491
11.9SummaryofFormulas
9
TheChi-SquareProbabilityDistribution
m=dfands=
p
2df
Goodness-of-FitHypothesisTest
 Usegoodness-of-fittotestwhetheradatasetfitsaparticularprobabilitydistribution.
 Thedegreesoffreedomarenumberofcellsorcategories-1.
 TheteststatisticisS
k
(O E)
2
E
,whereO=observedvalues(data),E=expectedvalues(fromtheory),
andk=thenumberofdifferentdatacellsorcategories.
 Thetestisright-tailed.
TestofIndependence
 Usethetestofindependencetotestwhethertwofactorsareindependentornot.
 Thedegreesoffreedomareequalto(numberofcolumns-1)(numberofrows-1).
 Theteststatisticis S
(ij)
(O E)
2
E
whereO=observedvalues,E=expectedvalues,i=thenumberofrows
inthetable,andj=thenumberofcolumnsinthetable.
 Thetestisright-tailed.
 Ifthenullhypothesisistrue,theexpectednumberE=
(rowtotal)(columntotal)
totalsurveyed
.
TestofHomogeneity
 Usethetestforhomogeneitytodecideiftwopopulationswithunknowndistributionshavethesame
distributionaseachother.
 Thedegreesoffreedomareequaltonumberofcolumns-1.
 Theteststatisticis S
(ij)
(O E)
2
E
whereO=observedvalues,E=expectedvalues,i=thenumberofrows
inthetable,andj=thenumberofcolumnsinthetable.
 Thetestisright-tailed.
 Ifthenullhypothesisistrue,theexpectednumberE=
(rowtotal)(columntotal)
totalsurveyed
.
NOTE
:Theexpectedvalueforeachcellneedstobeatleast5inordertousetheGoodness-of-Fit,
IndependenceandHomogeneitytests.
TestofaSingleVariance
 Usethetesttodeterminevariation.
 Thedegreesoffreedomarethenumberofsamples-1.
 The teststatisticis
(n 1)s2
s2
, where n=thetotalnumberofdata, s
2
=sample variance, ands
2
=
populationvariance.
 Thetestmaybeleft,right,ortwo-tailed.
9
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492
CHAPTER11. THECHI-SQUAREDISTRIBUTION
11.10Practice1:Goodness-of-FitTest
10
11.10.1StudentLearningOutcomes
 Thestudentwillconductagoodness-of-fittest.
11.10.2Given
Thefollowingdataarereal. ThecumulativenumberofAIDScasesreportedfor r SantaClaraCountyis
brokendownbyethnicityasfollows:(Source: HIV/AIDSEpidemiologySantaClaraCounty,SantaClara
CountyPublicHealthDepartment,May2011)
Ethnicity
NumberofCases
White
2229
Hispanic
1157
Black/African-American
457
Asian,PacificIslander
232
Total=4075
Table11.14
ThepercentageofeachethnicgroupinSantaClaraCountyisasfollows:
Ethnicity
Percentageoftotalcountypop-
ulation
Number expected (round to 2
decimalplaces)
White
42.9%
1748.18
Hispanic
26.7%
Black/African-American
2.6%
Asian,PacificIslander
27.8%
Total=100%
Table11.15
11.10.3ExpectedResults
IftheethnicityofAIDSvictimsfollowedtheethnicityofthetotalcountypopulation,fillintheexpected
numberofcasesperethnicgroup.
11.10.4Goodness-of-FitTest
Performagoodness-of-fittesttodeterminewhetherthemake-upofAIDScasesfollowstheethnicityofthe
generalpopulationofSantaClaraCounty.
Exercise11.10.1
H
o
:
10
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493
Exercise11.10.2
H
a
:
Exercise11.10.3
Isthisaright-tailed,left-tailed,ortwo-tailedtest?
Exercise11.10.4
(Solutiononp.518.)
degreesoffreedom=
Exercise11.10.5
(Solutiononp.518.)
Chi
2
teststatistic=
Exercise11.10.6
(Solutiononp.518.)
p-value=
Exercise11.10.7
Graphthesituation.Labelandscalethehorizontalaxis.Markthemeanandteststatistic. Shade
intheregioncorrespondingtothep-value.
Leta=0.05
Decision:
ReasonfortheDecision:
Conclusion(writeoutincompletesentences):
11.10.5DiscussionQuestion
Exercise11.10.8
DoesitappearthatthepatternofAIDScasesinSantaClaraCountycorrespondstothedistribu-
tionofethnicgroupsinthiscounty?Whyorwhynot?
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494
CHAPTER11. THECHI-SQUAREDISTRIBUTION
11.11Practice2:ContingencyTables
11
11.11.1StudentLearningOutcomes
 Thestudentwillconductatestforindependenceusingcontingencytables.
Conductahypothesistesttodetermineifsmokinglevelandethnicityareindependent.
11.11.2CollecttheData
CopythedataprovidedinProbabilityTopicsPractice1:ContingencyTables intothetablebelow.
SmokingLevelsbyEthnicity(Observed)
Smoking
Level
Per
Day
African
American
Native
Hawaiian
Latino
Japanese
Americans
White
TOTALS
1-10
11-20
21-30
31+
TOTALS
Table11.16
11.11.3Hypothesis
Statethehypotheses.
 H
o
:
 H
a
:
11.11.4ExpectedValues
Enterexpectedvaluesintheabovebelow.Roundtotwodecimalplaces.
11.11.5AnalyzetheData
Calculatethefollowingvalues:
Exercise11.11.1
(Solutiononp.518.)
Degreesoffreedom=
Exercise11.11.2
(Solutiononp.518.)
Chi
2
teststatistic=
Exercise11.11.3
(Solutiononp.518.)
p-value=
Exercise11.11.4
(Solutiononp.518.)
Isthisaright-tailed,left-tailed,ortwo-tailedtest?Explainwhy.
11
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495
11.11.6GraphtheData
Exercise11.11.5
Graphthesituation.Labelandscalethehorizontalaxis.Markthemeanandteststatistic. Shade
intheregioncorrespondingtothep-value.
11.11.7Conclusions
Statethedecisionandconclusion(inacompletesentence)forthefollowingpreconceivedlevelsofa.
Exercise11.11.6
(Solutiononp.518.)
a=0.05
a. Decision:
b. Reasonforthedecision:
c. Conclusion(writeoutinacompletesentence):
Exercise11.11.7
a=0.01
a. Decision:
b. Reasonforthedecision:
c. Conclusion(writeoutinacompletesentence):
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496
CHAPTER11. THECHI-SQUAREDISTRIBUTION
11.12Practice3:TestofaSingleVariance
12
11.12.1StudentLearningOutcomes
 Thestudentwillconductatestofasinglevariance.
11.12.2Given
Supposeanairlineclaimsthatitsflightsareconsistentlyontimewithanaveragedelayofatmost15min-
utes.Itclaimsthattheaveragedelayissoconsistentthatthevarianceisnomorethan150minutes.Doubt-
ingtheconsistencypartoftheclaim,adisgruntledtravelercalculatesthedelaysforhisnext25flights.The
averagedelayforthose25flightsis22minuteswithastandarddeviationof15minutes.
11.12.3SampleVariance
Exercise11.12.1
Isthetravelerdisputingtheclaimabouttheaverageoraboutthevariance?
Exercise11.12.2
(Solutiononp.518.)
Asamplestandarddeviationof15minutesisthesameasasamplevarianceof__________min-
utes.
Exercise11.12.3
Isthisaright-tailed,left-tailed,ortwo-tailedtest?
11.12.4HypothesisTest
Performahypothesistestontheconsistencypartoftheclaim.
Exercise11.12.4
H
o
:
Exercise11.12.5
H
a
:
Exercise11.12.6
(Solutiononp.518.)
Degreesoffreedom=
Exercise11.12.7
(Solutiononp.518.)
Chi
2
teststatistic=
Exercise11.12.8
(Solutiononp.518.)
p-value=
Exercise11.12.9
Graphthesituation.Labelandscalethehorizontalaxis.Markthemeanandteststatistic. Shade
thep-value.
12
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497
Exercise11.12.10
Leta=0.05
Decision:
Conclusion(writeoutinacompletesentence):
11.12.5DiscussionQuestions
Exercise11.12.11
Howdidyouknowtotestthevarianceinsteadofthemean?
Exercise11.12.12
Ifanadditionaltestweredoneontheclaimoftheaveragedelay,whichdistributionwouldyou
use?
Exercise11.12.13
Ifanadditionaltestwasdoneontheclaimoftheaveragedelay,but45flightsweresurveyed,
whichdistributionwouldyouuse?
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498
CHAPTER11. THECHI-SQUAREDISTRIBUTION
11.13Homework
13
Exercise11.13.1
a. Explainwhythe“goodnessoffit”testandthe“testforindependence”aregenerallyrighttailed
tests.
b. Ifyoudidaleft-tailedtest,whatwouldyoubetesting?
11.13.1WordProblems
Foreachwordproblem, use asolutionsheet to solvethe hypothesistestproblem. . Goto o TheTableof
Contents14.Appendixforthechi-squaresolutionsheet.Roundexpectedfrequencytotwodecimalplaces.
Exercise11.13.2
A6-sideddieisrolled120times.Fillintheexpectedfrequencycolumn.Then,conductahypoth-
esistesttodetermineifthedieisfair.Thedatabelowaretheresultofthe120rolls.
FaceValue
Frequency
ExpectedFrequency
1
15
2
29
3
16
4
15
5
30
6
15
Table11.17
Exercise11.13.3
(Solutiononp.518.)
ThemaritalstatusdistributionoftheU.S.malepopulation,age15andolder,isasshownbelow.
(Source:U.S.CensusBureau,CurrentPopulationReports)
MaritalStatus
Percent
ExpectedFrequency
nevermarried
31.3
married
56.1
widowed
2.5
divorced/separated
10.1
Table11.18
Suppose thatarandomsample of400U.S.youngadultmales, 18–24yearsold, yieldedthe
followingfrequencydistribution.Weareinterestedinwhetherthisagegroupofmalesfitsthedis-
tributionoftheU.S.adultpopulation.Calculatethefrequencyonewouldexpectwhensurveying
400people.Fillintheabovetable,roundingtotwodecimalplaces.
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499
MaritalStatus
Frequency
nevermarried
140
married
238
widowed
2
divorced/separated
20
Table11.19
Thenexttwoquestionsrefertothefollowinginformation. Thecolumnsinthechartbelowcontainthe
Race/EthnicityofU.S.PublicSchoolsforarecentyear,thepercentagesfortheAdvancedPlacementExami-
neePopulationforthatclassandtheOverallStudentPopulation.(Source:http://www.collegeboard.com).
Supposetherightcolumncontainstheresultofasurveyof1000localstudentsfromthatyearwhotookan
APExam.
Race/Ethnicity
AP Examinee Popula-
tion
OverallStudentPopu-
lation
SurveyFrequency
Asian,AsianAmerican
orPacificIslander
10.2%
5.4%
113
BlackorAfricanAmeri-
can
8.2%
14.5%
94
HispanicorLatino
15.5%
15.9%
136
American Indian or
AlaskaNative
0.6%
1.2%
10
White
59.4%
61.6%
604
Notreported/other
6.1%
1.4%
43
Table11.20
Exercise11.13.4
Performagoodness-of-fittesttodeterminewhetherthelocalresultsfollowthedistributionofthe
U.S.OverallStudentPopulationbasedonethnicity.
Exercise11.13.5
(Solutiononp.518.)
Performagoodness-of-fittesttodeterminewhetherthelocalresultsfollowthedistributionofU.
S.APExamineePopulation,basedonethnicity.
Exercise11.13.6
TheCityofSouthLakeTahoe,CA,hasanAsianpopulationof1419people,outofatotalpopu-
lationof23,609(Source:U.S.CensusBureau).Supposethatasurveyof1419self-reportedAsians
inManhattan,NY,areayieldedthedatainthetablebelow. Conductagoodnessoffittesttode-
termineiftheself-reportedsub-groupsofAsiansintheManhattanareafitthatoftheLakeTahoe
area.
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500
CHAPTER11. THECHI-SQUAREDISTRIBUTION
Race
LakeTahoeFrequency
ManhattanFrequency
AsianIndian
131
174
Chinese
118
557
Filipino
1045
518
Japanese
80
54
Korean
12
29
Vietnamese
9
21
Other
24
66
Table11.21
Thenexttwoquestionsrefertothefollowinginformation: UCLA conducted asurveyofmore than
263,000collegefreshmenfrom385collegesinfall2005. Theresultsofstudentexpectedmajorsbygender
were reportedinThe Chronicle ofHigherEducation(2/2/2006). . Supposeasurveyof5000graduating
femalesand5000graduatingmaleswasdoneasafollow-uplastyeartodeterminewhattheiractualmajor
was. TheresultsareshowninthetablesforExercises7and8. . Thesecondcolumnineachtabledoesnot
addto100%becauseofrounding.
Exercise11.13.7
(Solutiononp.519.)
Conductahypothesistesttodetermineiftheactualcollegemajorofgraduatingfemalesfitsthe
distributionoftheirexpectedmajors.
Major
Women-ExpectedMajor
Women-ActualMajor
Arts&Humanities
14.0%
670
BiologicalSciences
8.4%
410
Business
13.1%
685
Education
13.0%
650
Engineering
2.6%
145
PhysicalSciences
2.6%
125
Professional
18.9%
975
SocialSciences
13.0%
605
Technical
0.4%
15
Other
5.8%
300
Undecided
8.0%
420
Table11.22
Exercise11.13.8
Conductahypothesistesttodetermineiftheactualcollegemajorofgraduatingmalesfitsthe
distributionoftheirexpectedmajors.
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