Chapter 11Us 665 Random Sample 100 Workers Were

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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$2000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.

$2300 $2250 $1700 $1875 $1735

$2750 $2175 $1670 $2390 $3000

$1995 $2000 $2075 $2000 $1780

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$1500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.

$1800 $1750 $1200 $1375 $1235

$2250 $1675 $1170 $1890 $2500

$1495 $1500 $1575 $1500 $1280

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235

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

rarelybreakparof65.Thescoresfromarandomsampleof20professionalgolfersarelistedbelow.Findthe

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

64 62 65 65 68 67 59 71 65 70

62 64 66 66 73 71 65 67 68 58

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$6.10.Inarandomsampleof100workers,68werepaidlessthan$6.10,10werepaid

$6.10,andtherestmorethan$6.10.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11hoursperweek.In

arandomsampleof100students,63workedmorethan11hours,10workedexactly11hoursandtherest

workedlessthan11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,64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) –3.9 B) –4.006 C) –4.7 D) –4.8

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61.Arandom

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

44 60 71 76 43 67 82 39 45 50

59 66 66 56 56 61 61 71 66 50

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$2000permonth.Therentsforarandomsampleof15one–bedroomapartments

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

$2300 $2250 $1700 $1875 $1735

$2750 $2175 $1670 $2390 $3000

$1995 $2000 $2075 $2000 $1780

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

than70.Thescoresfromarandomsampleof20professionalgolfersarelistedbelow.Testtheclub

professionalʹsclaim.Useα=0.05.

69 67 70 70 73 72 64 76 70 75

67 69 71 71 78 76 70 72 73 63

Page236

14) Acardealerclaimsthatitsnewmodelcarstillgetsatleast26 milespergallonofgas.Tencarsaretested.The

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

20.8 18.6 24.8 19.9 23 25.2 28.3 22.9 17.7 24

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

U.S.is$6.50.Inarandomsampleof100workers,68werepaidlessthan$6.50,10werepaid

$6.50,andtherestmorethan$6.50.Testthegovernmentʹsclaim.Useα=0.05.

16) Testthehypothesisthatthemedianageofstatisticsteachersis41 years.Arandomsampleof60statistics

teachersfound25above41yearsand35below41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 1080 1180 860 840 970 1140 1160 930 1010

AfterScore 1100 1180 850 880 1000 1150 1150 970 1030

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 214 212 279 206 260 242 254 268 237

After 224 217 279 204 267 257 259 263 242

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 188 208 194 264 202 173 269 245 259193

After 173 203 202 254 197 173 239 227 257178

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237

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 510 06780796

DaysAbsent

AfterIncentive 4904661676

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13hoursperweek.In

arandomsampleof100students,62workedmorethan13hours,10workedexactly13hoursandtherest

workedlessthan13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 $943.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$943 41

$943 5

Morethan$943 54

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.)

Page238

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 1020 1200 950 970 1060 820 1000 850 920

AfterScore 1040 1200 940 1010 1090 830 990 890 940

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 279 246 233 244 220 254 257 276 272

After 289 251 233 242 227 269 262 271 277

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 214 264 248 199 258 268 206 200 228170

After 199 259 256 189 253 268 176 182 226155

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 98 83 92 99 81 87 97 84 8294

After 95 77 92 91 75 76 100 74 7878

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

Page239

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

550 520 770

480 750 530

580 780 610

590 730 750

490 440 680

430 710 590

690 550 530

630 640 540

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 14 26 20 22 23 30

Women 13 16 13 18 30 16

Men 18 26 16 23 27 28

Women 38 12 14 17 21 31

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 17 29 23 25 26 33

Women 16 19 16 21 33 19

Men 21 29 19 26 30 31

Women 41 15 17 20 24 34

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 1020 990 850 1010 1200 1000 1030 970 950

AfterScore 1040 990 840 1050 1230 1010 1020 1010 970

Page240

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 195 180 208 253 226 210 202 276 229

After 205 185 208 251 233 225 207 271 234

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 208 230 184 209 255 249 259 185 227234

After 193 225 192 199 250 249 229 167 225219

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 95 94 85 97 96 82 87 90 8083

After 92 88 85 89 90 71 90 80 7667

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

Page241

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 587410 92457

DaysAbsent

AfterIncentive 377297053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

510 480 730

440 710 490

540 740 570

550 690 710

450 400 640

390 670 550

650 510 490

590 600 500

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 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

Page242

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.

16) 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

16 13 11

17 10 17

14 7 9

20 8 13

18 9 14

13 5 9

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

Page243

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

10 9 10 7

91279

10 9 8 7

11 9 7 8

11 9 6 9

10 10 7 8

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1 Fertilizer2 Fertilizer3 Fertilizer4

7674

6946

7654

8645

8636

7745

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

Page244

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1 Fertilizer2 Fertilizer3 Fertilizer4

10 9 10 7

91279

10 9 8 7

11 9 7 8

11 9 6 9

10 10 7 8

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

25 38 41

30 43 47

29 40 38

54 49 63

39 44 47

27 33 37

Page245

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

3.5 2.1 3

2.9 2.4 3.3

2.8 3.6 3.1

3.8 2.2 2.8

3.4 2.6 2.8

2.4 2.3 2.7

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

46 54 42

45 43 39

46 52 43

44 47 45

47 49 44

45 46 41

46 50 51

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

2170 1800 1550 2420

2000 1560 1690 2370

2020 1710 1600 2620

2230 1670 1620 2170

1920 1720 1520 2020

2070 1770 2220

1670 2370

2270

Page246

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.

11) 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

12) 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 395 480 485 390 470

Math 480 550 575 510 560

Verbal 500 400 450 520 410

Math 525 610 530 400 410

A) 0.648 B) 0.564 C) 0.794 D) –0.648

Page247

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 445 530 535 440 520

Math 530 600 625 560 610

Verbal 550 450 500 570 460

Math 575 660 580 450 460

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 2 3 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 345 430 435 340 420

Math 430 500 525 460 510

Verbal 450 350 400 470 360

Math 475 560 480 350 360

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248

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spend

studying?Useα=0.01.

Hours 3528244563

Score 65 80 60 88 66 78 85 90 90 71

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 73 86 92 91 89 99 76 101 81

Absences 4 8 11 11 9 16 5 16 6

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) 6714 11 12 19 7515

Sales(thousandsofdollars) 35 37 82 66 69 65 52 59 124

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249

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.

13) 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

14) 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250

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

15) 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

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.

3) 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

Page251

4) 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.

5) 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

6) 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

7) 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

8) 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

Page252

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.

9) 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

10) 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

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

11.3 TheKruskal–WallisTest

1 PerformaKruskal–WallisTest

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

11.4 RankCorrelation

1 TestaClaimAboutaCorrelation

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

11.5 TheRunsTest

1 FindtheNumberofRuns

2 PerformaRunsTest

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

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