Chapter 06 Assume The Population Standard Deviation Find The

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Ch.6 ConfidenceIntervals

6.1 ConfidenceIntervalsfortheMean(σKnown)

1 FindaCriticalValue

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

Provideanappropriateresponse.

1) Findthecriticalvaluezcthatcorrespondstoa94%confidencelevel.

A) ±1.88 B) ±1.645 C) ±1.96 D) ±2.33

2 FindtheMarginofError

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

Provideanappropriateresponse.

1) Determinethesamplingerrorifthegradepointaveragesfor10randomlyselectedstudentsfromaclassof125

studentshasameanofx=2.8.Assumethegradepointaverageofthe125studentshasameanof

μ=3.5.

A) 0.7 B) 3.15 C) –0.7 D) 2.45

2) Findthemarginoferrorforthegivenvaluesofc,σ,andn.

c=0.90,σ=11.5,n=120

A) 1.73 B) 0.16 C) 1.05 D) 0.94

3) Findthemarginoferrorforthegivenvaluesofc,σ,andn.

c=0.98,σ=0.78,n=150

A) 0.15 B) 0.08 C) 0.11 D) 0.12

4) Findthemarginoferrorforthegivenvaluesofc,σ,andn.

c=0.95,σ=677,n=40

A) $210 B) $77 C) $2891 D) $7

3 ConstructandInterpretConfidenceIntervalsforthePopulationMean

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

Provideanappropriateresponse.

1) Arandomsampleof150studentshasagradepointaveragewithameanof2.86.Assumethepopulation

standarddeviationis0.78.Constructtheconfidenceintervalforthepopulationmean,μ,ifc=0.98.

A) (2.71,3.01) B) (2.51,3.53) C) (2.43,3.79) D) (2.31,3.88)

2) Arandomsampleof40studentshasatestscorewithx=81.5.Assumethepopulationstandarddeviationis

10.2.Constructtheconfidenceintervalforthepopulationmean,μifc=0.90.

A) (78.8,84.2) B) (51.8,92.3) C) (66.3,89.1) D) (71.8,93.5)

3) Arandomsampleof40studentshasameanannualearningsof$3120.Assumethepopulationstandard

deviationis$677.Constructtheconfidenceintervalforthepopulationmean,μifc=0.95.

A) ($2910,$3330) B) ($210,$110) C) ($4812,$5342) D) ($1987,$2346)

4) Arandomsampleof56fluorescentlightbulbshasameanlifeof645hours.Assumethepopulationstandard

deviationis31hours.Constructa95%confidenceintervalforthepopulationmean.

A) (636.9,653.1) B) (539.6,551.2) C) (112.0,118.9) D) (712.0,768.0)

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5) Agroupof49randomlyselectedstudentshasameanageof22.4years.Assumethepopulationstandard

deviationis3.8.Constructa98%confidenceintervalforthepopulationmean.

A) (21.1,23.7) B) (20.3,24.5) C) (19.8,25.1) D) (18.8,26.3)

6) Agroupof40bowlersshowedthattheiraveragescorewas192.Assumethepopulationstandarddeviationis

8.Findthe95%confidenceintervalofthemeanscoreofallbowlers.

A) (189.5,194.5) B) (186.5,197.5) C) (188.5,195.6) D) (187.3,196.1)

7) Inarandomsampleof60computers,themeanrepaircostwas$150.Assumethepopulationstandard

deviationis$36.Constructa90%confidenceintervalforthepopulationmean.

A) ($142,$158) B) ($138,$162) C) ($141,$159) D) ($537,$654)

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

8) Inarandomsampleof60computers,themeanrepaircostwas$150.Assumethepopulationstandard

deviationis$36.

a)Constructthe99%confidenceintervalforthepopulationmeanrepaircost.

b)Ifthelevelofconfidencewasloweredto95%,whatwillbetheeffectontheconfidenceinterval?

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

9) Inarecentstudyof42eighthgraders,themeannumberofhoursperweekthattheywatchedtelevisionwas

19.6.Assumethepopulationstandarddeviationis5.8hours.Findthe98%confidenceintervalforthe

populationmean.

A) (17.5,21.7) B) (14.1,23.2) C) (18.3,20.9) D) (19.1,20.4)

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

10) Inarecentstudyof92eighthgraders,themeannumberofhoursperweekthattheywatchedtelevisionwas

22.6.Assumethepopulationstandarddeviationis5.2hours.

a)Findthe99%confidenceintervalofthemean.

b)Ifthestandarddeviationisdoubledto10.4,whatwillbetheeffectontheconfidenceinterval?

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

11) Inasampleof10randomlyselectedwomen,itwasfoundthattheirmeanheightwas63.4inches.From

previousstudies,itisassumedthatthestandarddeviationσis2.4andthatthepopulationofheight

measurementsisnormallydistributed.Constructthe95%confidenceintervalforthepopulationmean.

A) (61.9,64.9) B) (58.1,67.3) C) (59.7,66.5) D) (60.8,65.4)

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

12) Inasampleof10randomlyselectedwomen,itwasfoundthattheirmeanheightwas63.4inches.From

previousstudies,itisassumedthatthestandarddeviation,σ,is2.4inchesandthatthepopulationofheight

measurementsisnormallydistributed.

a)Constructthe99%confidenceintervalforthepopulationmeanheightofwomen.

b)Ifthesamplesizewasdoubledto20women,whatwillbetheeffectontheconfidenceinterval?

Page136

13) Thenumbersofadvertisementsseenorheardinoneweekfor30randomlyselectedpeopleintheUnitedStates

arelistedbelow.Constructa95%confidenceintervalforthetruemeannumberofadvertisements.Assume

thatσis159.5.

598 494 441 595 728 690 684 486 735 808

481 298 135 846 764 317 649 732 582 677

734 588 590 540 673 727 545 486 702 703

14) Thenumberofwinsinaseasonfor32randomlyselectedprofessionalfootballteamsarelistedbelow.

Constructa90%confidenceintervalforthetruemeannumberofwinsinaseason.Assumethatσis2.6.

99981

972

11106411988

12107512643

1299710775

4 DeterminetheMinimumSampleSize

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

Provideanappropriateresponse.

1) ThestandardIQtesthasameanof96andastandarddeviationof14.Wewanttobe99%certainthatweare

within4IQpointsofthetruemean.Determinetherequiredsamplesize.

A) 82 B) 10 C) 178 D) 1

2) Anurseatalocalhospitalisinterestedinestimatingthebirthweightofinfants.Howlargeasamplemustshe

selectifshedesirestobe95%confidentthatthetruemeaniswithin3ouncesofthesamplemean?The

standarddeviationofthebirthweightsisknowntobe6ounces.

A) 16 B) 15 C) 4 D) 3

3) Inordertosetrates,aninsurancecompanyistryingtoestimatethenumberofsickdaysthatfulltimeworkers

atanautorepairshoptakeperyear.Apreviousstudyindicatedthatthestandarddeviationwas2.8days.How

largeasamplemustbeselectedifthecompanywantstobe95%confidentthatthetruemeandiffersfromthe

samplemeanbynomorethan1day?

A) 31 B) 141 C) 512 D) 1024

4) Inordertoefficientlybidonacontract,acontractorwantstobe95%confidentthathiserrorislessthantwo

hoursinestimatingtheaveragetimeittakestoinstalltileflooring.Previouscontractsindicatethatthe

standarddeviationis4.5hours.Howlargeasamplemustbeselected?

A) 20 B) 4 C) 5 D) 19

5) Inordertofairlysetflatratesforautomechanics,ashopforemanneedstoestimatetheaveragetimeittakesto

replaceafuelpumpinacar.Howlargeasamplemustheselectifhewantstobe99%confidentthatthetrue

averagetimeiswithin15minutesofthesampleaverage?Assumethestandarddeviationofalltimesis30

minutes.

A) 27 B) 26 C) 6 D) 5

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

6) Inordertosetrates,aninsurancecompanyistryingtoestimatethenumberofsickdaysthatfulltimeworkers

atanautorepairshoptakeperyear.Apreviousstudyindicatedthatthestandarddeviationwas2.8days.a)

Howlargeasamplemustbeselectedifthecompanywantstobe95%confidentthatthetruemeandiffersfrom

thesamplemeanbynomorethan1day?b)Repeatpart(a)usinga98%confidenceinterval.Whichlevelof

confidencerequiresalargersamplesize?Explain.

5DeterminetheFinitePopulationCorrectionFactor

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

Provideanappropriateresponse.

1) Therewere800mathinstructorsatamathematicsconvention.Fortyinstructorswererandomlyselectedand

givenanIQtest.Thescoresproducedameanof130withastandarddeviationof10.Finda95%confidence

intervalforthemeanofthe800instructors.Usethefinitepopulationcorrectionfactor.

6Concepts

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

Provideanappropriateresponse.

1) Arandomsampleof200highschoolseniorsisgiventheSAT–Vtest.Themeanscoreforthissampleisx=451.

Whatcanyousayaboutthemeanscoreμofallhighschoolseniors?

2) Thegradepointaveragesfor10randomlyselectedstudentsinastatisticsclasswith125studentsarelisted

below.Whatcanyousayaboutthemeanscoreμofall125students?

3.62.62.02.12.73.42.91.93.93.7

3) Acertainconfidenceinintervalis7.75<μ<9.45.FindthesamplemeanxandtheerrorofestimateE.

4) Giventhesamesamplestatistics,whichlevelofconfidencewillproducethenarrowestconfidenceinterval:

75%,85%,90%,or95%?Explainyourreasoning.

5) Thegradepointaveragesfor10randomlyselectedstudentsinastatisticsclasswith125studentsarelisted

below.

2.03.21.82.90.94.03.32.93.60.8

Whatistheeffectonthewidthoftheconfidenceintervalifthesamplesizeisincreasedto20?Explainyour

reasoning.

6.2 ConfidenceIntervalsfortheMean(σUnknown)

1 FindaCriticalValue

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

Provideanappropriateresponse.

1) Findthecriticalvalue,tcforc=0.99andn=10.

A) 3.250 B) 3.169 C) 2.262 D) 1.833

2) Findthecriticalvalue,tc

forc=0.95andn=16.

A) 2.131 B) 2.120 C) 2.602 D) 2.947

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3) Findthecriticalvalue,tc

forc=0.90andn=15.

A) 1.761 B) 1.753 C) 2.145 D) 2.624

2 ConstructandInterpretConfidenceIntervalsforthePopulationMean

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

Provideanappropriateresponse.

1) FindthevalueofE,themarginoferror,forc=0.99

n=16ands=2.6.

A) 1.92 B) 0.48 C) 1.69 D) 0.42

2) FindthevalueofE,themarginoferror,forc=0.90

n=10ands=3.1.

A) 1.80 B) 1.36 C) 0.57 D) 1.78

3) FindthevalueofE,themarginoferror,forc=0.95

n=15ands=5.6.

A) 3.10 B) 3.19 C) 2.55 D) 0.80

4) Constructtheindicatedconfidenceintervalforthepopulationmeanμusingthet–distribution.

c=0.95,x=645,s=31,n=16

A) (628.5,661.5) B) (876.2,981.5) C) (531.2,612.9) D) (321.7,365.8)

5) Constructtheindicatedconfidenceintervalforthepopulationmeanμusingthet–distribution.

c=0.99,x=22.4,s=3.8,n=19

A) (19.9,24.9) B) (16.3,26.9) C) (17.2,23.6) D) (18.7,24.1)

6) Usetheconfidenceintervaltofindthemarginoferrorandthesamplemean.

(12,20)

A) E=4,x=16 B) E=8,x=16 C) E=8,x=12 D) E=4,x=20

7) Inarandomsampleof28families,theaverageweeklyfoodexpensewas$95.60withastandarddeviationof

$22.50.Determinewhetheranormaldistributionorat–distributionshouldbeusedorwhetherneitherofthese

canbeusedtoconstructaconfidenceinterval.Assumethedistributionofweeklyfoodexpensesisnormally

shaped.

A) Usethet–distribution.

B) Usenormaldistribution.

C) Cannotusenormaldistributionort–distribution.

8) Forasampleof20IQscoresthemeanscoreis105.8.Thestandarddeviation,σ

is15.Determinewhethera

normaldistributionorat–distributionshouldbeusedorwhetherneitherofthesecanbeusedtoconstructa

confidenceinterval.AssumethatIQscoresarenormallydistributed.

A) Usenormaldistribution.

B) Usethet–distribution.

C) Cannotusenormaldistributionort–distribution.

9) Arandomsampleof40collegestudentshasameanearningsof$3120withastandarddeviationof$677over

thesummermonths.Determinewhetheranormaldistributionorat–distributionshouldbeusedorwhether

neitherofthesecanbeusedtoconstructaconfidenceinterval.

A) Usenormaldistribution.

B) Usethet–distribution.

C) Cannotusenormaldistributionort–distribution.

Page139

10) Arandomsampleof15statisticstextbookshasameanpriceof$105withastandarddeviationof$30.25.

Determinewhetheranormaldistributionorat–distributionshouldbeusedorwhetherneitherofthesecanbe

usedtoconstructaconfidenceinterval.Assumethedistributionofstatisticstextbookpricesisnotnormally

distributed.

A) Cannotusenormaldistributionort–distribution.

B) Usenormaldistribution.

C) Usethet–distribution.

11) Constructa95%confidenceintervalforthepopulationmean,μ.Assumethepopulationhasanormal

distribution.Asampleof20collegestudentshadmeanannualearningsof$3120withastandarddeviationof

$677.

A) ($2803,$3437) B) ($1324,$1567) C) ($2135,$2567) D) ($2657,$2891)

12) Constructa90%confidenceintervalforthepopulationmean,μ.Assumethepopulationhasanormal

distribution.Asampleof15randomlyselectedstudentshasagradepointaverageof2.86withastandard

deviationof0.78.

A) (2.51,3.21) B) (2.41,3.42) C) (2.37,3.56) D) (2.28,3.66)

13) Constructa95%confidenceintervalforthepopulationmean,μ.Assumethepopulationhasanormal

distribution.Asampleof25randomlyselectedstudentshasameantestscoreof81.5withastandarddeviation

of10.2.

A) (77.29,85.71) B) (56.12,78.34) C) (66.35,69.89) D) (87.12,98.32)

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

14) Constructa98%confidenceintervalforthepopulationmean,μ.Assumethepopulationhasanormal

distribution.Arandomsampleof20collegestudentshasmeanannualearningsof$3140withastandard

deviationof$686.

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

15) Constructa95%confidenceintervalforthepopulationmean,μ.Assumethepopulationhasanormal

distribution.Arandomsampleof16fluorescentlightbulbshasameanlifeof645hourswithastandard

deviationof31hours.

A) (628.5,661.5) B) (876.2,981.5) C) (531.2,612.9) D) (321.7,365.8)

16) Constructa99%confidenceintervalforthepopulationmean,μ.Assumethepopulationhasanormal

distribution.Agroupof19randomlyselectedstudentshasameanageof22.4yearswithastandarddeviation

of3.8years.

A) (19.9,24.9) B) (16.3,26.9) C) (17.2,23.6) D) (18.7,24.1)

17) Constructa98%confidenceintervalforthepopulationmean,μ.Assumethepopulationhasanormal

distribution.Astudyof14bowlersshowedthattheiraveragescorewas192withastandarddeviationof8.

A) (186.3,197.7) B) (222.3,256.1) C) (328.3,386.9) D) (115.4,158.8)

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

18) Constructa95%confidenceintervalforthepopulationmean,μ.Assumethepopulationhasanormal

distribution.Inarandomsampleof26computers,themeanrepaircostwas$166withastandarddeviationof

$39.

Page140

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

19) Constructa90%confidenceintervalforthepopulationmean,μ.Assumethepopulationhasanormal

distribution.Inarecentstudyof22eighthgraders,themeannumberofhoursperweekthattheywatched

televisionwas19.6withastandarddeviationof5.8hours.

A) (17.47,21.73) B) (18.63,20.89) C) (5.87,7.98) D) (19.62,23.12)

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

20) a)Constructa95%confidenceintervalforthepopulationmean,μ.Assumethepopulationhasanormal

distribution.Inarandomsampleof26computers,themeanrepaircostwas$143withastandarddeviationof

$35.

b)Supposeyoudidsomeresearchonrepaircostsforcomputersandfoundthatthestandarddeviationis

σ=35.Usethenormaldistributiontoconstructa95%confidenceintervalforthepopulationmean,μ.Compare

theresults.

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

21) Arandomsampleof10parkingmetersinabeachcommunityshowedthefollowingincomesforaday.

Assumetheincomesarenormallydistributed.

$3.60$4.50$2.80$6.30$2.60$5.20$6.75$4.25$8.00$3.00

Findthe95%confidenceintervalforthetruemean.

A) ($3.39,$6.01) B) ($2.11,$5.34) C) ($4.81,$6.31) D) ($1.35,$2.85)

22) Thegradepointaveragesfor10randomlyselectedhighschoolstudentsarelistedbelow.Assumethegrade

pointaveragesarenormallydistributed.

2.03.21.82.90.94.03.32.93.60.8

Finda98%confidenceintervalforthetruemean.

A) (1.55,3.53) B) (0.67,1.81) C) (2.12,3.14) D) (3.11,4.35)

23) Alocalbankneedsinformationconcerningthecheckingaccountbalancesofitscustomers.Arandomsample

of15accountswaschecked.Themeanbalancewas$686.75withastandarddeviationof$256.20.Finda98%

confidenceintervalforthetruemean.Assumethattheaccountbalancesarenormallydistributed.

A) ($513.17,$860.33) B) ($238.23,$326.41) C) ($326.21,$437.90) D) ($487.31,$563.80)

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

24) Amanufacturerreceivesanorderforfluorescentlightbulbs.Theorderrequiresthatthebulbshaveameanlife

spanof850hours.Themanufacturerselectsarandomsampleof25fluorescentlightbulbsandfindsthatthey

haveameanlifespanof845hourswithastandarddeviationof15hours.Testtoseeifthemanufactureris

makingacceptablelightbulbs.Usea95%confidencelevel.Assumethedataarenormallydistributed.

25) Acoffeemachineissupposedtodispense12ouncesofcoffeeineachcup.Aninspectorselectsarandom

sampleof40cupsofcoffeeandfindstheyhaveanaverageamountof12.2ounceswithastandarddeviationof

0.3ounce.Usea99%confidenceintervaltotestwhetherthemachineisdispensingacceptableamountsof

coffee.

Page141

26) Thenumbersofadvertisementsseenorheardinoneweekfor30randomlyselectedpeopleintheUnitedStates

arelistedbelow.Constructa95%confidenceintervalforthetruemeannumberofadvertisements.

598 494 441 595 728 690 684 486 735 808

481 298 135 846 764 317 649 732 582 677

734 588 590 540 673 727 545 486 702 703

27) Thenumberofwinsinaseasonfor32randomlyselectedprofessionalfootballteamsarelistedbelow.

Constructa90%confidenceintervalforthetruemeannumberofwinsinaseason.

99981

972

11106411988

12107512643

1299710775

6.3 ConfidenceIntervalsforPopulationProportions

1 FindaPointEstimate

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

Provideanappropriateresponse.

1) When410collegestudentsweresurveyed,150 saidtheyowntheircar.Findapointestimateforp,the

populationproportionofstudentswhoowntheircars.

A) 0.366 B) 0.634 C) 0.577 D) 0.268

2) Asurveyof100fatalaccidentsshowedthat16 werealcoholrelated.Findapointestimateforp,thepopulation

proportionofaccidentsthatwerealcoholrelated.

A) 0.16 B) 0.84 C) 0.190 D) 0.138

3) Asurveyof700non–fatalaccidentsshowedthat231 involvedtheuseofacellphone.Findapointestimatefor

p,thepopulationproportionofnon–fatalaccidentsthatinvolvedtheuseofacellphone.

A) 0.330 B) 0.670 C) 0.493 D) 0.248

4) Asurveyof250homelesspersonsshowedthat11 wereveterans.Findapointestimatep,forthepopulation

proportionofhomelesspersonswhoareveterans.

A) 0.044 B) 0.956 C) 0.046 D) 0.042

5) Asurveyof2210golfersshowedthat389 ofthemareleft–handed.Findapointestimateforp,thepopulation

proportionofgolfersthatareleft–handed.

A) 0.176 B) 0.824 C) 0.214 D) 0.150

Page142

2 ConstructandInterpretConfidenceIntervalsforthePopulationProportion

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

Provideanappropriateresponse.

1) Inasurveyof2480golfers,15%saidtheywereleft–handed.Thesurveyʹsmarginoferrorwas3%.Constructa

confidenceintervalfortheproportionofleft–handedgolfers.

A) (0.12,0.18) B) (0.18,0.21) C) (0.12,0.15) D) (0.11,0.19)

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

2) TheFederalBureauofLaborStatisticssurveyed50,000peopleandfoundtheunemploymentratetobe5.8%.

Themarginoferrorwas0.2%.Constructaconfidenceintervalfortheunemploymentrate.

3) When490collegestudentsweresurveyed,130 saidtheyowntheircar.Constructa95%confidenceintervalfor

theproportionofcollegestudentswhosaytheyowntheircars.

4) Asurveyof300fatalaccidentsshowedthat123werealcoholrelated.Constructa98%confidenceintervalfor

theproportionoffatalaccidentsthatwerealcoholrelated.

5) Asurveyof400non–fatalaccidentsshowedthat181 involvedtheuseofacellphone.Constructa99%

confidenceintervalfortheproportionoffatalaccidentsthatinvolvedtheuseofacellphone.

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

6) Asurveyof280homelesspersonsshowedthat63wereveterans.Constructa90%confidenceintervalforthe

proportionofhomelesspersonswhoareveterans.

A) (0.184,0.266) B) (0.176,0.274) C) (0.167,0.283) D) (0.161,0.289)

7) Asurveyof2450golfersshowedthat281ofthemareleft–handed.Constructa98%confidenceintervalforthe

proportionofgolfersthatareleft–handed.

A) (0.100,0.130) B) (0.203,0.293) C) (0.369,0.451) D) (0.683,0.712)

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

8) Inasurveyof10golfers,2werefoundtobeleft–handed.Isitpracticaltoconstructthe90%confidenceinterval

forthepopulationproportion,p?Explain.

9) TheUSATodayclaimsthat44%ofadultswhoaccesstheInternetreadtheinternationalnewsonline.You

wanttochecktheaccuracyoftheirclaimbysurveyingarandomsampleof120adultswhoaccesstheInternet

andaskingthemiftheyreadtheinternationalnewsonline.Fifty–twoadultsrespondedʺyes.ʺUsea95%

confidenceintervaltotestthenewspaperʹsclaim.

3DeterminetheMinimumSampleSize

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

Provideanappropriateresponse.

1) AresearcheratamajorhospitalwishestoestimatetheproportionoftheadultpopulationoftheUnitedStates

thathashighbloodpressure.Howlargeasampleisneededinordertobe90%confidentthatthesample

proportionwillnotdifferfromthetrueproportionbymorethan5%?

A) 271 B) 9 C) 542 D) 164

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2) ApollsterwishestoestimatetheproportionofUnitedStatesvoterswhofavorcapitalpunishment.Howlargea

sampleisneededinordertobe98%confidentthatthesampleproportionwillnotdifferfromthetrue

proportionbymorethan3%?

A) 1509 B) 1068 C) 20 D) 3017

3) Aprivateopinionpollisconductedforapoliticiantodeterminewhatproportionofthepopulationfavors

decriminalizingmarijuanapossession.Howlargeasampleisneededinordertobe95%confidentthatthe

sampleproportionwillnotdifferfromthetrueproportionbymorethan4%?

A) 601 B) 423 C) 1201 D) 13

4) Amanufacturerofgolfequipmentwishestoestimatethenumberofleft–handedgolfers.Howlargeasampleis

neededinordertobe95%confidentthatthesampleproportionwillnotdifferfromthetrueproportionby

morethan4%?Apreviousstudyindicatesthattheproportionofleft–handedgolfersis9%.

A) 197 B) 139 C) 217 D) 19

5) Aresearcherwishestoestimatethenumberofhouseholdswithtwocars.Howlargeasampleisneededin

ordertobe95%confidentthatthesampleproportionwillnotdifferfromthetrueproportionbymorethan5%?

Apreviousstudyindicatesthattheproportionofhouseholdswithtwocarsis22%.

A) 264 B) 186 C) 339 D) 4

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

6) Astatehighwaypatrolofficialwishestoestimatethenumberofdriversthatexceedthespeedlimittravelinga

certainroad.

a)Howlargeasampleisneededinordertobe90%confidentthatthesampleproportionwillnotdifferfrom

thetrueproportionbymorethan3%?

b)Repeatpart(a)assumingpreviousstudiesfoundthat80%ofdriversonthisroadexceededthespeedlimit.

6.4 ConfidenceIntervalsforVarianceandStandardDeviatio

1 FindCriticalValues

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

Provideanappropriateresponse.

1) Findthecriticalvalues,X2

Rand X 2

L,forc=0.95andn=12.

A) 3.816and21.920 B) 3.053and24.725 C) 4.575and26.757 D) 2.603and19.675

2) Findthecriticalvalues,X2

Rand X2

L,forc=0.90andn=15.

A) 6.571and23.685 B) 4.075and31.319 C) 4.660and29.131 D) 5.629and26.119

3) Findthecriticalvalues,X2

Rand X 2

L,forc=0.98andn=20.

A) 7.633and36.191 B) 6.844and27.204 C) 8.907and38.582 D) 10.117and32.852

4) Findthecriticalvalues,X2

Rand X 2

L,forc=0.99andn=10.

A) 1.735and23.587 B) 2.156and25.188 C) 2.088and21.666 D) 2.558and23.209

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2 ConstructConfidenceIntervalsforthePopulationVarianceandPopulationStandardDeviation

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

Assumethesampleistakenfromanormallydistributedpopulationandconstructtheindicatedconfidenceinterval.

1) Constructtheindicatedconfidenceintervalsforthepopulationvarianceσ2andthepopulationstandard

deviationσ.Assumethesampleisfromanormallydistributedpopulation

c=0.95,s2=15.3,n=25

A) (9.33

29.61);(3.05

5.44) B) (13.9

26.52);(3.73

5.15)

C) (9.03

27.99);(3

5.29) D) (9.75

25.13);(3.12

5.01)

2) Constructtheindicatedconfidenceintervalsforthepopulationvarianceσ2andthepopulationstandard

deviationσ.Assumethesampleisfromanormallydistributedpopulation

c=0.99,s=225.8,n=61

A) (33,268.86

86,090.46);(182.4

293.41) B) (36,725.23

75,567.87);(191.64

274.9)

C) (38,683.12

70,833.06);(196.68

266.14) D) (41,119.11

65,845.98);(202.78

256.6)

3) Constructa95%confidenceintervalforthepopulationstandarddeviationσ ofarandomsampleof15men

whohaveameanweightof165.2poundswithastandarddeviationof10.5pounds.Assumethepopulationis

normallydistributed.

A) (7.7

16.6) B) (59.1

274.2) C) (2.4

5.1) D) (8.1

15.3)

4) Arandomsampleof16menhaveameanheightof67.5inchesandastandarddeviationof2.4inches.

Constructa99%confidenceintervalforthepopulationstandarddeviation,σ.

A) (1.6

4.3) B) (1.7

4.5) C) (1.0

2.8) D) (1.7

4.1)

5) Arandomsampleof20womenhaveameanheightof62.5inchesandastandarddeviationof1.8 inches.

Constructa98%confidenceintervalforthepopulationvariance,σ2.

A) (1.7

8.1) B) (1.3

2.8) C) (0.9

4.5) D) (1.8

8.5)

6) Theheights(ininches)of20randomlyselectedadultmalesarelistedbelow.Constructa99%confidence

intervalforthevariance,σ2.

70727170697369687071

67717074696871717172

A) (1.47,8.27) B) (21.61,69.06) C) (1.35,8.43) D) (2.16,71.06)

7) Thegradepointaveragesfor10randomlyselectedstudentsarelistedbelow.Constructa90%confidence

intervalforthepopulationstandarddeviation,σ.

2.03.21.82.90.94.03.32.93.60.8

A) (0.81,1.83) B) (0.32,0.85) C) (0.53,1.01) D) (1.10,2.01)

8) Themeanreplacementtimeforarandomsampleof12microwaveovensis8.6yearswithastandarddeviation

of5.3years.Constructthe98%confidenceintervalforthepopulationvariance,σ

A) (12.5

101.2) B) (3.5

10.1) C) (2.4

19.1) D) (11.8

86.5)

9) Astudentrandomlyselects10CDsatastore.Themeanis$13.75withastandarddeviationof$1.50.Construct

a95%confidenceintervalforthepopulationstandarddeviation,σ.

A) ($1.03,$2.74) B) ($0.99,$2.50) C) ($1.06,$7.51) D) ($0.84,$2.24)

Page145

10) Thestem–and–leafplotshowsthetestscoresof16randomlyselectedstudents.Constructa99%confidence

intervalforthepopulationstandarddeviation.



9

9

583

74429

5835

317

A) (7.61,20.33) B) (57.97,413.27) C) (62.18,363.63) D) (7.89,19.07)

11) Thedotplotshowstheweights(inpounds)of15dogsselectedrandomlyfromthoseadoptedoutbyananimal

shelterlastweek.Constructa98%confidenceintervalforthepopulationvariance.

Weights

15 20 25

Pounds

A) (3.03,18.97) B) (1.74,4.36) C) (3.38,15.70) D) (2.89,16.91)

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

12) Acontainerofcaroilissupposedtocontain1000millilitersofoil.Aqualitycontrolmanagerwantstobesure

thatthestandarddeviationoftheoilcontainersislessthan20milliliters.Herandomlyselects10cansofoil

withameanof997millilitersandastandarddeviationof32milliliters.Usethesesampleresultstoconstructa

95%confidenceintervalforthetruevalueofσ.Doesthisconfidenceintervalsuggestthatthevariationintheoil

containersisatanacceptablelevel?

Page146

Ch.6 ConfidenceIntervals

AnswerKey

6.1 ConfidenceIntervalsfortheMean(σKnown)

1 FindaCriticalValue

2 FindtheMarginofError

3 ConstructandInterpretConfidenceIntervalsforthePopulationMean

4 DeterminetheMinimumSampleSize

5DeterminetheFinitePopulationCorrectionFactor

6 Concepts

6.2 ConfidenceIntervalsfortheMean(σUnknown)

1 FindaCriticalValue

Page147

2 ConstructandInterpretConfidenceIntervalsforthePopulationMean

6.3 ConfidenceIntervalsforPopulationProportions

1 FindaPointEstimate

2 ConstructandInterpretConfidenceIntervalsforthePopulationProportion

3 DeterminetheMinimumSampleSize

Page148

6.4 ConfidenceIntervalsforVarianceandStandardDeviatio

1 FindCriticalValues

2 ConstructConfidenceIntervalsforthePopulationVarianceandPopulationStandardDeviation

Page149