591
Figure13.2
Probabilitystatement:p-value=P(F>2.23)=0.1241
Compareaandthevalue:a=0.01
p-value=0.1241
a<p-value
Makeadecision:Sincea<p-value,youcannotrejectH
o
.
Conclusion:Thereisnotsufficientevidencetoconcludethatthereisadifferenceamongthemean
gradesforthesororities.
TI-83+orTI84: PutthedataintolistsL1,L2,L3,andL4. Press
andarrowoverto
.
Arrowdownto
.Press
andEnter(
). TheFstatisticis2.2303andthe
p-valueis0.1241.df(numerator)=3(under
)anddf(denominator)=16(under
).
Example13.3
Afourthgradeclassisstudyingtheenvironment.Oneoftheassignmentsistogrowbeanplants
indifferentsoils.Tommychosetogrowhisbeanplantsinsoilfoundoutsidehisclassroommixed
withdryerlint. Tarachosetogrowherbeanplantsinpottingsoilboughtatthelocalnursery.
Nickchosetogrowhisbeanplantsinsoilfromhismother’sgarden. Nochemicalswereused
ontheplants,onlywater. Theyweregrowninsidetheclassroomnexttoalargewindow. Each
childgrew5plants. Attheendofthegrowingperiod,eachplantwasmeasured,producingthe
followingdata(ininches):
Tommy’sPlants
Tara’sPlants
Nick’sPlants
24
25
23
21
31
27
23
23
22
30
20
30
23
28
20
Table13.5
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592
CHAPTER13. FDISTRIBUTIONANDONE-WAYANOVA
Problem1
Doesitappearthatthethreemediainwhichthebeanplantsweregrownproducethesamemean
height?Testata3%levelofsignificance.
Solution
Thistime,wewillperformthecalculationsthatleadtotheF’statistic.Noticethateachgrouphas
thesamenumberofplantssowewillusetheformulaF=
ns_
x
2
s2
pooled
.
First,calculatethesamplemeanandsamplevarianceofeachgroup.
Tommy’sPlants
Tara’sPlants
Nick’sPlants
SampleMean
24.2
25.4
24.4
SampleVariance
11.7
18.3
16.3
Table13.6
Next,calculatethevarianceofthethreegroupmeans(Calculatethevarianceof24.2,25.4,and
24.4).Varianceofthegroupmeans=0.413=s
x
2
ThenMS
between
=ns
x
2
=(5)(0.413)wheren=5isthesamplesize(numberofplantseachchild
grew).
Calculatethemeanofthethreesamplevariances(Calculatethemeanof11.7,18.3,and16.3).Mean
ofthesamplevariances=15.433=s
2
pooled
ThenMS
within
=s
2
pooled
=15.433.
TheFstatistic(orFratio)isF=
MS
between
MS
within
=
ns_
x
2
s2
pooled
=
(5)(0.413)
15.433
=0.134
Thedfsforthenumerator=thenumberofgroups 1=3 1=2
Thedfsforthedenominator=thetotalnumberofsamples thenumberofgroups=15 5 3=12
ThedistributionforthetestisF
2,12
andtheFstatisticisF=0.134
Thep-valueisP(F>0.134)=0.8759.
Decision:Sincea=0.03andthep-value=0.8759,donotrejectH
o
.(Why?)
Conclusion:Witha3%thelevelofsignificance,fromthesampledata,theevidenceisnotsufficient
toconcludethatthemeanheightsofthebeanplantsaredifferent.
(Thisexperimentwasactuallydonebythreeclassmatesofthesonofoneoftheauthors.)
Anotherfourthgraderalsogrewbeanplantsbutthistimeinajelly-likemass. Theheightswere
(ininches)24,28,25,30,and32.
Problem2
(Solutiononp.610.)
DoaOne-WayANOVA teston the4groups. You mayuse yourcalculator or computer to
performthetest.Aretheheightsofthebeanplantsdifferent?Useasolutionsheet(Section14.5.4).
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593
13.4.1OptionalClassroomActivity
Fromtheclass,createfourgroupsofthesamesizeasfollows:menunder22,menatleast22,womenunder
22,womenatleast22. HaveeachmemberofeachgrouprecordthenumberofstatesintheUnitedStates
heorshehasvisited.RunanANOVAtesttodetermineiftheaveragenumberofstatesvisitedinthefour
groupsarethesame.Testata1%levelofsignificance.Useoneofthesolutionsheets(Section14.5.4)atthe
endofthechapter(afterthehomework).
13.5TestofTwoVariances
6
AnotheroftheusesoftheFdistributionistestingtwovariances. It t isoftendesirable tocomparetwo
variancesratherthantwoaverages.Forinstance,collegeadministratorswouldliketwocollegeprofessors
gradingexamstohavethesamevariationintheirgrading.Inorderforalidtofitacontainer,thevariation
inthelidandthecontainershouldbethesame. Asupermarketmightbeinterestedinthevariabilityof
check-outtimesfortwocheckers.
InordertoperformaFtestoftwovariances,itisimportantthatthefollowingaretrue:
1. Thepopulationsfromwhichthetwosamplesaredrawnarenormallydistributed.
2. Thetwopopulationsareindependentofeachother.
Supposewesamplerandomlyfromtwoindependentnormalpopulations.Lets
2
1
ands
2
2
bethepopulation
variancesands
2
1
ands
2
2
bethesamplevariances.Letthesamplesizesben
1
andn
2
.Sinceweareinterested
incomparingthetwosamplevariances,weusetheFratio
F=
"
(
s
1
)
2
(
s
1
)
2
#
"
(
s
2
)
2
(
s
2
)
2
#
FhasthedistributionFF(n
1
1,n
2
1)
wheren
1
1arethedegreesoffreedomforthenumeratorandn
2
1arethedegreesoffreedomforthe
denominator.
Ifthenullhypothesisiss
2
1
=s
2
2
,thentheF-RatiobecomesF=
"
(
s
1
)
2
(
s
1
)
2
#
"
(
s
2
)
2
(
s
2
)
2
#
=
(s
1
)
2
(s
2
)
2
.
NOTE
:TheFratiocouldalsobe
(s
2
)
2
(s
1
)
2
.ItdependsonH
a
andonwhichsamplevarianceislarger.
Ifthetwopopulationshaveequalvariances,thens2
1
ands2
2
arecloseinvalueandF=
(s
1
)
2
(s
2
)
2
iscloseto1.But
ifthetwopopulationvariancesareverydifferent,s
2
1
ands
2
2
tendtobeverydifferent,too. Choosings
2
1
as
thelargersamplevariancecausestheratio
(s
1
)
2
(s
2
)
2
tobegreaterthan1.Ifs
2
1
ands
2
2
arefarapart,thenF=
(s
1
)
2
(s
2
)
2
isalargenumber.
Therefore,ifFiscloseto1,theevidencefavorsthenullhypothesis(thetwopopulationvariancesareequal).
ButifFismuchlargerthan1,thentheevidenceisagainstthenullhypothesis.
6
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594
CHAPTER13. FDISTRIBUTIONANDONE-WAYANOVA
Atestoftwovariancesmaybeleft,right,ortwo-tailed.
Example13.4
Twocollege instructorsareinterestedinwhether ornotthereisanyvariationinthe waythey
grademathexams.Theyeachgradethesamesetof30exams.Thefirstinstructor’sgradeshavea
varianceof52.3.Thesecondinstructor’sgradeshaveavarianceof89.9.
Problem
Testtheclaimthatthefirstinstructor’svarianceissmaller. (Inmostcolleges,itisdesirablefor
thevariancesofexamgradestobenearlythesameamonginstructors.)Thelevelofsignificanceis
10%.
Solution
Let1and2bethesubscriptsthatindicatethefirstandsecondinstructor,respectively.
n
1
=n
2
=30.
H
o
:s
2
1
=s
2
2
andH
a
:s
2
1
<s
2
2
Calculatetheteststatistic:Bythenullhypothesis
s
2
1
=s
2
2
,theFstatisticis
F=
"
(
s
1
)
2
(
s
1
)
2
#
"
(
s
2
)
2
(
s
2
)
2
#
=
(s
1
)
2
(s
2
)
2
=
52.3
89.9
=0.5818
Distributionforthetest:F
29,29
wheren
1
1=29andn
2
1=29.
Graph:
Thistestislefttailed.
Drawthegraphlabelingandshadingappropriately.
Figure13.3
Probabilitystatement:p-value=P(F<0.5818)=0.0753
Compareaandthep-value:a=0.10
a>p-value.
Makeadecision:Sincea>p-value,rejectH
o
.
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595
Conclusion:Witha10%levelofsignificance,fromthedata,thereissufficientevidencetoconclude
thatthevarianceingradesforthefirstinstructorissmaller.
TI-83+andTI-84: Press
andarrowoverto
.Arrowdownto
. Press
.Arrowto
andpress
.For
,
,
,and
,enter
p
(52.3),
,
p
(89.9),and
. Press
aftereach. Arrowtos
and<s
. Press
.Arrowdownto
and
press
.F0.5818andp-value =0.0753. . Dotheprocedureagainandtry
insteadof
.
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596
CHAPTER13. FDISTRIBUTIONANDONE-WAYANOVA
13.6Summary
7
 AOne-WayANOVAhypothesistestdeterminesifseveralpopulationmeansareequal.Thedistribu-
tionforthetestistheFdistributionwith2differentdegreesoffreedom.
Assumptions:
a. Eachpopulationfromwhichasampleistakenisassumedtobenormal.
b. Eachsampleisrandomlyselectedandindependent.
c. Thepopulationsareassumedtohaveequalstandarddeviations(orvariances)
 ATestofTwoVarianceshypothesistestdeterminesiftwovariancesarethesame. Thedistribution
forthehypothesistestistheFdistributionwith2differentdegreesoffreedom.
Assumptions:
a. Thepopulationsfromwhichthetwosamplesaredrawnarenormallydistributed.
b. Thetwopopulationsareindependentofeachother.
7
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597
13.7Practice:One-WayANOVA
8
13.7.1StudentLearningOutcome
 ThestudentwillconductaOne-WayANOVAhypothesistest.
13.7.2Given
Supposeagroupisinterestedindeterminingwhetherteenagersobtaintheirdriverslicensesatapproxi-
matelythesameaverageageacrossthecountry. Supposethatthefollowingdataarerandomlycollected
fromfiveteenagersineachregionofthecountry. Thenumbersrepresenttheageatwhichteenagersob-
tainedtheirdriverslicenses.
Northeast
South
West
Central
East
16.3
16.9
16.4
16.2
17.1
16.1
16.5
16.5
16.6
17.2
16.4
16.4
16.6
16.5
16.6
16.5
16.2
16.1
16.4
16.8
x=
________
________
________
________
________
s2=
________
________
________
________
________
Table13.7
13.7.3Hypothesis
Exercise13.7.1
Statethehypotheses.
H
o
:
H
a
:
13.7.4DataEntry
Enterthedataintoyourcalculatororcomputer.
Exercise13.7.2
(Solutiononp.610.)
degreesoffreedom-numerator:df(n)=
Exercise13.7.3
(Solutiononp.610.)
degreesoffreedom-denominator:df(d)=
Exercise13.7.4
(Solutiononp.610.)
Fteststatistic=
Exercise13.7.5
(Solutiononp.610.)
p-value=
8
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598
CHAPTER13. FDISTRIBUTIONANDONE-WAYANOVA
13.7.5DecisionsandConclusions
Statethedecisionsandconclusions(incompletesentences)forthefollowingpreconceivedlevelsofa.
Exercise13.7.6
a=0.05
Decision:
Conclusion:
Exercise13.7.7
a=0.01
Decision:
Conclusion:
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599
13.8Homework
9
D
IRECTIONS
: Useasolutionsheettoconductthefollowinghypothesistests. Thesolutionsheet
canbefoundintheTableofContents14.Appendix.
Exercise13.8.1
(Solutiononp.610.)
Threestudents,Linda,Tuan,andJavier,aregiven5laboratoryratseachforanutritionalexperi-
ment.Eachrat’sweightisrecordedingrams.LindafeedsherratsFormulaA,Tuanfeedshisrats
FormulaB,andJavierfeedshisratsFormulaC.Attheendofaspecifiedtimeperiod,eachratis
weighedagainandthenetgainingramsisrecorded. Usingasignificancelevelof10%,testthe
hypothesisthatthethreeformulasproducethesamemeanweightgain.
WeightsofStudentLabRats
Linda’srats
Tuan’srats
Javier’srats
43.5
47.0
51.2
39.4
40.5
40.9
41.3
38.9
37.9
46.0
46.3
45.0
38.2
44.2
48.6
Table13.8
Exercise13.8.2
A grassrootsgroup opposedtoa a proposed increase e inthe gas s taxclaimed that t the increase
wouldhurtworking-classpeoplethemost,sincetheycommutethefarthesttowork. Suppose
thatthegrouprandomlysurveyed24individualsandaskedthemtheirdailyone-waycommut-
ingmileage. Theresultsarebelow. Usinga5%significancelevel,testthehypothesisthatthe3
meancommutingmileagesarethesame.
working-class
professional(middleincomes)
professional(wealthy)
17.8
16.5
8.5
26.7
17.4
6.3
49.4
22.0
4.6
9.4
7.4
12.6
65.4
9.4
11.0
47.1
2.1
28.6
19.5
6.4
15.4
51.2
13.9
9.3
Table13.9
Exercise13.8.3
(Solutiononp.610.)
RefertoExercise13.8.1. Determinewhetherornotthevarianceinweightgainisstatisticallythe
sameamongJavier’sandLinda’srats.
9
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600
CHAPTER13. FDISTRIBUTIONANDONE-WAYANOVA
Exercise13.8.4
RefertoExercise13.8.2above(Exercise13.8.2).Determinewhetherornotthevarianceinmileage
drivenisstatisticallythesameamongtheworkingclassandprofessional(middleincome)groups.
Forthenexttwoproblems,refertothedatafromTerriVogel’sLogBook.
http://cnx.org/content/m17132/latest/?collection=col10522/latest/
10
Exercise13.8.5
(Solutiononp.610.)
Examinethe7practicelaps.Determinewhetherthemeanlaptimeisstatisticallythesameforthe
7practicelaps,orifthereisatleastonelapthathasadifferentmeantimefromtheothers.
Exercise13.8.6
Examinepracticelaps3and4. Determinewhetherornotthevarianceinlaptimeisstatistically
thesameforthosepracticelaps.
Forthenextfourproblems,refertothefollowingdata.
Thefollowingtableliststhenumberofpagesinfourdifferenttypesofmagazines.
homedecorating
news
health
computer
172
87
82
104
286
94
153
136
163
123
87
98
205
106
103
207
197
101
96
146
Table13.10
Exercise13.8.7
(Solutiononp.610.)
Usingasignificancelevelof5%,testthehypothesisthatthefourmagazinetypeshavethesame
meanlength.
Exercise13.8.8
Eliminateonemagazinetypethatyounowfeelhasameanlengthdifferentthantheothers.Redo
thehypothesistest,testingthattheremainingthreemeansarestatisticallythesame. Useanew
solutionsheet.Basedonthistest,arethemeanlengthsfortheremainingthreemagazinesstatisti-
callythesame?
Exercise13.8.9
Whichtwomagazinetypesdoyouthinkhavethesamevarianceinlength?
Exercise13.8.10
Whichtwomagazinetypesdoyouthinkhavedifferentvariancesinlength?
Exercise13.8.11
(Solutiononp.611.)
Aresearcherwantstoknowifthemeantime(inminutes)thatpeoplewatchtheirfavoritenews
stationarethesame.Supposethatthetablebelowshowstheresultsofastudy.
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