421
a. Graph:
Figure9.10
b. Determinethep-value:
7. Doyouordoyounotrejectthenullhypothesis?Why?
8. Writeaclearconclusionusingacompletesentence.
9.18.4JeansSurvey
Supposethatyoungadultsownanaverageof3pairsofjeans.Survey8peoplefromyourclasstodetermine
iftheaverageishigherthan3.
1. H
o
:
2. H
a
:
3. Inwords,definetherandomvariable.__________=
4. Thedistributiontouseforthetestis:
5. Determinetheteststatisticusingyourdata.
6. Drawagraphandlabelitappropriately.Shadetheactuallevelofsignificance.
a. Graph:
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422
CHAPTER9. HYPOTHESISTESTING:SINGLEMEANANDSINGLE
PROPORTION
Figure9.11
b. Determinethep-value:
7. Doyouordoyounotrejectthenullhypothesis?Why?
8. Writeaclearconclusionusingacompletesentence.
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423
SolutionstoExercisesinChapter9
SolutionstoPractice1:SingleMean,KnownPopulationStandardDeviation
SolutiontoExercise9.13.1(p.397)
Means
SolutiontoExercise9.13.2(p.397)
a: H
o
:m=2.5(or,H
o
:m2.5)
b: H
a
:m>2.5
SolutiontoExercise9.13.3(p.397)
right-tailed
SolutiontoExercise9.13.4(p.397)
X
SolutiontoExercise9.13.5(p.397)
Themeantimespentinjailfor26firsttimeconvictedburglars
SolutiontoExercise9.13.6(p.397)
Yes,1.5
SolutiontoExercise9.13.7(p.397)
a. 3
b. 1.5
c. 1.8
d. 26
SolutiontoExercise9.13.8(p.397)
s
SolutiontoExercise9.13.9(p.397)
X~N
2.5,
1.5
p
26
SolutiontoExercise9.13.11(p.398)
0.0446
SolutiontoExercise9.13.12(p.398)
a. Rejectthenullhypothesis
SolutionstoPractice2:SingleMean,UnknownPopulationStandardDeviation
SolutiontoExercise9.14.1(p.399)
averages
SolutiontoExercise9.14.2(p.399)
a. H
o
:m=15
b. H
a
:m6=15
SolutiontoExercise9.14.3(p.399)
two-tailed
SolutiontoExercise9.14.4(p.399)
X
SolutiontoExercise9.14.5(p.399)
themeantimespentondeathrowforthe75inmates
SolutiontoExercise9.14.6(p.399)
No
SolutiontoExercise9.14.7(p.399)
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424
CHAPTER9. HYPOTHESISTESTING:SINGLEMEANANDSINGLE
PROPORTION
a. 17.4
b. s
c. 75
SolutiontoExercise9.14.8(p.399)
test
SolutiontoExercise9.14.9(p.399)
t
74
SolutiontoExercise9.14.11(p.400)
0.0015
SolutiontoExercise9.14.12(p.400)
a. Rejectthenullhypothesis
SolutionstoPractice3:SingleProportion
SolutiontoExercise9.15.1(p.401)
Proportions
SolutiontoExercise9.15.2(p.401)
a. H
o
:p=0.095
b. H
a
:p<0.095
SolutiontoExercise9.15.3(p.401)
left-tailed
SolutiontoExercise9.15.4(p.401)
P’
SolutiontoExercise9.15.5(p.401)
theproportionofpeopleinthattownsurveyedsufferingfromdepressionoradepressiveillness
SolutiontoExercise9.15.6(p.401)
a. 7
b. 100
c. 0.07
SolutiontoExercise9.15.7(p.401)
0.0293
SolutiontoExercise9.15.8(p.401)
Normal
SolutiontoExercise9.15.10(p.402)
0.1969
SolutiontoExercise9.15.11(p.402)
a. Donotrejectthenullhypothesis
SolutionstoHomework
SolutiontoExercise9.16.1(p.403)
a. H
o
:m=34;H
a
:m6=34
c. H
o
:m100,000;H
a
:m<100,000
d. H
o
:p=0.29;H
a
:p6=0.29
g. H
o
:p=0.50;H
a
:p6=0.50
i. H
o
:p0.11;H
a
:p<0.11
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425
SolutiontoExercise9.16.2(p.403)
a. TypeIerror:Weconcludethatthemeanisnot34years,whenitreallyis34years.TypeIIerror: Wedo
notconcludethatthemeanisnot34years,whenitisnotreally34years.
c. TypeIerror:Weconcludethatthemeanislessthan$100,000,whenitreallyisatleast$100,000.TypeII
error:Wedonotconcludethatthemeanislessthan$100,000,whenitisreallylessthan$100,000.
d. TypeIerror: Weconcludethattheproportionofh.s. . seniorswhogetdrunkeachmonthisnot29%,
whenitreallyis29%. TypeIIerror: Wedonotconcludethattheproportionofh.s. . seniorsthatget
drunkeachmonthisnot29%,whenitisreallynot29%.
i. TypeIerror:Weconcludethattheproportionislessthan11%,whenitisreallyatleast11%.TypeIIerror:
Wedonotconcludethattheproportionislessthan11%,whenitreallyislessthan11%.
SolutiontoExercise9.16.5(p.403)
e. z2.71
f. 0.0034
h. Decision:Rejectnull;Conclusion:m<19
i. (17.449,18.757)
SolutiontoExercise9.16.7(p.404)
e. 3.5
f. 0.0005
h. Decision:Rejectnull;Conclusion:m>4.5
i. (4.7553,5.4447)
SolutiontoExercise9.16.9(p.404)
e. 2.7
f. 0.0042
h. Decision:RejectNull
i. (80.789,85.211)
SolutiontoExercise9.16.11(p.404)
d. t
11
e. 1.96
f. 0.0380
h. Decision:Rejectnullwhena=0.05;donotrejectnullwhena=0.01
i. (3.8865,5.9468)
SolutiontoExercise9.16.13(p.404)
e. -1.64
f. 0.1000
h. Decision:Donotrejectnull
i. (0.3216,0.4784)
SolutiontoExercise9.16.15(p.405)
d. t
9
e. -1.33
f. 0.1086
h. Decision:Donotrejectnull
i. (51.886,62.114)
SolutiontoExercise9.16.19(p.406)
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426
CHAPTER9. HYPOTHESISTESTING:SINGLEMEANANDSINGLE
PROPORTION
e. 1.65
f. 0.0984
h. Decision:Donotrejectnull
i. (0.6836,0.8533)
SolutiontoExercise9.16.21(p.407)
e. -2.39
f. 0.0093
h. Decision:Rejectnull
i. (91.854,100.15)
SolutiontoExercise9.16.23(p.408)
e. -1.82
f. 0.0345
h. Decision:Donotrejectnull
i. (0.5331,0.7685)
SolutiontoExercise9.16.25(p.410)
e. z2.99
f. 0.0014
h. Decision:Rejectnull;Conclusion: p<.70
i. (0.4529,0.6582)
SolutiontoExercise9.16.27(p.411)
e. 0.57
f. 0.7156
h. Decision:Donotrejectnull
i. (0.3415,0.6185)
SolutiontoExercise9.16.29(p.412)
e. 1.32
f. 0.1873
h. Decision:Donotrejectnull
i. (0.65,0.90)
SolutiontoExercise9.16.31(p.412)
e. 9.98
f. 0.0000
h. Decision:Rejectnull
i. (28.8,30.0)
SolutiontoExercise9.16.33(p.412)
e. -44.7
f. 0.0000
h. Decision:Rejectnull
i. (0.60,0.90)-inyears
SolutiontoExercise9.16.34(p.413)
B
SolutiontoExercise9.16.35(p.413)
D
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427
SolutiontoExercise9.16.36(p.413)
C
SolutiontoExercise9.16.37(p.413)
C
SolutiontoExercise9.16.38(p.414)
A
SolutiontoExercise9.16.39(p.414)
C
SolutiontoExercise9.16.40(p.414)
D
SolutiontoExercise9.16.41(p.414)
D
SolutiontoExercise9.16.42(p.415)
D
SolutiontoExercise9.16.43(p.415)
C
SolutiontoExercise9.16.44(p.415)
B
SolutionstoReview
SolutiontoExercise9.17.1(p.416)
D
SolutiontoExercise9.17.2(p.416)
Nosolutionprovided.Thereareseveralwaysinwhichthehistogramcouldbeconstructed.
SolutiontoExercise9.17.3(p.416)
a. 2.8
b. 1.48
c. 90%
SolutiontoExercise9.17.4(p.416)
M=3;Q1=1;Q3=4
SolutiontoExercise9.17.5(p.416)
1and4
SolutiontoExercise9.17.6(p.416)
D
SolutiontoExercise9.17.7(p.417)
C
SolutiontoExercise9.17.8(p.417)
A
SolutiontoExercise9.17.9(p.417)
B
SolutiontoExercise9.17.10(p.417)
B
SolutiontoExercise9.17.11(p.417)
B
SolutiontoExercise9.17.12(p.417)
b. P(20)
c. 0.1122
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428
CHAPTER9. HYPOTHESISTESTING:SINGLEMEANANDSINGLE
PROPORTION
SolutiontoExercise9.17.13(p.417)
CI:(5.52,8.48)
SolutiontoExercise9.17.14(p.417)
a. uniform
b. exponential
c. normal
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Chapter10
HypothesisTesting:TwoMeans,Paired
Data,TwoProportions
10.1HypothesisTesting: TwoPopulationMeansandTwoPopulation
Proportions
1
10.1.1StudentLearningOutcomes
Bytheendofthischapter,thestudentshouldbeableto:
 Classifyhypothesistestsbytype.
 Conductandinterprethypothesistestsfortwopopulationmeans,populationstandarddeviations
known.
 Conductandinterprethypothesistestsfortwopopulationmeans,populationstandarddeviations
unknown.
 Conductandinterprethypothesistestsfortwopopulationproportions.
 Conductandinterprethypothesistestsformatchedorpairedsamples.
10.1.2Introduction
Studiesoftencompare two groups. . Forexample, , researchersareinterested intheeffect aspirinhasin
preventingheartattacks.Overthelastfewyears,newspapersandmagazineshavereportedaboutvarious
aspirinstudiesinvolvingtwogroups. Typically,onegroupisgivenaspirinandtheothergroupisgivena
placebo.Then,theheartattackrateisstudiedoverseveralyears.
Thereareothersituationsthatdealwiththecomparisonoftwogroups.Forexample,studiescomparevar-
iousdietandexerciseprograms. Politicianscomparetheproportionofindividualsfromdifferentincome
bracketswhomightvoteforthem. StudentsareinterestedinwhetherSATorGREpreparatorycourses
reallyhelpraisetheirscores.
Inthepreviouschapter,youlearnedtoconducthypothesistestsonsinglemeansandsingleproportions.
Youwillexpanduponthatinthischapter. Youwillcomparetwomeansortwoproportionstoeachother.
Thegeneralprocedureisstillthesame,justexpanded.
1
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429
430
CHAPTER10. HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO
PROPORTIONS
Tocomparetwomeansortwoproportions,youworkwithtwogroups.Thegroupsareclassifiedeitheras
independentormatchedpairsIndependentgroupsmeanthatthetwosamplestakenareindependent,
thatis,samplevaluesselectedfromonepopulationarenotrelatedinanywaytosamplevaluesselected
fromtheotherpopulation.Matchedpairsconsistoftwosamplesthataredependent.Theparametertested
usingmatchedpairsisthepopulationmean. Theparameterstestedusingindependentgroupsareeither
populationmeansorpopulationproportions.
NOTE
:Thischapterreliesoneitheracalculatororacomputertocalculatethedegreesoffreedom,
theteststatistics,andp-values.TI-83+andTI-84instructionsareincludedaswellastheteststatis-
ticformulas.WhenusingtheTI-83+/TI-84calculators,wedonotneedtoseparatetwopopulation
means,independentgroups,populationvariancesunknownintolargeandsmallsamplesizes.
However,moststatisticalcomputersoftwarehastheabilitytodifferentiatethesetests.
Thischapterdealswiththefollowinghypothesistests:
Independentgroups(samplesareindependent)
• Testoftwopopulationmeans.
• Testoftwopopulationproportions.
Matchedorpairedsamples(samplesaredependent)
• Becomesatestofonepopulationmean.
10.2 Comparing Two Independent PopulationMeans with Unknown
PopulationStandardDeviations
2
1. Thetwoindependentsamplesaresimplerandomsamplesfromtwodistinctpopulations.
2. Bothpopulationsarenormallydistributedwiththepopulationmeansandstandarddeviationsun-
knownunlessthesamplesizesaregreaterthan30.Inthatcase,thepopulationsneednotbenormally
distributed.
NOTE
: Thetestcomparingtwoindependentpopulationmeanswithunknownandpossiblyun-
equalpopulationstandarddeviationsiscalledtheAspin-Welcht-test. Thedegreesoffreedom
formulawasdevelopedbyAspin-Welch.
Thecomparisonoftwopopulationmeansisverycommon.Adifferencebetweenthetwosamplesdepends
onboththemeansandthestandarddeviations. Verydifferentmeanscanoccurbychanceifthereisgreat
variationamongtheindividualsamples. Inordertoaccountforthevariation,wetakethedifferenceof
thesamplemeans,
X
1
-
X
2
,anddividebythestandarderror(shownbelow)inordertostandardizethe
difference.Theresultisat-scoreteststatistic(shownbelow).
Becausewe donot know thepopulationstandard deviations, we estimatethemusingthe two sample
standarddeviationsfromourindependentsamples. Forthehypothesistest,wecalculate e theestimated
standarddeviation,orstandarderror,ofthedifferenceinsamplemeans,
X
1
-
X
2
.
Thestandarderroris:
s
(S
1
)
2
n
1
+
(S
2
)
2
n
2
(10.1)
Theteststatistic(t-score)iscalculatedasfollows:
2
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