Ch.10 Chi–SquareTestsandtheF–Distribution
10.1 Goodness–o
f
–FitTest
1 FindExpectedFrequencies
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Provideanappropriateresponse.
1) Thefrequencydistributionshowstheagesforasampleof100employees.Findtheexpectedfrequenciesfor
eachclasstodetermineiftheemployeeagesarenormallydistributed.
Classboundaries Frequency,f
29.5–39.5 14
39.5–49.5 29
49.5–59.5 31
59.5–69.5 18
69.5–79.5 8
2 PerformaChi–SquareGoodness–of–FitTest
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) Manytrackrunnersbelievethattheyhaveabetterchanceofwinningiftheystartintheinsidelanethatis
closesttothefield.Forthedatabelow,thelaneclosesttothefieldisLane1,thenextlaneisLane2,andsoon
untiltheoutermostlane,Lane6.Thedataliststhenumberofwinsfortrackrunnersinthedifferentstarting
positions.Calculatethechi–squareteststatisticχ2totesttheclaimthatthenumberofwinsisuniformly
distributedacrossthedifferentstartingpositions.Theresultsarebasedon240wins.
StartingPosition 123456
NumberofWins 45 36 33 50 32 44
A) 6.750 B) 9.326 C) 12.592 D) 15.541
2) Manytrackrunnersbelievethattheyhaveabetterchanceofwinningiftheystartintheinsidelanethatis
closesttothefield.Forthedatabelow,thelaneclosesttothefieldisLane1,thenextlaneisLane2,andsoon
untiltheoutermostlane,Lane6.Thedataliststhenumberofwinsfortrackrunnersinthedifferentstarting
positions.Findthecriticalvalueχ 2
0totesttheclaimthatthenumberofwinsisuniformlydistributedacross
thedifferentstartingpositions.Theresultsarebasedon240wins.
StartingPosition 123456
NumberofWins 32 50 44 33 36 45
A) 11.070 B) 9.236 C) 15.086 D) 12.833
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SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
3) Manytrackrunnersbelievethattheyhaveabetterchanceofwinningiftheystartintheinsidelanethatis
closesttothefield.Forthedatabelow,thelaneclosesttothefieldisLane1,thenextlaneisLane2,andsoon
untiltheoutermostlane,Lane6.Thedataliststhenumberofwinsfortrackrunnersinthedifferentstarting
positions.Testtheclaimthatthenumberofwinsisuniformlydistributedacrossthedifferentstarting
positions.Theresultsarebasedon240wins.
StartingPosition 123456
NumberofWins 33 44 45 32 36 50
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
4) Acoffeehousewishestoseeifcustomershaveanypreferenceamong5differentbrandsofcoffee.Asampleof
200customersprovidedthedatabelow.Calculatethechi–squareteststatisticχ2totesttheclaimthatthe
distributionisuniform..
Brand 12345
Customers 30 65 18 32 55
A) 37.45 B) 45.91 C) 48.91 D) 55.63
5) Acoffeehousewishestoseeifcustomershaveanypreferenceamong5differentbrandsofcoffee.Asampleof
200customersprovidedthedatabelow.Findthecriticalvalueχ2
0totesttheclaimthatthedistributionis
uniform.Useα=0.01.
Brand 12345
Customers 30 32 55 65 18
A) 13.277 B) 9.488 C) 11.143 D) 14.860
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
6) Anewcoffeehousewishestoseewhethercustomershaveanypreferenceamong5differentbrandsofcoffee.A
sampleof200customersprovidedthedatabelow.Testtheclaimthatthedistributionisuniform.Useα
=0.01.
Brand 12345
Customers 55 32 30 65 18
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
7) Ateacherfiguresthatfinalgradesinthestatisticsdepartmentaredistributedas:A,25%;B,25%;C,40%;D,5%;
F,5%.Attheendofarandomlyselectedsemester,thefollowingnumberofgradeswererecorded.Calculate
thechi–squareteststatisticχ2todetermineifthegradedistributionforthedepartmentisdifferentthan
expected.
Grade ABCDF
Number 36 42 60 814
A) 5.25 B) 6.87 C) 3.41 D) 4.82
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8) Ateacherfiguresthatfinalgradesinthestatisticsdepartmentaredistributedas:A,25%;B,25%;C,40%;D,5%;
F,5%.Attheendofarandomlyselectedsemester,thefollowingnumberofgradeswererecorded.Findthe
criticalvalueχ 2
0todetermineifthegradedistributionforthedepartmentisdifferentthanexpected.Useα
=
0.01.
Grade ABCDF
Number 42 36 60 814
A) 13.277 B) 11.143 C) 9.488 D) 7.779
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
9) Ateacherfiguresthatfinalgradesinthestatisticsdepartmentaredistributedas:A,25%;B,25%;C,40%;D,5%;
F,5%.Attheendofarandomlyselectedsemester,thefollowingnumberofgradeswererecorded.Determineif
thegradedistributionforthedepartmentisdifferentthanexpected.Useα=0.01.
Grade ABCDF
Number 36 42 60 14 8
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
10) Eachsideofastandardsix–sideddieshouldappearapproximately1
6ofthetimewhenthedieisrolled.A
playersuspectsthatacertaindieisloaded.Thesuspecteddieisrolled90times.Theresultsareshownbelow.
Calculatethechi–squareteststatisticχ2totesttheplayerʹsclaim.
Number 123456
Frequency 15 12 16 19 17 11
A) 3.067 B) 2.143 C) 5.013 D) 4.312
11) Eachsideofastandardsix–sideddieshouldappearapproximately1
6ofthetimewhenthedieisrolled.A
playersuspectsthatacertaindieisloaded.Thesuspecteddieisrolled90times.Theresultsareshownbelow.
Findthecriticalvalueχ 2
0totesttheplayerʹsclaim.Useα=0.10.
Number 123456
Frequency 11 15 12 16 17 19
A) 9.236 B) 1.610 C) 12.833 D) 11.071
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
12) Eachsideofastandardsix–sideddieshouldappearapproximately1
6ofthetimewhenthedieisrolled.A
playersuspectsthatacertaindieisloaded.Thesuspecteddieisrolled90times.Theresultsareshownbelow.
Testtheplayerʹsclaim.Useα=0.10.
Number 123456
Frequency 19 12 15 17 16 11
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MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
13) Arandomsampleof160carcrashesareselectedandcategorizedbyage.Theresultsarelistedbelow.Theage
distributionofdriversforthegivencategoriesis18%fortheunder26group,39%forthe26–45group,31%for
the45–65group,and12%forthegroupover65.Calculatethechi–squareteststatisticχ2totesttheclaimthat
allageshavecrashratesproportionaltotheirdrivingrates.
Age Under26 26–45 46–65 Over65
Drivers 66 39 25 30
A) 75.101 B) 85.123 C) 101.324 D) 95.431
14) Arandomsampleof160carcrashesareselectedandcategorizedbyage.Theresultsarelistedbelow.Theage
distributionofdriversforthegivencategoriesis18%fortheunder26group,39%forthe26–45group,31%for
the45–65group,and12%forthegroupover65.Findthecriticalvalueχ2
0totesttheclaimthatallageshave
crashratesproportionaltotheirdrivingrates.Useα=0.05.
Age Under26 26–45 46–65 Over65
Drivers 66 39 25 30
A) 7.815 B) 6.251 C) 11.143 D) 9.348
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
15) Arandomsampleof160carcrashesareselectedandcategorizedbyage.Theresultsarelistedbelow.Theage
distributionofdriversforthegivencategoriesis18%fortheunder26group,39%forthe26–45group,31%for
the46–65group,and12%forthegroupover65.Testtheclaimthatallageshavecrashratesproportionalto
theirdrivingrates.Useα=0.05.
Age Under26 26–45 46–65 Over65
Drivers 66 39 25 30
16) Asociologistbelievesthatthelevelsofeducationalattainmentofhomelesspersonsarenotuniformly
distributed.Totestthisclaim,yourandomlysurvey100homelesspersonsandrecordtheeducational
attainmentofeach.Theresultsareshowninthefollowingtable.Calculatethechi–squareteststatisticχ2to
testthesociologistʹsclaim.
Response Frequency,f
Lessthanhighschool 38
Highschoolgraduate/G.E.D. 34
Morethanhighschool 28
17) Asociologistbelievesthatthelevelsofeducationalattainmentofhomelesspersonsarenotuniformly
distributed.Totestthisclaim,yourandomlysurvey100homelesspersonsandrecordtheeducational
attainmentofeach.Theresultsareshowninthefollowingtable.Findthecriticalvalueχ02totestthe
sociologistʹsclaim.Useα=0.10.
Response Frequency,f
Lessthanhighschool 38
Highschoolgraduate/G.E.D. 34
Morethanhighschool 28
Page235
18) Asociologistbelievesthatthelevelsofeducationalattainmentofhomelesspersonsarenotuniformly
distributed.Totestthisclaim,yourandomlysurvey100homelesspersonsandrecordtheeducational
attainmentofeach.Theresultsareshowninthefollowingtable.Atα
=0.10,isthereevidencetosupportthe
sociologistʹsclaimthatthedistributionisnotuniform?
Response Frequency,f
Lessthanhighschool 38
Highschoolgraduate/G.E.D. 34
Morethanhighschool 28
3 TestforNormality
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Provideanappropriateresponse.
1) Thefrequencydistributionshowstheagesforasampleof100employees.Aretheagesofemployeesnormally
distributed?Useα=0.05.
Classboundaries Frequency,f
29.5–39.5 14
39.5–49.5 29
49.5–59.5 31
59.5–69.5 18
69.5–79.5 8
Page236
10.2 Independence
1 FindExpectedFrequencies
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthemarginalfrequenciesforthegivencontingencytable.
1)
BloodType
 Sex
OABAB
F 110 87 20 8
M71 74 16 4
A)
110 87 20 8 225
71 74 16 4 165
181 161 36 12 390
B)
110 87 20 8 225
71 74 16 4 160
181 161 36 12 390
C)
110 87 20 8 235
71 74 16 4 165
181 161 36 12 400
D)
110 87 20 8 215
71 74 16 4 175
181 161 36 17 390
Page237
2) Thetablebelowdescribesthesmokinghabitsofagroupofasthmasufferers.
Nonsmoker Lightsmoker Heavysmoker
Men 393 90 77
Women 355 80 76

A)
Non–
smoker
Light
smoker
Heavy
smoker Total
Men 393 90 77 560
Women 355 80 76 511
Total 748 170 153 1071
B)
Non–
smoker
Light
smoker
Heavy
smoker Total
Men 393 90 77 560
Women 355 80 76 511
Total 748 170 118 1071
C)
Non–
smoker
Light
smoker
Heavy
smoker Total
Men 393 90 77 560
Women 355 80 76 583
Total 748 170 153 1071
D)
Non–
smoker
Light
smoker
Heavy
smoker Total
Men 393 90 77 415
Women 355 80 76 511
Total 748 182 153 1071
3) Thefollowingcontingencytablewasbasedonarandomsampleofdriversandclassifiesdriversbyagegroup
andnumberofaccidentsinthepastthreeyears.

Numberof
accidents <25
Age
25–45 >45
096 145 287
164 67 62
>120 826
A)
Numberof
accidents <25
Age
25–45 >45 Total
096 145 287 528
164 67 62 193
>120 826 54
Total 180 220 375 775
B)
Numberof
accidents <25
Age
25–45 >45 Total
096 145 287 528
164 67 62 193
>120 826 54
Total 180 220 385 785
C)
Numberof
accidents <25
Age
25–45 >45 Total
096 145 287 528
164 67 62 225
>120 826 54
Total 180 220 375 775
D)
Numberof
accidents <25
Age
25–45 >45 Total
096 145 287 500
164 67 62 193
>120 826 54
Total 180 215 375 770
Page238
Findtheindicatedexpectedfrequency.
4) Thecontingencytablebelowshowstheresultsofarandomsampleof400 staterepresentativesthatwas
conductedtoseewhethertheiropinionsonabillarerelatedtotheirpartyaffiliation.
Opinion
Party
Republican
Democrat
Independent
Approve Disapprove NoOpinion
84 40 28
100 48 36
20 32 12
FindtheexpectedfrequencyforthecellE2,2.Roundtothenearesttenthifnecessary.
A) 55.2 B) 45.6 C) 93.84 D) 34.96
5) Aresearcherwantstodeterminewhetherthenumberofminutesadultsspendonlineperdayisrelatedto
gender.Arandomsampleof315adultswasselectedandtheresultsareshownbelow.Findtheexpected
frequencyforthecellE2,2todetermineifthereisenoughevidencetoconcludethatthenumberofminutes
spentonlineperdayisrelatedtogender.Roundtothenearesttenthifnecessary.
Minutesspentonlineperday
Gender
Male
Female
0–30 30–60 60–90 90+
23 35 76 46
31 42 46 16
A) 33 B) 52.3 C) 44 D) 69.7
6) Amedicalresearcherisinterestedindeterminingifthereisarelationshipbetweenadultsover50whowalk
regularlyandlow,moderate,andhighbloodpressure.Arandomsampleof236adultsover50isselectedand
theresultsaregivenbelow.FindtheexpectedfrequencyE2,2totesttheclaimthatwalkingandlow,moderate,
andhighbloodpressurearenotrelated.Roundtothenearesttenthifnecessary.
BloodPressure Low Moderate High
Walkers 38 64 20
Non–walkers 24 63 27
A) 61.3 B) 29.9 C) 65.7 D) 22.7
7) Asportsresearcherisinterestedindeterminingifthereisarelationshipbetweenthenumberofhometeamand
visitingteamwinsanddifferentsports.Arandomsampleof526gamesisselectedandtheresultsaregiven
below.FindtheexpectedfrequencyforE2,2totesttheclaimthatthenumberofhometeamandvisitingteam
winsareindependentofthesport.Roundtothenearesttenthifnecessary.
Football Basketball Soccer Baseball
Hometeamwins 41 154 27 81
Visitingteamwins 30 98 20 75
A) 106.8 B) 19.9 C) 145.2 D) 27.1
Page239
2 PerformaChi–SquareTestforIndependence
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) Thecontingencytablebelowshowstheresultsofarandomsampleof200staterepresentativesthatwas
conductedtoseewhethertheiropinionsonabillarerelatedtotheirpartyaffiliation.
Opinion
Party
Republican
Democrat
Independent
Approve Disapprove NoOpinion
42 20 14
50 24 18
10 16 6
Findthecriticalvalueχ 2
0,totesttheclaimofindependenceusingα=0.05.
A) 9.488 B) 7.779 C) 11.143 D) 13.277
2) Thecontingencytablebelowshowstheresultsofarandomsampleof200staterepresentativesthatwas
conductedtoseewhethertheiropinionsonabillarerelatedtotheirpartyaffiliation.
Opinion
Party
Republican
Democrat
Independent
Approve Disapprove NoOpinion
42 20 14
50 24 18
10 16 6
Findthechi–squareteststatistic,χ2,totesttheclaimofindependence.
A) 8.030 B) 11.765 C) 7.662 D) 9.483
3) Aresearcherwantstodeterminewhetherthenumberofminutesadultsspendonlineperdayisrelatedto
gender.Arandomsampleof315adultswasselectedandtheresultsareshownbelow.Findthecriticalvalue
χ2
0todetermineifthereisenoughevidencetoconcludethatthenumberofminutesspentonlineperdayis
relatedtogender.Useα=0.05.
Minutesspentonlineperday
Gender
Male
Female
0–30 30–60 60–90 90+
25 35 75 45
30 45 45 15
A) 7.815 B) 9.348 C) 11.345 D) 6.251
4) Aresearcherwantstodeterminewhetherthenumberofminutesadultsspendonlineperdayisrelatedto
gender.Arandomsampleof315adultswasselectedandtheresultsareshownbelow.Calculatethechi –square
teststatisticχ2todetermineifthereisenoughevidencetoconcludethatthenumberofminutesspentonline
perdayisrelatedtogender.
Minutesspentonlineperday
Gender
Male
Female
0–30 30–60 60–90 90+
25 35 75 45
30 45 45 15
A) 18.146 B) 19.874 C) 20.912 D) 21.231
Page240
5) Amedicalresearcherisinterestedindeterminingifthereisarelationshipbetweenadultsover50whowalk
regularlyandlow,moderate,andhighbloodpressure.Arandomsampleof236adultsover50isselectedand
theresultsaregivenbelow.Findthecriticalvalueχ2
0totesttheclaimthatwalkingandlow,moderate,and
highbloodpressurearenotrelated.Useα=0.01.
BloodPressure Low Moderate High
Walkers 35 62 25
Non–walkers 21 65 28
A) 9.210 B) 9.348 C) 6.251 D) 0.584
6) Amedicalresearcherisinterestedindeterminingifthereisarelationshipbetweenadultsover50whowalk
regularlyandlow,moderate,andhighbloodpressure.Arandomsampleof236adultsover50isselectedand
theresultsaregivenbelow.Calculatethechi–squareteststatisticχ2totesttheclaimthatwalkingandlow,
moderate,andhighbloodpressurearenotrelated.
BloodPressure Low Moderate High
Walkers 35 62 25
Non–walkers 21 65 28
A) 3.473 B) 16.183 C) 18.112 D) 6.003
7) Asportsresearcherisinterestedindeterminingifthereisarelationshipbetweenthenumberofhometeamand
visitingteamwinsanddifferentsports.Arandomsampleof526gamesisselectedandtheresultsaregiven
below.Findthecriticalvalueχ 2
0totesttheclaimthatthenumberofhometeamandvisitingteamwinsis
independentofthesport.Useα=0.01.
Football Basketball Soccer Baseball
Hometeamwins 39 156 25 83
Visitingteamwins 31 98 19 75
A) 11.345 B) 12.838 C) 7.815 D) 9.348
8) Asportsresearcherisinterestedindeterminingifthereisarelationshipbetweenthetypeofsportandtypeof
teamwinning(hometeamversusvisitingteam).Arandomsampleof526gamesisselectedandtheresultsare
givenbelow.Calculatethechi–squareteststatisticχ2totesttheclaimthatthetypeofteamwinningis
independentofthesport.
Football Basketball Soccer Baseball
Hometeamwins 39 156 25 83
Visitingteamwins 31 98 19 75
A) 3.290 B) 2.919 C) 5.391 D) 4.192
Page241
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Performtheindicatedhypothesistest.Besuretodothefollowing:identifytheclaimandstatethenullandalternative
hypotheses.Determinethecriticalvalue.Calculatetheteststatistic.Decidetorejectortofailtorejectthenull
hypothesisandinterpretthedecisioninthecontextoftheoriginalclaim.
9) Thecontingencytablebelowshowsthepartyandopinionsonabillofarandomsampleof200state
representatives.
Party Opinion
Republican
Democrat
Independent
Approve Disapprove NoOpinion
42 20 14
50 24 18
10 16 6
Testtheclaimthatopinionsonthebillaredependentonpartyaffiliation.Useα=0.05.
10) Aresearcherisinterestedindeterminingifthereisarelationshipbetweenexerciseandbloodpressurefor
adultsover50.Arandomsampleof236adultsover50isselectedandtheresultsaregivenbelow.
BloodPressure
Exercise
Low Moderate High
Walkers 35 62 25
Non–walkers 21 65 28

Testtheclaimthatwalkingandbloodpressureareindependent.Useα
=0.01.
11) Aresearcherwantstodeterminewhetherthetimespentonlineperdayisrelatedtogender.Arandomsample
of315adultswasselectedandtheresultsareshownbelow.
Gender Timespentonlineperday(minutes)
Male
Female
0–30 30–60 60–90 90+
25 35 75 45
30 45 45 15
Testthehypothesisthatthetimespentonlineperdayisrelatedtogender.Useα=0.05.
12) Asportsresearcherisinterestedindeterminingifthereisarelationshipbetweenthetypeofsportandtypeof
teamwinning(hometeamversusvisitingteam).Arandomsampleof526gamesisselectedandtheresultsare
givenbelow.
Football Basketball Soccer Baseball
Hometeamwins 39 156 25 83
Visitingteamwins 31 98 19 75
Testtheclaimthatthetypeofteamwinningisindependentofthetypeofsport.Useα=0.01.
Page242
13) Thedatabelowshowstheageandfavoritetypeofmusicof779randomlyselectedpeople.
Age Country Rock Pop Classical
15–21 21 45 90 33
21–30 68 55 42 48
30–40 65 47 31 57
40–50 60 39 25 53
Testtheclaimthatpreferredmusictypeisdependentonage.Useα=0.05.
3 PerformaHomogeneityofProportionsTest
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Performahomogeneityofproportionstesttotestwhetherthepopulationproportionsareequal.Thehypothesestobe
testedare:
H0:Theproportionsareequal
Ha:Atleastoneoftheproportionsisdifferentfromtheothers.
1) Randomsamplesof400menand400womenwereobtainedandparticipantswereaskedwhetherthey
plannedtovoteinthenextelection.Theresultsarelistedbelow.Performahomogeneityofproportionstestto
testtheclaimthattheproportionofmenwhoplantovoteinthenextelectionisthesameastheproportionof
womenwhoplantovote.Useα=0.05.
Men Women
Plantovote 230 255
Donotplantovote 170 145
2) Arandomsampleof100studentsfrom5differentcollegeswasrandomlyselected,andthenumberwhosmoke
wasrecorded.Theresultsarelistedbelow.Performahomogeneityofproportionstesttotesttheclaimthatthe
proportionofstudentswhosmokeisthesameinall5colleges.Useα
=0.01.
Colleges
12345
Smoker 18 25 12 33 22
Nonsmoker 82 75 88 67 78
10.3 ComparingTwoVariances
1 FindtheCriticalF–Value
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) FindthecriticalvalueF0foratwo–tailedtestusingα =0.05,d.f.N=5,andd.f.D=10.
A) 4.24 B) 4.07 C) 4.47 D) 6.62
2) FindthecriticalvalueF0foraone–tailedtestusingα =0.01,d.f.N=3,andd.f.D=20.
A) 4.94 B) 25.58 C) 5.82 D) 3.09
3) FindthecriticalvalueF0foraone–tailedtestusingα =0.05,d.f.N=6,andd.f.D=16.
A) 2.74 B) 3.94 C) 2.66 D) 2.19
4) FindthecriticalvalueF0foratwo–tailedtestusingα =0.02,d.f.N=5,andd.f.D=10.
A) 5.64 B) 10.05 C) 5.99 D) 5.39
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5) FindthecriticalvalueF0totesttheclaimthatσ 2
1=σ 2
2.Twosamplesarerandomlyselectedfrompopulations
thatarenormal.Thesamplestatisticsaregivenbelow.Useα=0.05.
n1=25 n2=30
s2
1=3.61 s 2
2=2.25
A) 2.15 B) 2.21 C) 2.14 D) 2.09
6) FindthecriticalvalueF0totesttheclaimthatσ 2
1=σ 2
2.Twosamplesarerandomlyselectedfrompopulations
thatarenormal.Thesamplestatisticsaregivenbelow.Useα=0.02.
n1=13 n2=12
s2
1=7.84 s 2
2=6.25
A) 4.40 B) 2.79 C) 4.25 D) 3.43
7) FindthecriticalvalueF0totesttheclaimthatσ 2
1>σ 2
2.Twosamplesarerandomlyselectedfrompopulations
thatarenormal.Thesamplestatisticsaregivenbelow.Useα=0.01.
n1=16 n2=13
s2
1=1600 s 2
2=625
A) 4.01 B) 3.18 C) 2.62 D) 3.67
8) FindthecriticalvalueF0totesttheclaimthatσ 2
1≤σ 2
2.Twosamplesarerandomlyselectedfrompopulations
thatarenormal.Thesamplestatisticsaregivenbelow.Useα=0.05.
n1=16 n2=15
s2
1=8.41 s 2
2=7.84
A) 2.46 B) 3.66 C) 2.95 D) 2.40
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9) FindthecriticalvalueF0totesttheclaimthatσ 2
1≠σ 2
2.Twosamplesarerandomlyselectedfrompopulations
thatarenormal.Thesamplestatisticsaregivenbelow.Useα=0.02.
n1=11 n2=18
s2
1=0.578 s 2
2=0.260
A) 3.59 B) 2.45 C) 2.92 D) 4.56
2 TestaClaimAbouttheDifferencesBetweenTwoPopulationVariances
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) CalculatetheteststatisticFtotesttheclaimthatσ2
1=σ 2
2.Twosamplesarerandomlyselectedfrom
populationsthatarenormal.Thesamplestatisticsaregivenbelow.
n1=25 n2=30
s2
1=6.498 s 2
2=4.05
A) 1.604 B) 1.267 C) 0.623 D) 2.573
2) CalculatetheteststatisticFtotesttheclaimthatσ2
1=σ 2
2.Twosamplesarerandomlyselectedfrom
populationsthatarenormal.Thesamplestatisticsaregivenbelow.
n1=13 n2=12
s2
1=17.248 s 2
2=13.75
A) 1.254 B) 0.797 C) 1.120 D) 1.573
3) CalculatetheteststatisticFtotesttheclaimthatσ2
1≥σ 2
2.Twosamplesarerandomlyselectedfrom
populationsthatarenormal.Thesamplestatisticsaregivenbelow.
n1=13 n2=12
s2
1=35.224 s 2
2=28.083
A) 1.254 B) 0.797 C) 1.573 D) 1.120
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4) CalculatetheteststatisticFtotesttheclaimthatσ2
1>σ 2
2.Twosamplesarerandomlyselectedfrom
populationsthatarenormal.Thesamplestatisticsaregivenbelow.
n1=16 n2=13
s2
1=4800 s 2
2=1875
A) 2.560 B) 1.600 C) 0.391 D) 6.554
5) CalculatetheteststatisticFtotesttheclaimthatσ2
1≤σ 2
2.Twosamplesarerandomlyselectedfrom
populationsthatarenormal.Thesamplestatisticsaregivenbelow.
n1=16 n2=15
s2
1=12.615 s 2
2=11.76
A) 1.073 B) 0.932 C) 1.036 D) 1.151
6) CalculatetheteststatisticFtotesttheclaimthatσ2
1≠σ 2
2.Twosamplesarerandomlyselectedfrom
populationsthatarenormal.Thesamplestatisticsaregivenbelow.
n1=11 n2=18
s2
1=1.156 s 2
2=0.52
A) 2.223 B) 1.490 C) 4.933 D) 0.450
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Performtheindicatedhypothesistest.Besuretodothefollowing:identifytheclaimandstatethenullandalternative
hypotheses.Determinethecriticalvalueandrejectionregion.Calculatetheteststatistic.Decidetorejectortofailto
rejectthenullhypothesisandinterpretthedecisioninthecontextoftheoriginalclaim.Assumethatthesamplesare
independentandthateachpopulationhasanormaldistribution.
7) Testtheclaimthatσ 2
1=σ 2
2.Twosamplesarerandomlyselectedfrompopulationsthatarenormal.The
samplestatisticsaregivenbelow.Useα=0.05.
n1=25 n2=30
s2
1=5.776 s 2
2=3.6
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8) Testtheclaimthatσ 2
1=σ 2
2.Twosamplesarerandomlyselectedfrompopulationsthatarenormal.The
samplestatisticsaregivenbelow.Useα=0.02.
n1=13 n2=12
s2
1=10.976 s 2
2=8.75
9) Testtheclaimthatσ 2
1>σ 2
2.Twosamplesarerandomlyselectedfrompopulationsthatarenormal.The
samplestatisticsaregivenbelow.Useα=0.01.
n1=16 n2=13
s2
1=3200 s 2
2=1250
10) Testtheclaimthatσ 2
1≤σ 2
2.Twosamplesarerandomlyselectedfrompopulationsthatarenormal.The
samplestatisticsaregivenbelow.Useα=0.05.
n1=16 n2=15
s2
1=21.866 s 2
2=20.384
11) Testtheclaimthatσ 2
1≠σ 2
2.Twosamplesarerandomlyselectedfrompopulationsthatarenormal.The
samplestatisticsaregivenbelow.Useα=0.02.
n1=11 n2=18
s2
1=2.89 s 2
2=1.3
12) Alocalbankhasthereputationofhavingavarianceinwaitingtimesaslowasthatofanybankinthearea.A
competitorbankintheareachecksthewaitingtimeatbothbanksandclaimsthatitsvarianceofwaitingtimes
islowerthanatthelocalbank.Thesamplestatisticsarelistedbelow.Testthecompetitorʹsclaim.Useα
=0.05.
LocalBank CompetitorBank
n1=13 n2=16
s1=1.69minutes s2=1.43minutes
13) Astudywasconductedtodetermineifthevariancesofelementaryschoolteachersalariesfromtwo
neighboringdistrictswereequal.Asampleof25teachersfromeachdistrictwasselected.Thefirstdistricthad
astandarddeviationofs1=$4830,andtheseconddistricthadastandarddeviations2=$4410.Testtheclaim
thatthevariancesofthesalariesfrombothdistrictsareequal.Useα=0.05
Page247
14) Arandomsampleof21womenhadbloodpressurelevelswithavarianceof553.6.Arandomsampleof18men
hadbloodpressurelevelswithavarianceof368.64.Testtheclaimthatthebloodpressurelevelsforwomen
havealargervariancethanthoseformen.Useα=0.01.
15) Theweightsofarandomsampleof121womenbetweentheagesof25and34hadastandarddeviationof28
pounds.Theweightsof121womenbetweentheagesof55and64hadastandarddeviation21pounds.Test
theclaimthattheolderwomenarefromapopulationwithastandarddeviationlessthanthatforwomenin
the25to34agegroup.Useα=0.05.
16) Atacollege,61femalestudentswererandomlyselectedanditwasfoundthattheirmonthlyincomehada
standarddeviationof$218.70.For121malestudents,thestandarddeviationwas$303.66.Testtheclaimthat
varianceofmonthlyincomesishigherformalestudentsthanitisforfemalestudents.Useα
=0.01.
17) Amedicalresearchersuspectsthatthevarianceofthepulserateofsmokersishigherthanthevarianceofthe
pulserateofnon–smokers.Usethesamplestatisticsbelowtotesttheresearcherʹssuspicion.Use
α=0.05.
Smokers Non–smokers
n1=61 n2=121
s1=7.8 s2=5.3
18) Astatisticsteacherbelievesthatthevarianceoftestscoresofstudentsinhereveningstatisticsclassislower
thanthevarianceoftestscoresofstudentsinherdayclass.Theresultsofanexam,giventothedayand
eveningstudents,areshownbelow.Cantheteacherconcludethatthescoresofeveningstudentshavealower
variance?Useα=0.01.
DayStudents EveningStudents
n1=31 n2=41
s1=9.8 s2=5.3
19) Astatisticsteacherwantstoseewhetherthereisasignificantdifferencebetweenthevarianceoftheagesofday
studentsandthevarianceoftheagesofnightstudents.Arandomsampleof31studentsisselectedfromeach
group.Thedataaregivenbelow.Testtheclaimthatthereisnodifferencebetweenthevariancesofthetwo
groups.
Useα=0.05.
DayStudents
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
EveningStudents
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
Page248
20) Alocalbankhasthereputationofhavingavarianceinwaitingtimesaslowasthatofanybankinthearea.A
competitorbankintheareachecksthewaitingtimeatbothbanksandclaimsthatitsvarianceofwaitingtimes
islowerthanatthelocalbank.Thesamplestatisticsarelistedbelow.Testthecompetitorʹsclaim.Useα
=0.05.
LocalBank CompetitorBank
n1=41 n2=61
s1=2minutes s2=1.1minutes
3 FindtheCriticalF–valueforaLeft–tailedTest
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Provideanappropriateresponse.
1) Findtheleft–tailedandright–tailedcriticalF–valuesforatwo–tailedtest.Letα =0.02,d.f.N=7,and
d.f.D=5.
2) Findtheleft–tailedandright–tailedcriticalF–valuesforatwo–tailedtest.Usethesamplestatisticsbelow.Let
α=0.05.
n1=5n
2=6
s2
1=5.8 s2
2=2.7
4 ConstructtheIndicatedConfidenceInterval
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Provideanappropriateresponse.
1) Theweightsofarandomsampleof25womenbetweentheagesof25and34hadastandarddeviationof28
pounds.Theweightsofarandomsampleof41womenbetweentheagesof55and64hadastandard
deviationof21pounds.Constructa95%confidenceintervalforσ12
σ22,whereσ12andσ22arethevariancesof
theweightsofwomenbetweentheages25and34andtheweightsofwomenbetweentheagesof55and64
respectively.
10.4 AnalysisofVariance
1 PerformaOne–WayANOVATest
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) FindthecriticalF0–valuetotesttheclaimthatthepopulationshavethesamemean.Useα=0.05.
Brand1 Brand2 Brand3
n=8n=8n=8
x=3.0 x=2.6 x=2.6
s=0.50 s=0.60 s=0.55
A) 3.47 B) 3.210 C) 2.819 D) 1.892
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2) FindtheteststatisticFtotesttheclaimthatthepopulationshavethesamemean.
Brand1 Brand2 Brand3
n=8n=8n=8
x=3.0 x=2.6 x=2.6
s=0.50 s=0.60 s=0.55
A) 1.403 B) 1.021 C) 1.182 D) 0.832
3) Amedicalresearcherwishestotrythreedifferenttechniquestolowerbloodpressureofpatientswithhigh
bloodpressure.Thesubjectsarerandomlyselectedandassignedtooneofthreegroups.Group1isgiven
medication,Group2isgivenanexerciseprogram,andGroup3isassignedadietprogram.Attheendofsix
weeks,eachsubjectʹsbloodpressureisrecorded.FindthecriticalvalueF0totesttheclaimthatthereisno
differenceamongthemeans.Useα=0.05.
Group1 Group2 Group3
984
12 2 12
11 3 4
15 5 8
13 4 9
806
A) 3.68 B) 19.43 C) 4.77 D) 39.43
4) Amedicalresearcherwishestotrythreedifferenttechniquestolowerbloodpressureofpatientswithhigh
bloodpressure.Thesubjectsarerandomlyselectedandassignedtooneofthreegroups.Group1isgiven
medication,Group2isgivenanexerciseprogram,andGroup3isassignedadietprogram.Attheendofsix
weeks,eachsubjectʹsbloodpressureisrecorded.FindtheteststatisticFtotesttheclaimthatthereisno
differenceamongthemeans.
Group1 Group2 Group3
13 8 6
12 5 12
11 3 4
15 2 8
949
804
A) 11.095 B) 9.812 C) 8.369 D) 12.162
5) Fourdifferenttypesoffertilizersareusedonraspberryplants.Thenumberofraspberriesoneachrandomly
selectedplantisgivenbelow.FindthecriticalvalueF0totesttheclaimthatthetypeoffertilizermakesno
differenceinthemeannumberofraspberriesperplant.Useα=0.01.
Fertilizer1Fertilizer2Fertilizer3Fertilizer4
68 63
55 35
65 33
75 44
75 24
66 35
A) 4.94 B) 4.43 C) 26.69 D) 4.22
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6) Fourdifferenttypesoffertilizersareusedonraspberryplants.Thenumberofraspberriesoneachrandomly
selectedplantisgivenbelow.FindtheteststatisticFtotesttheclaimthatthetypeoffertilizermakesno
differenceinthemeannumberofraspberriesperplant.
Fertilizer1Fertilizer2Fertilizer3Fertilizer4
68 63
75 35
65 23
55 44
75 35
66 34
A) 8.357 B) 8.123 C) 7.123 D) 6.912
7) Aresearcherwishestodeterminewhetherthereisadifferenceintheaverageageofelementaryschool,high
school,andcommunitycollegeteachers.Teachersarerandomlyselected.Theiragesarerecordedbelow.Find
thecriticalvalueF0totesttheclaimthatthereisnodifferenceintheaverageageofeachgroup.Useα
=0.01.
ElementaryTeachers HighSchoolTeachers CommunityCollegeTeachers
23 41 39
28 38 45
27 36 36
37 47 61
25 42 45
52 31 35
A) 6.36 B) 5.09 C) 9.43 D) 5.42
8) Aresearcherwishestodeterminewhetherthereisadifferenceintheaverageageofelementaryschool,high
school,andcommunitycollegeteachers.Teachersarerandomlyselected.Theiragesarerecordedbelow.Find
theteststatisticFtotesttheclaimthatthereisnodifferenceintheaverageageofeachgroup.
ElementaryTeachers HighSchoolTeachers CommunityCollegeTeachers
23 36 45
28 38 45
27 41 36
52 47 61
25 42 39
37 31 35
A) 2.517 B) 2.913 C) 3.189 D) 4.312
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Performtheindicatedone–wayANOVAtest.Besuretodothefollowing:identifytheclaimandstatethenulland
alternativehypotheses.Determinethecriticalvalueandrejectionregion.CalculatetheteststatisticF.Decidetoreject
ortofailtorejectthenullhypothesisandinterpretthedecisioninthecontextoftheoriginalclaim.Assumethateach
populationisnormallydistributedandthatthepopulationvariancesareequal.
9) Testtheclaimthatthepopulationshavethesamemean.Useα=0.05.
Brand1 Brand2 Brand3
n=8n=8n=8
x=3.0 x=2.6 x=2.6
s=0.50 s=0.60 s=0.55
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10) Amedicalresearcherwishestotrythreedifferenttechniquestolowerbloodpressureofpatientswithhigh
bloodpressure.Thesubjectsarerandomlyselectedandassignedtooneofthreegroups.Group1isgiven
medication,Group2isgivenanexerciseprogram,andGroup3isassignedadietprogram.Attheendofsix
weeks,eachsubjectʹsbloodpressureisrecorded.Testtheclaimthatthereisnodifferenceamongthemeans.
Useα=0.05.
Group1 Group2 Group3
11 8 6
12 2 12
13 3 4
15 5 8
949
804
11) Fourdifferenttypesoffertilizersareusedonraspberryplants.Thenumberofraspberriesoneachrandomly
selectedplantisgivenbelow.Testtheclaimthatthetypeoffertilizermakesnodifferenceinthemeannumber
ofraspberriesperplant.Useα=0.01.
Fertilizer1Fertilizer2Fertilizer3Fertilizer4
65 63
58 35
75 43
65 24
75 34
66 35
12) Aresearcherwishestodeterminewhetherthereisadifferenceintheaverageageofelementaryschool,high
school,andcommunitycollegeteachers.Teachersarerandomlyselectedfromeachgroup.Theiragesare
recordedbelow.Testtheclaimthatatleastonemeanisdifferentfromtheothers.Useα
=0.01.
ElementaryTeachers HighSchoolTeachers CommunityCollegeTeachers
23 41 39
28 36 45
27 38 36
25 47 61
52 42 45
37 31 35
13) Thegradepointaveragesofstudentsparticipatinginsportsatalocalcollegearetobecompared.Thedataare
listedbelow.Testtheclaimthatatleastonegrouphasadifferentmean.Useα=0.05.
Tennis Golf Swimming
2.6 1.8 2.7
3.2 3.3 2.5
2.5 1.9 2.8
3.5 2.1 3.0
3.1 2.5
2.1
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14) Aresearcherwishestocomparetimesittakesworkerstoassembleacertaincomputercomponentusing
differentmachines.Workersarerandomlyselectedandrandomlyassignedtooneofthreedifferentmachines.
Thetime(inminutes)ittakeseachworkertoassemblethecomponentisrecorded.Testtheclaimthatthereis
nodifferenceinthemeanassemblytimesforthethreemachines.Useα
=0.01.
Machine1Machine2Machine3
32 40 31
32 29 28
31 38 29
30 33 25
33 36
31 32
35
15) Arealtorwishestocomparethesquarefootageofhousesofsimilarpricesin4differentcities.Thedataare
listedbelow.Cantherealtorconcludethatthemeansquarefootageisthesameinallfourcities?Useα
=0.01.
City#1 City#2 City#3 City#4
2150 1780 1530 2400
1980 1540 1670 2350
2000 1690 1580 2600
2210 1650 1750 2150
1900 1500 2000
1600 2200
2350
16) Amedicalresearcherwishestotrythreedifferenttechniquestolowerbloodpressureofpatientswithhigh
blood–pressure.Thesubjectsarerandomlyselectedandrandomlyassignedtooneofthreegroups.Group1is
givenmedication,Group2isgivenanexerciseprogram,andGroup3isassignedadietprogram.Attheendof
sixweeks,eachsubjectʹsbloodpressureisrecorded.PerformaScheffe
´Testtodeterminewhichmeanshavea
significancedifference.Useα=0.05.
Group1 Group2 Group3
11 8 6
12 2 12
13 5 4
15 3 8
949
804
17) Fourdifferenttypesoffertilizersareusedonraspberryplants.Thenumberofraspberriesoneachrandomly
selectedplantisgivenbelow.PerformaScheffe
´Testtodeterminewhichmeanshaveasignificancedifference.
Useα=0.01.
Fertilizer1Fertilizer2Fertilizer3Fertilizer4
6563
7835
5543
6534
7525
6634
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Ch.10 Chi–SquareTestsandtheF–Distribution
AnswerKey
10.1 Goodness–o
f
–FitTest
1 FindExpectedFrequencies
2 PerformaChi–SquareGoodness–of–FitTest
3TestforNormality
10.2 Independence
1 FindExpectedFrequencies
2 PerformaChi–SquareTestforIndependence
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3PerformaHomogeneityofProportionsTest
10.3 ComparingTwoVariances
1 FindtheCriticalF–Value
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2 TestaClaimAbouttheDifferencesBetweenTwoPopulationVariances
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3FindtheCriticalF–valueforaLeft–tailedTest
4 ConstructtheIndicatedConfidenceInterval
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10.4 AnalysisofVariance
1 PerformaOne–WayANOVATest
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