors

accordingtotheirperformance.Thedataarelistedbelow.A10isthehighestrankinganda1thelowest

ranking.Testtheclaimthatthereisnocorrelationbetweenprofessorsʹandstudentsʹrankings.Useα



=0.05.

Actor 1 2 3 4 5678

Professors 236108154

Students 4 3 1 4 5796

Page274

7) Thefinalexamscoresof10randomlyselectedstatisticsstudentsandthenumberofhourstheystudiedforthe

examaregivenbelow.Canyouconcludethatthereisacorrelationbetweenscoresonthetestandtimespent

studying?Useα=0.01.

Hours 79612 688910 7

Score 69 84 64 92 70 82 89 94 94 75

8) Thetemperaturesonrandomlychosendaysduringasummerclassandthenumberofabsencesfromclasson

thosedaysarelistedbelow.Canyouconcludethatthereisacorrelationbetweentemperatureandthenumber

absent?Useα=0.01.

Temp 75 88 94 93 91 101 78 103 83

Absences 6 10 13 13 11 18 7 18 8

9) Theagesandsystolicbloodpressuresof9randomlyselectedadultsaregivenbelow.Canyouconcludethat

thereisacorrelationbetweenageandbloodpressure?Useα=0.05.

Age 38 41 45 48 51 53 57 61 65

BloodPressure(mmHg) 116 120 123 131 142 145 148 150 152

10) Thenumberofabsencesandthefinalgradesof9randomlyselectedstudentsfromastatisticsclassaregiven

below.Canyouconcludethatthereisacorrelationbetweenfinalgradeandnumberofabsences?Useα

=0.01.

NumberofAbsences 03649215 85

FinalGrade 98 86 80 82 71 92 55 76 82

11) Amanagerwishestodeterminewhetherthereisarelationshipbetweenthedistancetraveledpermonthbythe

salesrepresentativesandtheamountofsalesgeneratedpermonth.Canyouconcludethatthereisacorrelation

betweendistancetraveledandsalesgenerated?Useα=0.05.

Distance(hundredsofmiles) 11 12 19 16 17 24 12 10 20

Sales(thousandsofdollars) 40 42 87 71 74 70 57 64 129

12) InanareaoftheMidwest,recordswerekeptontherainfallandtheyieldofwheat.Canyouconcludethat

thereisacorrelationbetweenrainfallandyieldofwheat?Useα=0.01.

Rainfall(inches) 13.5 11.8 16.4 15.5 21.8 13.3 10.0 18.6 19.0

Yield(bushelsperacre) 53.5 49.2 61.8 62.0 85.4 52.2 34.9 79.0 81.8

Page275

2 TesttheRankCorrelationCoefficientforn>30

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

Provideanappropriateresponse.

1) ThetablebelowliststheverbalandmathSATscoresof35studentsselectedatrandom.Findthecriticalvalues

totesttheclaimofnocorrelationbetweenverbalandmathSATscores.Useα

=0.05.

Verbal 295 380 385 290 370 400 300 350 420 310

Math 380 450 475 410 460 425 510 430 300 310

Verbal 295 340 410 520 360 400 660 530 700 610

Math 440 500 400 480 410 380 500 540 580 620

Verbal 290 470 510 380 390 550 420 430 330 370

Math 380 480 490 510 440 560 440 500 410 300

Verbal 430 390 530 380 390

Math 430 410 560 400 360

A) ±0.336 B) ±0.282 C) ±0.442 D) ±0.400

2) ThetablebelowliststheverbalandmathSATscoresof35studentsselectedatrandom.Findtheteststatistic,

rs,totesttheclaimofnocorrelationbetweenverbalandmathSATscores.

Verbal 295 380 385 290 370 400 300 350 420 310

Math 380 450 475 410 460 45 510 430 300 310

Verbal 295 340 410 520 360 400 660 530 700 610

Math 440 500 400 480 410 380 500 540 580 620

Verbal 290 470 510 380 390 550 420 430 330 370

Math 380 480 490 510 440 560 440 500 410 300

Verbal 430 390 530 380 390

Math 430 410 560 400 360

A) 0.545 B) 0.616 C) 0.731 D) 0.819

Page276

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

3) ThetablebelowliststheverbalandmathSATscoresof35studentsselectedatrandom.Testtheclaimofno

correlationbetweenverbalandmathSATscores.Useα=0.05.

Verbal 295 380 385 290 370 400 300 350 420 310

Math 380 450 475 410 460 45 510 430 300 310

Verbal 295 340 410 520 360 400 660 530 700 610

Math 440 500 400 480 410 380 500 540 580 620

Verbal 290 470 510 380 390 550 420 430 330 370

Math 380 480 490 510 440 560 440 500 410 300

Verbal 430 390 530 380 390

Math 430 410 560 400 360

11.5 TheRunsTest

1 FindtheNumberofRuns

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

Provideanappropriateresponse.

1) Twopokerplayersaredealtcardsinthesequenceshown,whereBrepresentsablackcardandRrepresentsa

redcard.FindtheteststatisticGtotestforrandomness.

BBBRBRB

RRBBBRR

2) Atelemarketersolicitedhouseholdstochangetheirlong–distancecarrier.Theresultsforoneafternoonare

shown,whereCrepresentsthehouseholdsthatchangedtheircarrierandSrepresentsthehouseholdsthatkept

theirsamecarrier.FindtheteststatisticGtotestforrandomness.

CCCS CS CS S CC

SCSCSCSSCSC

3) Usethesequencetofindthevaluesofn1andn2

LLLRLRLRRLLLRR

4) Usethesequencetofindthevaluesofn1andn2.

HHHLHLHLLHHLHLHLHLLHLH

Page277

2 PerformaRunsTest

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

Provideanappropriateresponse.

1) Anairportsecurityofficermanuallysearchedcarry–onbagsofpeopleboardingaplaneinthesequenceshown,

whereSrepresentsasearchedbagandNrepresentsabagnotsearched.Findtheteststatisticztotestfor

randomness.

NNNNNS NNS S

NNNNNNNS NN

NNS S S NNNS N

NS NNNNS S NN

NNNNNNS NNN

2) Apitchingmachinethrowsbaseballsthatareeitherstrikes(S)orballs(B).Acoachrecordswhethereachball

thrownduringbattingpracticeisastrikeoraball.Findthestandardizedteststatisticztotestforrandomness.

BBSBSSBBSB

BBBSBSSSSS

BBBBSBBSBB

BSSSBSSSBB

BBBSSSSSSS

Performarunstestforrandomness.Besuretodothefollowing:Identifytheclaimmathematicallyandstatethenull

andalternativehypotheses.Determinethecriticalvalueandfindtheteststatistic.Decidewhethertorejectorfailto

rejectthenullhypothesisandinterpretthedecisioninthecontextoftheoriginalclaim.

3) Twopokerplayersaredealtcardsinthesequenceshown,whereBrepresentsablackcardandRrepresentsa

redcard.Canyouconcludethatthedealingofcardswasnotrandom?Useα

=0.05.

BBBRBRB

RRBBBRR

4) Atelemarketersolicitedhouseholdstochangetheirlong–distancecarrier.Theresultsforoneafternoonare

shown,whereCrepresentsthehouseholdsthatchangedtheircarrierandSrepresentsthehouseholdsthatkept

theirexistingcarrier.Canyouconcludethatthesequenceofresultsisrandom?Useα=0.05.

CCCS CS CS S CC

SCSCSCSSCSC

5) Anairportsecurityofficersearchedcarry–onbagsofpeopleboardingaplaneinthesequenceshown,whereS

representsasearchedbagandNrepresentsabagnotsearched.Canyouconcludethattheofficerconducteda

randomsearch?Useα=0.05.

NNNNNS NNS S

NNNNNNNS NN

NNS S S NNNS N

NS NNNNS S NN

NNNNNNS NNN

Page278

6) Apitchingmachinethrowsbaseballsthatareeitherstrikes(S)orballs(B).Acoachrecordswhethereachball

thrownduringbattingpracticeisastrikeoraball.Canyouconcludethatthesequenceofpitchesisrandom?

Useα=0.05.

BBSBSSBBSB

BBBSBSSSSS

BBBBSBBSBB

BSSSBSSSBB

BBBSSSSSSS

3 PerformaRunsTestwithQuantitativeData

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

Provideanappropriateresponse.

1) Thesequenceshowsacompanyʹsdailysales,inthousandsofdollars,forthebusinessdaysduringthemonthof

September.Findthestandardizedteststatistic,G,totestforrandomness.

10 10 15 12.5 20 12.5 10 10 12.5 20

8101020 159 12.5 12.5 20 17.5

2) Thesequenceshowsacompanyʹsdailysales,inthousandsofdollars,forthebusinessdaysduringthemonthof

September.Testtheclaimthatthesalesarerandom.

10 10 15 12.5 20 12.5 10 10 12.5 20

8101020 159 12.5 12.5 20 17.5

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Ch.11 NonparametricTests

AnswerKey

11.1 TheSignTest

1 PerformaSignTest

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11.2 TheWilcoxonTests

1 PerformaWilcoxonTest

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2PerformaWilcoxonSigned–RankTestforn>30

11.3 TheKruskal–WallisTest

1 PerformaKruskal–WallisTest

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2ComparetheResultsofaKruskal–WallisTestandaOne–WayANOVA

11.4 RankCorrelation

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2TesttheRankCorrelationCoefficientforn>30

11.5 TheRunsTest

1 FindtheNumberofRuns

2 PerformaRunsTest

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3 PerformaRunsTestwithQuantitativeData

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