Economics Chapter 09 Construct 99 Confidence Interval For The Proportion

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Ch.9 EstimatingtheValueofaParameter

9.1 EstimatingaPopulationProportio

1 Obtainapointestimateforthepopulationproportion.

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

Provideanappropriateresponse.

1) When385juniorcollegestudentsweresurveyed,170 saidthattheyhavepreviouslyownedamotorcycle.Find

apointestimateforp,thepopulationproportionofstudentswhohavepreviouslyownedamotorcycle.

A) 0.442 B) 0.558 C) 0.791 D) 0.306

2) Asurveyof100fatalaccidentsshowedthatin43 casesthedriveratfaultwasinadequatelyinsured.Finda

pointestimateforp,thepopulationproportionofaccidentswherethedriveratfaultwasinadequatelyinsured

A) 0.43 B) 0.57 C) 0.754 D) 0.301

3) Asurveyof700non–fatalaccidentsshowedthat143 involvedfaultyequipment.Findapointestimateforp,

thepopulationproportionofaccidentsthatinvolvedfaultyequipment.

A) 0.204 B) 0.796 C) 0.257 D) 0.170

4) Asurveyof250householdsshowed91 ownedatleastonesnowblower.Findapointestimateforp,the

populationproportionofhouseholdsthatownatleastonesnowblower.

A) 0.364 B) 0.636 C) 0.572 D) 0.267

5) Asurveyof2690musiciansshowedthat368 ofthemareleft–handed.Findapointestimateforp,the

populationproportionofmusiciansthatareleft–handed.

A) 0.137 B) 0.863 C) 0.158 D) 0.120

6) Amarketingresearchcompanyneedstoestimatewhichoftwomedicalplansitsemployeesprefer.Arandom

sampleofnemployeesproducedthefollowing98%confidenceintervalfortheproportionofemployeeswho

preferplanA:(0.241,0.561).Identifythepointestimateforestimatingthetrueproportionofemployeeswho

preferthatplan.

A) 0.401 B) 0.16 C) 0.241 D) 0.561

7) Manypeoplethinkthatanationallobbyʹssuccessfulfightagainstguncontrollegislationisreflectingthewillof

aminorityofAmericans.Arandomsampleof4000citizensyielded2130whoareinfavorofguncontrol

legislation.FindthepointestimateforestimatingtheproportionofallAmericanswhoareinfavorofgun

controllegislation.

A) 0.5325 B) 2130 C) 4000 D) 0.4675

2 Constructandinterpretaconfidenceintervalforthepopulationproportion.

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

Determinethecriticalvaluezα/2thatcorrespondstothegivenlevelofconfidence.

1) 88%

A) 1.555 B) 1.175 C) 0.81 D) 1.21

2) 95%

A) 1.96 B) 1.645 C) 0.83 D) 1.68

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

Provideanappropriateresponse.

3) When440juniorcollegestudentsweresurveyed,200 saidtheyhaveapassport.Constructa95%confidence

intervalfortheproportionofjuniorcollegestudentsthathaveapassport.Roundtothenearestthousandth.

4) Asurveyof700non–fatalaccidentsshowedthat163 involveduninsureddrivers.Constructa99%confidence

intervalfortheproportionoffatalaccidentsthatinvolveduninsureddrivers.Roundtothenearestthousandth.

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

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

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

6) AnarticleaFloridanewspaperreportedonthetopicsthatteenagersmostwanttodiscusswiththeirparents.

Thefindings,theresultsofapoll,showedthat46%wouldlikemorediscussionaboutthefamilyʹsfinancial

situation,37%wouldliketotalkaboutschool,and30%wouldliketotalkaboutreligion.Theseandother

percentageswerebasedonanationalsamplingof546teenagers.Estimatetheproportionofallteenagerswho

wantmorefamilydiscussionsaboutschool.Usea95%confidencelevel.Expresstheanswerintheformp

^±E

androundtothenearestthousandth.

A) 0.37±0.040 B) 0.37±0.002 C) 0.63±0.040 D) 0.63±0.002

7) Manypeoplethinkthatanationallobbyʹssuccessfulfightagainstguncontrollegislationisreflectingthewillof

aminorityofAmericans.Arandomsampleof4000citizensyielded2250whoareinfavorofguncontrol

legislation.EstimatethetrueproportionofallAmericanswhoareinfavorofguncontrollegislationusinga

99%confidenceinterval.Expresstheanswerintheformp

^±Eandroundtothenearestten–thousandth.

A) 0.5625±0.0202 B) 0.5625±0.6337 C) 0.4375±0.0202 D) 0.4375±0.6337

8) Auniversitydeanisinterestedindeterminingtheproportionofstudentswhoreceivesomesortoffinancial

aid.Ratherthanexaminetherecordsforallstudents,thedeanrandomlyselects200studentsandfindsthat118

ofthemarereceivingfinancialaid.Usea95%confidenceintervaltoestimatethetrueproportionofstudentson

financialaid.Expresstheanswerintheformp

^±Eandroundtothenearestthousandth.

A) 0.59±0.068 B) 0.59±0.002 C) 0.59±0.474 D) 0.59±0.005

9) TrueorFalse:Thegeneralformofalarge–sample(1–α)100%confidenceintervalforapopulationproportion

pisp

^±zα/2 p

^(1–p

n,wherep

^=x

nisthesampleproportionofobservationswiththecharacteristicof

interest.

A) True B) False

10) Whatisthebestpointestimateforpinordertoconstructaconfidenceintervalforp?

A) p

B) μpC) p D) p

11) Aconfidenceintervalforpcanbeconstructedusing

A) p

^±zα/2 p

^(1–p

nB) p±zσ

nC) p

^±zσ

nD) p±zα/2 p(1–p)

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3 Findthesamplesizeneededforestimatingapopulationproportionwithinagivenmarginoferror.

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clinicwishestoestimatetheproportionoftheadultpopulationoftheUnitedStatesthat

hassleepdeprivation.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) 13 C) 1201 D) 423

2) AsenatorwishestoestimatetheproportionofUnitedStatesvoterswhofavorabolishingtheElectoralCollege.

Howlargeasampleisneededinordertobe99%confidentthatthesampleproportionwillnotdifferfromthe

trueproportionbymorethan3%?

A) 1842 B) 1509 C) 22 D) 3684

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

addingmorenationalparks.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) 164 C) 542 D) 9

4) Apollsterwishestoestimatethenumberofleft–handedscientists.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6%?Aprevious

studyindicatesthattheproportionofleft–handedscientistsis9%.

A) 88 B) 62 C) 97 D) 27

5) Aresearcherwishestoestimatethenumberofhouseholdswithtwocomputers.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

2%?Apreviousstudyindicatesthattheproportionofhouseholdswithtwocomputersis20%.

A) 2172 B) 1537 C) 2715 D) 3

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legallyintoxicateddriversonacertainroad.

a)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6%?

b)Repeatpart(a)assumingpreviousstudiesfoundthat85%ofthedriversonthisroadarelegallyintoxicated.

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

7) Alocaloutdoorequipmentstoreisbeingsold.Thebuyersaretryingtoestimatethepercentageofitemsthat

areoutdated.Theywillrandomlysampleamongits100,000itemsinordertodeterminetheproportionof

merchandisethatisoutdated.Thecurrentownershaveneverdeterminedtheiroutdatedpercentageandcan

nothelpthebuyers.Approximatelyhowlargeasampledothebuyersneedinordertoinsurethattheyare95%

confidentthatthemarginoferroriswithin3%?

A) 1068 B) 2135 C) 4269 D) 545

8) Aconfidenceintervalwasusedtoestimatetheproportionofmathmajorsthatarefemale.Arandomsampleof

72mathmajorsgeneratedthefollowingconfidenceinterval:(0.438,0.642).Usingtheinformationabove,what

sizesamplewouldbenecessaryifwewantedtoestimatethetrueproportiontowithin3%using98%

reliability?

A) 1499 B) 1509 C) 1569 D) 1448

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9) Manypeoplethinkthatanationallobbyʹssuccessfulfightagainstguncontrollegislationisreflectingthewillof

aminorityofAmericans.Apreviousrandomsampleof4000citizensyielded2250whoareinfavorofgun

controllegislation.Howmanycitizenswouldneedtobesampledifa99%confidenceintervalwasdesiredto

estimatethetrueproportiontowithin1%?

A) 16,318 B) 17,240 C) 16,577 D) 15,251

10) Auniversitydeanisinterestedindeterminingtheproportionofstudentswhoreceivesomesortoffinancial

aid.Ratherthanexaminetherecordsforallstudents,thedeanrandomlyselects200studentsandfindsthat118

ofthemarereceivingfinancialaid.Ifthedeanwantedtoestimatetheproportionofallstudentsreceiving

financialaidtowithin1%with90%reliability,howmanystudentswouldneedtobesampled?

A) 6546 B) 1584 C) 3980 D) 66

11) Inacollegestudentpoll,itisofinteresttoestimatetheproportionp ofstudentsinfavorofchangingfroma

quarter–systemtoasemester–system.Howmanystudentsshouldbepolledsothatwecanestimatepto

within0.09usinga99%confidenceinterval?

A) 205 B) 182 C) 261 D) 114

12) TrueorFalse?WhenchoosingthesamplesizeforestimatingapopulationproportionptowithinEunitswith

confidence(1–α)100%,ifyoutakep≈0.5astheapproximationtop,youwillalwaysobtainasamplesizethat

isatleastaslargeasrequired.

A) True B) False

13) TrueorFalse?Ifnoestimateofpexistswhendeterminingthesamplesize,wecanuse0.5intheformulatoget

avalueforn.

A) True B) False

9.2 EstimatingaPopulationMea

1 Determinet–values.

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

Findthet–value.

1) Lett0beaspecificvalueoft.Findt0suchthatthestatementistrue:P(t≥t0)=0.005wheredf=20.

A) 2.845 B) –2.845 C) 2.861 D) –2.861

2) Findthet–valuesuchthattheareaintherighttailis0.2 with5 degreesoffreedom.

A) 0.920 B) 0.200 C) –0.92 D) 0.941

3) Findthet–valuesuchthatthearealeftofthet–valueis0.025 with17 degreesoffreedom.

A) –2.11 B) –3.222 C) 2.110 D) 2.120

4) Findthecriticalt–valuethatcorrespondsto99%confidenceandn=10.

A) 2.821 B) 3.250 C) 2.262 D) 1.833

5) Findthecriticalt–valuethatcorrespondsto95%confidenceandn=16.

A) 1.753 B) 2.131 C) 2.602 D) 2.947

6) Findthecriticalt–valuethatcorrespondsto90%confidenceandn=15.

A) 1.345 B) 1.761 C) 2.145 D) 2.624

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2 StatepropertiesofStudentʹst–distribution.

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

Provideanappropriateresponse.

1) WhichofthefollowingisnotacharacteristicofStudentsʹt–distribution?

A) meanof1

B) symmetricdistribution

C) dependsondegreesoffreedom.

D) Forlargesamples,thetandzdistributionsarenearlyequivalent.

2) Supposea90%confidenceintervalforμ turnsouttobe(1000,2100). Ifthisintervalwasbasedonasampleof

sizen=25,explainwhatassumptionsarenecessaryforthisintervaltobevalid.

A) Thepopulationmusthaveanapproximatelynormaldistribution.

B) Thesamplingdistributionofthesamplemeanmusthaveanormaldistribution.

C) Thepopulationofsalariesmusthaveanapproximatetdistribution.

D) Thesamplingdistributionmustbebiasedwith24 degreesoffreedom.

3) Acomputerpackagewasusedtogeneratethefollowingprintoutforestimatingthesalepriceofcondominiums

inaparticularneighborhood.

X=sale_price

SAMPLEMEANOFX=46,600

SAMPLESTANDARDDEV=13,747

SAMPLESIZEOFX=15

CONFIDENCE=99

UPPERLIMIT=57,166.70

SAMPLEMEANOFX=46,600

LOWERLIMIT=36,033.30

Whatassumptionsarenecessaryforanyinferencesderivedfromthisprintouttobevalid?

A) Thesamplewasrandomlyselectedfromanapproximatelynormalpopulation.

B) Thesamplevarianceequalsthepopulationvariance.

C) Thepopulationmeanhasanapproximatenormaldistribution.

D) Allofthesearenecessary.

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4) Acomputerpackagewasusedtogeneratethefollowingprintoutforestimatingthesalepriceofcondominiums

inaparticularneighborhood.

X=sale_price

SAMPLEMEANOFX=46,600

SAMPLESTANDARDDEV=13,747

SAMPLESIZEOFX=15

CONFIDENCE=90

UPPERLIMIT=52,850.60

SAMPLEMEANOFX=46,600

LOWERLIMIT=40,349.40

Afriendsuggeststhatthemeansalepriceofhomesinthisneighborhoodis$43,000.Commentonyourfriendʹs

suggestion.

A) Basedonthisprintout,allyoucansayisthatthemeansalepricemightbe$43

000.

B) Yourfriendiswrong,andyouare90%certain.

C) Yourfriendiscorrect,andyouare90%certain.

D) Yourfriendiscorrect,andyouare100%certain.

5) ToselectthecorrectStudentʹst–distributionrequiresknowingthedegreesoffreedom.Howmanydegreesof

freedomarethereforasampleofsizen?

A) n–1 B) n C) n+1D)

x–μ

s/ n

6) TrueorFalse:EveryStudentʹst–distributionwithn<N,nthenumberinthesampleandNthenumberinthe

population,willbelesspeakedandhavethinnertails.

A) False B) True

7) TheareaunderthegraphofeveryStudentʹst–distributionis

A) 1 B) Lessthanthestandardnormaldistribution

C) Greaterthanthestandardnormaldistribution D) Areaofthestandardnormaldistribution

s/ n

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3 Constructandinterpretaconfidenceintervalforapopulationmean.

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

Asimplerandomsampleofsizen<30hasbeenobtained.Fromthenormalprobabilityplot,judgewhetherat–interval

shouldbeconstructed.

A) Yes;thedataareroughlynormallydistributedandtherearenooutliers

B) No;thedataareroughlynormallydistributedbutthereareoutliers

C) No;thedataarenotnormallydistributedandthereareoutliers

D) No;therearenooutliers,butthedataarenotnormallydistributed

A) No;therearenooutliersbutthedataarenotnormallydistributed

B) Yes;thedataareroughlynormallydistributedandtherearenooutliers

C) No;thedataareroughlynormallydistributed,butthereareoutliers

D) No;thedataarenotnormallydistributedandthereareoutliers

Asimplerandomsampleofsizen<30hasbeenobtained.Fromtheboxplot,judgewhetherat–intervalshouldbe

constructed.

A) No,thoughtherearenooutliers,thedataarenotnormallydistributedbutrightskewed

B) No,thereareoutliersandthedataarenotnormallydistributedbutrightskewed

C) Yes;thedataarenormallydistributedandtherearenooutliers

D) No;thedataarenormallydistributed,butthereareoutliers

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A) No;thedataappearroughlynormallydistributedbutthereareoutliers

B) Yes;thedataappearnormallydistributedandtherearenooutliers

C) No,thereareoutliersandthedataarenotnormallydistributed

D) No,therearenooutliersbutthedataarenotnormallydistributed

A) Yes;thedataappearroughlynormallydistributedandtherearenooutliers

B) No;thedataappearroughlynormallydistributed,butthereareoutliers

C) No;thedataarenotnormallydistributedandthereareoutliers

D) No;therearenooutliers,butthedataarenotnormallydistributed

Provideanappropriateresponse.

6) Determinethepointestimateofthepopulationmeanandmarginoferrorfortheconfidenceintervalwith

lowerbound7andupperbound:25.

A) x=16,E=9B)x=16,E=18 C) x=7,E=18 D) x=25,E=9

7) Howmuchmoneydoestheaverageprofessionalhockeyfanspendonfoodatasinglehockeygame?That

questionwasposedto10randomlyselectedhockeyfans.Thesampledresultsshowthatsamplemeanand

standarddeviationwere$19.00and$2.6,respectively.Usethisinformationtocreatea95%confidenceinterval

forthemean.Expresstheanswerintheformx±tα/2(s/ n).

A) 19±2.262(2.6/ 10)B)19±2.228(2.6/ 10)C)19±2.201(2.6/ 10)D)19±1.833(2.6/ 10)

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

distribution.Asampleof20part–timeworkershadmeanannualearningsof$3120withastandarddeviation

of$677.Roundtothenearestdollar.

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

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

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

deviationof0.78.Roundtothenearesthundredth.

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

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

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

10.2.Roundtothenearesthundredth.

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

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

distribution.Arandomsampleof16lithiumbatterieshasameanlifeof645hourswithastandarddeviationof

31hours.Roundtothenearesttenth.

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

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

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

deviationof3.8years.Roundtothenearesttenth.

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

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

distribution.Astudyof14carownersshowedthattheiraveragerepairbillwas$192withastandarddeviation

of$8.Roundtothenearestcent.

A) ($186.33,$197.67) B) ($222.33,$256.10) C) ($328.33,$386.99) D) ($115.40,$158.80)

14) 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playedvideo

gameswas19.6withastandarddeviationof5.8hours.Roundtothenearesthundredth.

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

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

Assumetheincomesarenormallydistributed.Findthe95%confidenceintervalforthetruemean.Roundto

thenearestcent.

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

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

16) Thegradepointaveragesfor10randomlyselectedjuniorcollegestudentsarelistedbelow.Assumethegrade

pointaveragesarenormallydistributed.Finda98%confidenceintervalforthetruemean.Roundtothenearest

hundredth.

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

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

17) Alocalbankneedsinformationconcerningthesavings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.Roundtothe

nearestcent.

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

18) Tohelpconsumersassesstheriskstheyaretaking,theFoodandDrugAdministration(FDA)publishesthe

amountofnicotinefoundinallcommercialbrandsofcigarettes.Anewcigarettehasrecentlybeenmarketed.

TheFDAtestsonthiscigarettegaveameannicotinecontentof28.4milligramsandstandarddeviationof2.6

milligramsforasampleofn=9cigarettes.TheFDAclaimsthatthemeannicotinecontentexceeds31.0

milligramsforthisbrandofcigarette,andtheirstatedreliabilityis90%.Doyouagree?

A) No,sincethevalue31.0doesnotfallinthe90%confidenceinterval.

B) Yes,sincethevalue31.0doesfallinthe90%confidenceinterval.

C) Yes,sincethevalue31.0doesnotfallinthe90%confidenceinterval.

D) No,sincethevalue31.0doesfallinthe90%confidenceinterval.

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

19) Inarandomsampleof26laptopcomputers,themeanrepaircostwas$170 withastandarddeviationof$34.

Assumethepopulationhasanormaldistribution.Constructa95%confidenceintervalforthepopulation

mean,μ.Supposeyoudidsomeresearchonrepaircostsforlaptopcomputersandfoundthatthestandard

deviationisσ=34.Usethenormaldistributiontoconstructa95%confidenceintervalforthepopulationmean,

μ.Comparetheresults.Roundtothenearestcent.

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

20) Whateffectwillanoutlierhaveonaconfidenceintervalthatisbasedonasmallsamplesize?

A) Theconfidenceintervalwillbewiderthananintervalwithouttheoutlier.

B) Theintervalwillbesmallerthananintervalwithouttheoutlier.

C) Theintervalwillbethesamewithorwithouttheoutlier.

D) Theintervalwillrevealexclusionarydata.

4 Findthesamplesizeneededtoestimatethepopulationmeanwithinagiven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samplesizerequiredtoestimatethemeanscoreonastandardizedtestwithin4pointsofthe

truemeanwith98%confidence.Assumethats=14basedonearlierstudies.

A) 67 B) 9 C) 172 D) 1

2) Adoctor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90%confidentthatherestimateiswithin2ouncesofthetruemean?Assumethats=

5ouncesbasedonearlierstudies.

A) 17 B) 16 C) 5 D) 4

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

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

atalocalbanktakeperyear.Basedonearlierstudies,theywillassumedthats=2.3days. a)Howlargea

samplemustbeselectedifthecompanywantstobe98%confidentthattheirestimateiswithin1dayofthe

truemean?b)Repeatpart(a)usinga99%confidenceinterval.Whichlevelofconfidencerequiresalarger

samplesize?Explain.

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

4) Thegradepointaveragesfor10randomlyselectedstudentsinanalgebraclasswith125studentsarelisted

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

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

A) Thewidthdecreases.

B) Thewidthincreases.

C) Thewidthremainsthesame.

D) Itisimpossibletotellwithoutmoreinformation.

5) TheprincipalatRiversideHighSchoolwouldliketoestimatethemeanlengthoftimeeachdaythatittakesall

thebusestoarriveandunloadthestudents.Howlargeasampleisneedediftheprincipalwouldliketoassert

with90%confidencethatthesamplemeanisoffby,atmost,7minutes.Assumethats=14minutesbasedon

previousstudies.

A) 11 B) 10 C) 12 D) 13

6) TrueorFalse:Asthelevelofconfidenceincreasesthenumberofitemtobeincludedinasamplewilldecrease

whentheerrorandthestandarddeviationareheldconstant.

A) False B) True

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9.3 EstimatingaPopulationStandardDeviatio

1 Findcriticalvaluesforthechi–squaredistribution.

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

Provideanappropriateresponse.

1) Findthecriticalvalues,χ 2

1–α/2 andχ 2

α/2 ,for95%confidence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,χ 2

1–α/2 andχ 2

α/2 ,for90%confidence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,χ 2

1–α/2 andχ 2

α/2 ,for98%confidence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,χ 2

1–α/2 andχ 2

α/2 ,for99%confidence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

5) TrueorFalse:Thechi–squaredistributionisasymmetricdistributionforalldegreesoffreedom.

A) False B) True

6) TrueorFalse:Thechi–squaredistributionisasymmetricdistributionisnegativewhenthedegreesoffreedom

becomelarge.

A) False B) True

7) TrueorFalse:Asthenumberofdegreesoffreedomincreases,thechi–squaredistributionbecomesmore

nearlysymmetric.

A) True B) False

2 Constructandinterpretconfidenceintervalsforthepopulationvarianceandstandarddeviation.

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

Provideanappropriateresponse.

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

whichhaveameanweightof165.2poundsandastandarddeviationof14.5pounds.Assumethepopulationis

normallydistributed.

A) (10.6

22.9) B) (112.7

522.9) C) (2.8

6.0) D) (11.1

21.2)

2) Assumethattheheightsofbookcasesarenormallydistributed.Arandomsampleof16bookcasesinone

companyhaveameanheightof67.5inchesandastandarddeviationof2.1inches.Constructa99%confidence

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

A) (1.4

3.8) B) (1.5

3.9) C) (1.0

2.6) D) (1.5

3.6)

3) Assumethattheheightsoffemaleexecutivesarenormallydistributed.Arandomsampleof20female

executives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)

Page257

4) Themeanreplacementtimeforarandomsampleof12cdplayersis8.6yearswithastandarddeviationof3.5

years.Constructthe98%confidenceintervalforthepopulationvariance,σ2.Assumethedataarenormally

distributed

A) (5.4

44.1) B) (2.3

6.6) C) (1.6

12.6) D) (5.1

37.7)

5) Astudentrandomlyselects10paperbacksatastore.Themeanpriceis$8.75withastandarddeviationof$1.50.

Constructa95%confidenceintervalforthepopulationstandarddeviation,σ.Assumethedataarenormally

distributed.

A) ($1.03,$2.74) B) ($0.43,$1.32) C) ($1.43,$2.70) D) ($1.76,$3.10)

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

6) TheJulyutilitybills(indollars)of20randomlyselectedhomeownersinonecityarelistedbelow.Constructa

99%confidenceintervalforthevariance,σ2.Assumethepopulationisnormallydistributed.

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

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

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

7) TheJuneprecipitation(ininches)for10randomlyselectedcitiesarelistedbelow.Constructa90%confidence

intervalforthepopulationstandarddeviation,σ.Assumethedataarenormallydistributed.

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)

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

8) Acontainerofsodaissupposedtocontain1000millilitersofsoda.Aqualitycontrolmanagerwantstobesure

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

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

sodacontainersisatanacceptablelevel?

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

9) TrueorFalse?Whenconstructinga(1–α)100%confidenceintervalforapopulationvarianceσ2,the

populationfromwhichtherandomsampleisselectedcanhaveanydistribution.

A) False B) True

10) Thebestpointestimateforthestandarddeviationofapopulationis

A) Thestandarddeviationofthesample. B) Thevarianceofthepopulation.

C) Thevarianceofthesample. D) (n–1)s2

σ2.

Page258

9.4 PuttingItTogether:WhichProcedureDoIUse?

1 Determinetheappropriateconfidenceintervaltoconstruct.

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

Provideanappropriateresponse.

1) Inarandomsampleof60dogownersenrolledinobediencetraining,itwasdeterminedthatthemeanamount

ofmoneyspentperownerwas$109.33perclass.Assumingthepopulationstandarddeviationoftheamount

spentperowneris$12,constructandinterpreta95%confidenceintervalforthemeanamountspentperowner

foranobedienceclass.

A) ($106.29,$112.37);weare95%confidentthatthemeanamountspentperdogownerforasingle

obedienceclassisbetween$106.29and$112.37.

B) ($106.78,$111.88);weare95%confidentthatthemeanamountspentperdogownerforasingle

obedienceclassisbetween$106.78and$111.88.

C) ($106.23,$112.43);weare95%confidentthatthemeanamountspentperdogownerforasingle

obedienceclassisbetween$106.23and$112.43.

D) ($106.74,$111.92);weare95%confidentthatthemeanamountspentperdogownerforasingle

obedienceclassisbetween$106.74and$111.92.

2) Asurveyof1010collegeseniorsworkingtowardsanundergraduatedegreewasconducted.Eachstudentwas

asked,ʺAreyouplanningornotplanningtopursueagraduatedegree?ʺOfthe1010surveyed,658statedthat

theywereplanningtopursueagraduatedegree.Constructandinterpreta98%confidenceintervalforthe

proportionofcollegeseniorswhoareplanningtopursueagraduatedegree.

A) (0.616,0.686);weare98%confidentthattheproportionofcollegeseniorswhoareplanningtopursuea

graduatedegreeisbetween0.616and0.686.

B) (0.620,0.682);weare98%confidentthattheproportionofcollegeseniorswhoareplanningtopursuea

graduatedegreeisbetween0.620and0.682.

C) (0.621,0.680);weare98%confidentthattheproportionofcollegeseniorswhoareplanningtopursuea

graduatedegreeisbetween0.621and0.680.

D) (0.612,0.690);weare98%confidentthattheproportionofcollegeseniorswhoareplanningtopursuea

graduatedegreeisbetween0.612and0.690.

Constructa95%Z–intervalora95%t–intervalaboutthepopulationmean.Assumethedatacomefromapopulationthat

isapproximatelynormalwithnooutliers.

3) Theheightsof20–to29–year–oldfemalesareknowntohaveapopulationstandarddeviationσ

=2.7 inches.

Asimplerandomsampleofn=15females20to29yearsoldresultsinthefollowingdata:

63.1 67.9 64.8 62.2 65.4

63.3 66.2 68.2 69.7 64.1

68.4 69.9 67.3 64.5 70.2

A) (64.98,67.72);weare95%confidentthatthemeanheightof20 –to29–year–oldfemalesisbetween64.98

and67.72inches.

B) (64.85,67.85);weare95%confidentthatthemeanheightof20 –to29–year–oldfemalesisbetween64.85

and67.85inches.

C) (65.12,67.58);weare95%confidentthatthemeanheightof20 –to29–year–oldfemalesisbetween65.12

and67.58inches.

D) (65.20,67.50);weare95%confidentthatthemeanheightof20 –to29–year–oldfemalesisbetween65.20

and67.50inches.

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4) Fifteenrandomlyselectedmenwereaskedtorunonatreadmillfor6minutes.Afterthe6minutes,theirpulses

weremeasuredandthefollowingdatawereobtained:

105 94 98 88 104

101 99 85 84 124

122 114 97 101 90

A) (93.7,107.1);weare95%confidentthatthemeanpulserateofmenafter6minutesofexerciseisbetween

93.7and107.1beatsperminute.

B) (94.9,105.9);weare95%confidentthatthemeanpulserateofmenafter6minutesofexerciseisbetween

94.9and105.9beatsperminute.

C) (94.2,106.6);weare95%confidentthatthemeanpulserateofmenafter6minutesofexerciseisbetween

94.2and106.6beatsperminute.

D) (95.2,105.6);weare95%confidentthatthemeanpulserateofmenafter6minutesofexerciseisbetween

95.2and105.6beatsperminute.

9.5 EstimatingwithBootstrappin

1 Explainthegeneralbootstrapalgorithm.

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

Provideanappropriateresponse.

1) Ifwewishtoobtaina98%confidenceintervalofaparameterusingthebootstrapmethod,whichpercentilesof

theresampleddistributionwillformthelowerandupperboundsoftheinterval?

A) 1

99 B) 2

98 C) 2

99 D) 1

98

2) Arandomsampleof20electriciansisobtainedandthemonthlyincomeisrecordedforeachone.Aresearcher

planstousethebootstrapmethodwith1000resamplestoobtaina90%confidenceintervalforthemean

monthlyincomeofallelectricians.Whichofthefollowingisnottrueoftheresamples?

A) Eachresamplewillbeselectedfromthepopulation.

B) Eachresamplewillbeselectedwithreplacement.

C) Eachresamplewillbeselectedfromtheoriginalsample.

D) Eachresamplewillbeofthesamesizeastheoriginalsample.

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

3) Thetableshowsthemonthlyrents(indollars)for10studioapartmentsselectedrandomlyfromallstudio

apartmentsinonecity.

950 840 1330 1020 1180

880 990 7601250825

Explainthealgorithminusingthebootstrapmethodwith1000resamplestoobtaina90%confidenceinterval

forthemeanmonthlyrentofallstudioapartmentsinthecity.

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4) Acollegenurseobtainedarandomsampleof20studentsfromthecollege.Eachstudentwasaskediftheywere

takingantidepressants.Inthedata,a1indicatesthestudentistakingantidepressantsanda0indicatestheyare

nottakingantidepressants.

00101

10000

01000

00101

Treatthesedataasasimplerandomsampleofallstudentsatthecollege.Explainthealgorithminusingthe

bootstrapmethodwith1000resamplestoobtaina99%confidenceintervalfortheproportionofallstudentsat

thecollegewhoaretakingantidepressants.

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Ch.9 EstimatingtheValueofaParameter

AnswerKey

9.1 EstimatingaPopulationProportio

1 Obtainapointestimateforthepopulationproportion.

2 Constructandinterpretaconfidenceintervalforthepopulationproportion.

3 Findthesamplesizeneededforestimatingapopulationproportionwithinagivenmarginoferror.

9.2 EstimatingaPopulationMea

1 Determinet–values.

2 StatepropertiesofStudentʹst–distribution.

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3 Constructandinterpretaconfidenceintervalforapopulationmean.

4 Findthesamplesizeneededtoestimatethepopulationmeanwithinagivenmarginoferror.

9.3 EstimatingaPopulationStandardDeviatio

1 Findcriticalvaluesforthechi–squaredistribution.

2 Constructandinterpretconfidenceintervalsforthepopulationvarianceandstandarddeviation.

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9.4 PuttingItTogether:WhichProcedureDoIUse?

1 Determinetheappropriateconfidenceintervaltoconstruct.

9.5 EstimatingwithBootstrappin

1 Explainthegeneralbootstrapalgorithm.

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