321
Figure7.5
2. Drawasmoothcurvethroughthetopsofthebarsofthehistogram.Use1–2completesentencesto
describethegeneralshapeofthecurve.
7.10.5RepeattheProcedureforn=10
1. Forthesampleofn=10daysaveragedtogether,constructahistogramoftheaverages(yourmeans
togetherwiththemeansoftheothergroups). Makeintervalswithbarwidths=
1
2
day. Sketchthe
graphusingarulerandpencil.Scaletheaxes.
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CHAPTER7. THECENTRALLIMITTHEOREM
Figure7.6
2. Drawasmoothcurvethroughthetopsofthebarsofthehistogram.Use1–2completesentencesto
describethegeneralshapeofthecurve.
7.10.6DiscussionQuestions
1. Comparethethreehistogramsyouhavemade,theoneforthepopulationandthetwoforthesample
means.Inthreetofivesentences,describethesimilaritiesanddifferences.
2. Statethetheoretical(accordingtotheCLT)distributionsforthesamplemeans.
a. n=5:
X
b. n=10:
X
3. Arethesamplemeansforn=5andn=10“close”tothetheoreticalmean,m
x
? Explainwhyorwhy
not.
4. Whichofthetwodistributionsofsamplemeanshasthesmallerstandarddeviation?Why?
5. Asnchanged,whydidtheshapeofthedistributionofthedatachange?Use1–2completesentences
toexplainwhathappened.
NOTE
:ThislabwasdesignedandcontributedbyCarolOlmstead.
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323
SolutionstoExercisesinChapter7
SolutionstoPractice:TheCentralLimitTheorem
SolutiontoExercise7.6.1(p.302)
b. 3.5,4.25,0.2441
SolutiontoExercise7.6.2(p.302)
b. 0.7499
SolutiontoExercise7.6.3(p.303)
b. 4.49hours
SolutiontoExercise7.6.4(p.303)
b. 0.3802
SolutiontoExercise7.6.5(p.303)
b: 71.90
SolutionstoHomework
SolutiontoExercise7.7.1(p.305)
b. Xbar~N
60,
9
p
25
c. 0.5000
d. 59.06
e. 0.8536
f. 0.1333
h. 1530.35
i. 0.8536
SolutiontoExercise7.7.3(p.305)
a. N
36,
10
p
16
b. 1
c. 34.31
SolutiontoExercise7.7.5(p.306)
a. N
250,
50
p
49
b. 0.0808
c. 256.01feet
SolutiontoExercise7.7.7(p.306)
a. Thetotallengthoftimefor9criminaltrials
b. N(189,21)
c. 0.0432
d. 162.09
SolutiontoExercise7.7.9(p.306)
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324
CHAPTER7. THECENTRALLIMITTHEOREM
a. N
145,
14
p
49
b. 0.6247
c. 146.68
d. 145minutes
SolutiontoExercise7.7.11(p.307)
b. U(10,25)
c. 17.5
d.
q
225
12
=4.3301
f. N(17.5,0.5839)
h. N(962.5,32.11)
j. 0.0032
SolutiontoExercise7.7.13(p.308)
c. N
44,000,
6500
p
10
e. N
440,000,
p
10
(6500)
f. 0.9742
g. $52,330
h. $46,634
SolutiontoExercise7.7.15(p.308)
c. N
2.4,
0.9
p
80
e. N(192,8.05)
h. Individual
SolutiontoExercise7.7.17(p.309)
b. $20.71;$17.31;35
d. Exponentialdistribution,XExp(1/20.71)
f. $20.71;$11.14
j. N
20.71,
17.31
p
5
SolutiontoExercise7.7.19(p.310)
B
SolutiontoExercise7.7.20(p.310)
A
SolutiontoExercise7.7.21(p.310)
D
SolutiontoExercise7.7.22(p.310)
B
SolutiontoExercise7.7.23(p.310)
A
SolutiontoExercise7.7.24(p.310)
C
SolutiontoExercise7.7.25(p.311)
C. 0.0162
E. 0.0268
SolutiontoExercise7.7.26(p.311)
A. 0.2047
B. 0.9615
C. 0.0938
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325
SolutionstoReview
SolutiontoExercise7.8.1(p.312)
A
SolutiontoExercise7.8.2(p.312)
B
SolutiontoExercise7.8.3(p.312)
A
SolutiontoExercise7.8.4(p.312)
a. 0.7165
b. 4.16
c. 0
SolutiontoExercise7.8.5(p.312)
C
SolutiontoExercise7.8.6(p.313)
C
SolutiontoExercise7.8.7(p.313)
D
SolutiontoExercise7.8.8(p.313)
A
SolutiontoExercise7.8.9(p.313)
A
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326
CHAPTER7. THECENTRALLIMITTHEOREM
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Chapter8
ConfidenceIntervals
8.1ConfidenceIntervals
1
8.1.1StudentLearningOutcomes
Bytheendofthischapter,thestudentshouldbeableto:
 Calculateandinterpretconfidenceintervalsforonepopulationmeanandonepopulationproportion.
 Interpretthestudent-tprobabilitydistributionasthesamplesizechanges.
 Discriminatebetweenproblemsapplyingthenormalandthestudent-tdistributions.
8.1.2Introduction
Supposeyouaretryingtodeterminethemeanrentofatwo-bedroomapartmentinyourtown.Youmight
lookintheclassifiedsectionofthenewspaper,writedownseveralrentslisted,andaveragethemtogether.
Youwouldhaveobtainedapointestimateofthetruemean. Ifyouaretryingtodeterminethepercentof
timesyoumakeabasketwhenshootingabasketball,youmightcountthenumberofshotsyoumakeand
dividethatbythenumberofshotsyouattempted.Inthiscase,youwouldhaveobtainedapointestimate
forthetrueproportion.
Weusesampledatatomakegeneralizationsaboutanunknownpopulation.Thispartofstatisticsiscalled
inferentialstatistics.Thesampledatahelpustomakeanestimateofapopulationparameter.Werealize
thatthepointestimateismostlikelynottheexactvalueofthepopulationparameter,butclosetoit.After
calculatingpointestimates,weconstructconfidenceintervalsinwhichwebelievetheparameterlies.
Inthischapter,youwilllearntoconstructandinterpretconfidenceintervals. Youwillalsolearnanew
distribution,theStudent’s-t,andhowitisusedwiththeseintervals.Throughoutthechapter,itisimportant
tokeepinmindthattheconfidenceintervalisarandomvariable.Itistheparameterthatisfixed.
Ifyouworkedinthemarketingdepartmentofanentertainmentcompany,youmightbeinterestedinthe
meannumberofcompactdiscs(CD’s)aconsumerbuysper month. . Ifso, , youcouldconductasurvey
andcalculatethesample mean,
x, andthe samplestandarddeviation, s. Youwoulduse
xto estimate
thepopulationmeanand sto estimatethepopulationstandarddeviation. . Thesample e mean,
x, isthe
pointestimateforthepopulationmean,m.Thesamplestandarddeviation,s,isthepointestimateforthe
populationstandarddeviation,s.
1
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327
328
CHAPTER8. CONFIDENCEINTERVALS
Eachof
xandsisalsocalledastatistic.
Aconfidenceintervalisanothertypeofestimatebut,insteadofbeingjustonenumber,itisaninterval
ofnumbers. Theintervalofnumbersisarangeofvaluescalculatedfromagivensetofsampledata. The
confidenceintervalislikelytoincludeanunknownpopulationparameter.
SupposefortheCDexamplewedonotknowthepopulationmeanmbutwedoknowthatthepopulation
standarddeviationiss=1andoursamplesizeis100. ThenbytheCentralLimitTheorem,thestandard
deviationforthesamplemeanis
s
p
n
=
1
p
100
=0.1.
TheEmpiricalRule, whichappliestobell-shapeddistributions, saysthatinapproximately95% ofthe
samples,thesamplemean,
x,willbewithintwostandarddeviationsofthepopulationmeanm.ForourCD
example,twostandarddeviationsis(2)(0.1) = 0.2.Thesamplemean
xislikelytobewithin0.2unitsof
m.
Because
xiswithin0.2unitsofm,whichisunknown,thenmislikelytobewithin0.2unitsof
xin95%
ofthesamples. Thepopulationmeanmiscontainedinanintervalwhoselowernumberiscalculatedby
takingthesamplemeanandsubtractingtwostandarddeviations((2)(0.1))andwhoseuppernumberis
calculatedbytakingthesamplemeanandaddingtwostandarddeviations. Inotherwords,misbetween
0.2and
x+0.2in95%ofallthesamples.
FortheCDexample,supposethatasampleproducedasamplemean
x=2.Thentheunknownpopulation
meanmisbetween
0.2=0.2=1.8and
x+0.2=2+0.2=2.2
Wesaythatweare95%confidentthattheunknownpopulationmeannumberofCDsisbetween1.8and
2.2.The95%confidenceintervalis(1.8,2.2).
The95%confidenceintervalimpliestwopossibilities.Eithertheinterval(1.8,2.2)containsthetruemeanm
oroursampleproducedan
xthatisnotwithin0.2unitsofthetruemeanm.Thesecondpossibilityhappens
foronly5%ofallthesamples(100%-95%).
Rememberthataconfidenceintervaliscreatedforanunknownpopulationparameterlikethepopulation
mean,m.Confidenceintervalsforsomeparametershavetheform
(pointestimate-marginoferror,pointestimate+marginoferror)
Themarginoferrordependsontheconfidencelevelorpercentageofconfidence.
Whenyoureadnewspapersandjournals,somereportswillusethephrase"marginoferror."Otherreports
willnotusethatphrase,butincludeaconfidenceintervalasthepointestimate+or-themarginoferror.
Thesearetwowaysofexpressingthesameconcept.
NOTE
: Althoughthetextonlycoverssymmetricconfidenceintervals, , therearenon-symmetric
confidenceintervals(forexample,aconfidenceintervalforthestandarddeviation).
8.1.3OptionalCollaborativeClassroomActivity
Haveyourinstructorrecordthenumberofmealseachstudentinyourclasseatsoutinaweek. Assume
thatthestandarddeviationisknowntobe3meals.Constructanapproximate95%confidenceintervalfor
thetruemeannumberofmealsstudentseatouteachweek.
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329
1. Calculatethesamplemean.
2. s=3andn=thenumberofstudentssurveyed.
3. Constructtheinterval
2
s
p
n
,
x+2
s
p
n
Wesayweareapproximately95%confidentthatthetrueaveragenumberofmealsthatstudentseatoutin
aweekisbetween__________and___________.
8.2ConfidenceInterval,SinglePopulationMean,PopulationStandard
DeviationKnown,Normal
2
8.2.1CalculatingtheConfidenceInterval
Toconstructaconfidenceintervalforasingleunknownpopulationmean,wherethepopulationstan-
darddeviation isknown,we need
xasanestimate for mandweneedthemarginoferror. Here, , the
marginoferroriscalledtheerrorboundforapopulationmean(abbreviatedEBM).Thesamplemean
xis
thepointestimateoftheunknownpopulationmeanm
Theconfidenceintervalestimatewillhavetheform:
(pointestimate-errorbound,pointestimate+errorbound)or,insymbols,(
x EBM,
x+EBM)
Themarginoferrordependsontheconfidencelevel(abbreviatedCL).Theconfidencelevelisoftencon-
sideredthe probabilitythat thecalculatedconfidenceintervalestimate willcontainthetruepopulation
parameter. However,itismoreaccuratetostatethattheconfidencelevelisthepercentofconfidencein-
tervalsthatcontainthetruepopulationparameterwhenrepeatedsamplesaretaken. Mostoften,itisthe
choiceofthepersonconstructingthe confidenceintervaltochooseaconfidencelevelof90%or higher
becausethatpersonwantstobereasonablycertainofhisorherconclusions.
Thereisanotherprobabilitycalledalpha(a).aisrelatedtotheconfidencelevelCL.aistheprobabilitythat
theintervaldoesnotcontaintheunknownpopulationparameter.
Mathematically,a+CL=1.
Example8.1
Supposewehavecollecteddatafromasample.Weknowthesamplemeanbutwedonotknow
themeanfortheentirepopulation.
Thesamplemeanis7andtheerrorboundforthemeanis2.5.
x=7andEBM=2.5.
Theconfidenceintervalis(7 2.5,7+2.5);calculatingthevaluesgives(4.5,9.5).
Iftheconfidencelevel(CL)is95%,thenwesaythat"Weestimatewith95%confidencethatthe
truevalueofthepopulationmeanisbetween4.5and9.5."
Aconfidenceintervalforapopulationmeanwithaknownstandarddeviationisbasedonthefactthatthe
samplemeansfollowanapproximatelynormaldistribution.Supposethatoursamplehasameanof
x=10
andwehaveconstructedthe90%confidenceinterval(5,15)whereEBM=5.
Togeta90%confidenceinterval,wemustincludethecentral90%oftheprobabilityofthenormaldistri-
bution.Ifweincludethecentral90%,weleaveoutatotalofa=10%inbothtails,or5%ineachtail,ofthe
normaldistribution.
2
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330
CHAPTER8. CONFIDENCEINTERVALS
Tocapturethecentral90%,wemustgoout1.645"standarddeviations"oneitherside ofthecalculated
samplemean.1.645isthez-scorefromaStandardNormalprobabilitydistributionthatputsanareaof0.90
inthecenter,anareaof0.05inthefarlefttail,andanareaof0.05inthefarrighttail.
Itisimportantthatthe"standarddeviation"usedmustbeappropriatefortheparameterweareestimating.
Sointhissection,weneedtousethestandarddeviationthatappliestosamplemeans,whichis
s
p
n
.
s
p
n
is
commonlycalledthe"standarderrorofthemean"inordertoclearlydistinguishthestandarddeviationfor
ameanfromthepopulationstandarddeviations.
Insummary,asaresultoftheCentralLimitTheorem:
Xisnormallydistributed,thatis,
XN
m
X
,
s
p
n
.
 Whenthepopulationstandarddeviationsisknown,weuseaNormaldistributiontocalculate
theerrorbound.
CalculatingtheConfidenceInterval:
Toconstructaconfidenceintervalestimateforanunknownpopulationmean,weneeddatafromarandom
sample.Thestepstoconstructandinterprettheconfidenceintervalare:
 Calculatethesamplemean
xfromthesampledata.Remember,inthissection,wealreadyknowthe
populationstandarddeviations.
 FindtheZ-scorethatcorrespondstotheconfidencelevel.
 CalculatetheerrorboundEBM
 Constructtheconfidenceinterval
 Writeasentencethatinterpretstheestimateinthecontextofthesituationintheproblem. (Explain
whattheconfidenceintervalmeans,inthewordsoftheproblem.)
Wewillfirstexamineeachstepinmoredetail,andthenillustratetheprocesswithsomeexamples.
FindingzforthestatedConfidenceLevel
Whenweknowthepopulationstandarddeviations,weuseastandardnormaldistributiontocalculate
theerrorboundEBMandconstructtheconfidenceinterval.Weneedtofindthevalueofzthatputsanarea
equaltotheconfidencelevel(indecimalform)inthemiddleofthestandardnormaldistributionZN(0,1).
Theconfidencelevel,CL,istheareainthemiddleofthestandardnormaldistribution.CL=a.Soais
theareathatissplitequallybetweenthetwotails.Eachofthetailscontainsanareaequalto
a
2
.
Thez-scorethathasanareatotherightof
a
2
isdenotedbyz
a
2
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