Fundamentals of Statistics - 2e - Chapter11, Angielskie [EN](1)
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11
InferencesonTwo
Samples
CHAPTER
Outline
11.1
InferenceaboutTwoMeans:Dependent
Samples
11.2
InferenceaboutTwoMeans:Independent
Samples
11.3
InferenceaboutTwoPopulationProportions
"
ChapterReview
"
CaseStudy:ControlintheDesignofan
Experiment(OnCD)
DECISIONS
Supposethatyouhavejustreceivedaninheritanceof
$10,000anddecidethatthemoneyshouldbeinvested,rather
thanblownonfrivolousitems.Youhavedecidedthatyouwillin-
vestthemoneyinoneoftwotypesofmutualfunds.Butwhichtype?
SeetheDecisionsprojectonpage528.
PuttingItAllTogether
InChapters9and10wediscussedinferencesregardinga
singlepopulationparameter.Theinferentialmethodspre-
sentedinthesechapterswillbemodifiedslightlyinthis
chaptersothatwecancomparetwopopulationparameters.
Thefirsttwosectionsofthischapterdealwithtesting
forthedifferenceoftwopopulationmeans.Themethods
presentedinthischaptercanbeusedtodetermine
whetheracertaintreatmentresultsinsignificantlydiffer-
entsamplestatistics.Fromadesign-of-experimentspoint
ofview,themethodspresentedinSection11.1areusedto
handlematched-pairsdesigns(Section1.5,pages43–44)
withaquantitative-responsevariable.Forexample,we
mightwanttoknowwhethermarriedcoupleshavesimilar
IQs.Totestthistheory,wecouldrandomlyselect20mar-
riedcouplesanddeterminethedifferenceintheirIQs.
Section11.2presentsinferentialmethodsusedto
handlecompletelyrandomizeddesignswhenthereisa
singletreatmentthathastwolevelsandtheresponse
variableisquantitative.(Section1.5,pages42–43)For
example,wemightrandomlydivide100volunteerswho
havethecommoncoldintotwogroups,acontrolgroup
andanexperimentalgroup.Thecontrolgroupwouldre-
ceiveaplaceboandtheexperimentalgroupwouldre-
ceiveapredeterminedamountofsomeexperimental
drug.Theresponsevariablemightbethetimeuntilthe
coldsymptomsgoaway.
Section11.3discussesthedifferencebetweentwo
populationproportions.Again,wecanuseacompletely
randomizeddesigntocomparetwopopulationpropor-
tions.However,ratherthanhavingaquantitative
responsevariable,wewouldhaveabinomialresponse
variable;thatis,eithertheexperimentalunithasachar-
acteristicoritdoesnot.
507
508
Chapter11 InferencesonTwoSamples
11.1
InferenceaboutTwoMeans:DependentSamples
PreparingforThisSection
Beforegettingstarted,reviewthefollowing:
•
Matched-pairsdesign(Section1.5,pp.43–44)
•
Confidenceintervalsabout unknown(Section
9.2,pp.423–430)
•
Hypothesistestsabout unknown(Section10.3,
pp.480–486)
•
TypeIandTypeIIerrors(Section10.1,pp.457–458)
m
,
s
m
,
s
Objectives
Distinguishbetweenindependentanddependent
sampling
Testhypothesesregardingmatched-pairsdata
Constructandinterpretconfidenceintervalsaboutthe
populationmeandifferenceofmatched-pairsdata
DistinguishbetweenIndependent
andDependentSampling
Toperforminferenceonthedifferenceoftwopopulationmeans,wemustfirst
determinewhetherthedatacomefroman
independent
or
dependent
sample.A
samplingmethodis
independent
whentheindividualsselectedforonesample
donotdictatewhichindividualsaretobeinasecondsample.Asampling
methodis
dependent
whentheindividualsselectedtobeinonesampleareused
todeterminetheindividualstobeinthesecondsample.Forexample,ifweare
conductingastudythatcomparestheIQsofhusbandsandwives,onceahus-
bandisselectedtobeinthestudy,hiswifeisautomaticallymatchedwithhim,so
thisisdependentsampling.Dependentsamplesareoftenreferredtoas
matched-pairs
samples.
InOtherWords
Iftheindividualsintwosamplesare
somehowrelated(husband–wife,siblings,
similarcharacteristics,oreventhesame
person),thesamplingisdependent.
EXAMPLE1
DistinguishingbetweenIndependentandDependent
Sampling
Problem
:
Foreachofthefollowingexperiments,determinewhetherthesam-
plingmethodisindependentordependent.
(a)
ResearcherStevenJ.Sperber,MD,andhisassociateswantedtodetermine
theeffectivenessofanewmedication*inthetreatmentofdiscomfortasso-
ciatedwiththecommoncold.Theyrandomlydivided430subjectsintotwo
groups:Group1receivedthenewmedicationandGroup2receiveda
placebo.Thegoalofthestudywastodeterminewhetherthemeanofthe
symptomassessmentscoresoftheindividualsreceivingthenewmedication
(Group1)waslessthanthatoftheplacebogroup(Group2).
(b)
Inanexperimentconductedinabiologyclass,ProfessorAndyNeillmeas-
uredthetimerequiredfor12studentstocatchafallingmeterstickusing
theirdominanthandandnondominanthand.Thegoalofthestudywasto
determinewhetherthereactiontimeinanindividual’sdominanthandis
differentfromthereactiontimeinthenondominanthand.
Approach
:
Wemustdeterminewhethertheindividualsinonegroupwere
usedtodeterminetheindividualsintheothergroup.Ifso,thesamplingmethod
isdependent.Ifnot,thesamplingmethodisindependent.
Solution
(a)
ThesamplingmethodisindependentbecausetheindividualsinGroup1
werenotusedtodeterminewhichindividualsareinGroup2.
*Themedicationwasacombinationofpseudoephedrineandacetaminophen.Thestudyispub-
lishedinthe
ArchivesofFamilyMedicine
9(2000):979–985.
 Section11.1 InferenceaboutTwoMeans:DependentSamples
509
(b)
Thesamplingmethodisdependentbecausetheindividualsarerelated.The
measurementsforthedominantandnondominanthandareonthesame
individual.
NowWorkProblem5.
Inthissection,wewilldiscussinferenceonthedifferenceoftwomeans
fordependentsampling.Section11.2addressesinferencewhenthesamplingis
independent.
TestHypothesesRegardingMatched-PairsData
Inferenceonmatched-pairsdataisverysimilartoinferenceregardingapopula-
tionmeanwhenthepopulationstandarddeviationisunknown.Recallthatif
thepopulationfromwhichthesamplewasdrawnisnormallydistributedorthe
samplesizeislarge wesaidthat
Ú
30
2
followsStudent’s
t
-distributionwith degreesoffreedom.
Whenanalyzingmatched-pairsdata,wecomputethedifferenceineach
matchedpairandthenperforminferenceonthedifferenceddatausingthe
methodsofSection9.2or10.3.
InOtherWords
Statisticalinferencemethodson
matched-pairsdatausethesame
methodsasinferenceonasingle
populationmeanwith unknown,except
thatthe
differences
areanalyzed.
TestingHypothesesRegardingtheDifferenceofTwoMeans
UsingaMatched-PairsDesign
Totesthypothesesregardingthemeandifferenceofmatched-pairsdata,we
canusethefollowingsteps,providedthat
1.
thesampleisobtainedusingsimplerandomsampling;
2.
thesampledataarematchedpairs;
3.
thedifferencesarenormallydistributedwithnooutliersorthesample
size,
n
,islarge
Ú
30
2
Step1
:
Determinethenullandalternativehypotheses.Thehypothesescan
bestructuredinoneofthreeways,where isthepopulationmeandiffer-
enceofthematched-pairsdata.
Two-Tailed Left-Tailed Right-Tailed
0
:
m
0
:
m
0
:
m
1
:
m
1
:
m
1
:
m
Step2
:
Selectalevelofsignificance dependingontheseriousnessof
makingaTypeIerror.
Step3
:
Computetheteststatistic
whichapproximatelyfollowsStudent’s
t
-distributionwith degreesof
freedom.Thevaluesofandarethemeanandstandarddeviationofthe
differenceddata.
 510
Chapter11 InferencesonTwoSamples
ClassicalApproach
P-ValueApproach
Step4
:
UseTableVtodeterminethecriticalvalue
Step4
:
UseTableVtoestimatethe
P
-valueusing
using degreesoffreedom.
degreesoffreedom.
Two-Tailed Left-Tailed Right-Tailed Two-Tailed Left-Tailed Right-Tailed
Criticalvalue(s) and
Thearea
leftof
t
Thesumof
theareain
thetailsisthe
P
-value
isthe
P
-value
Thearearight
of
t
0
isthe
P
-value
Criticalregion(s)
Critical
Region
Critical
Region
Critical
Region
Critical
Region
"&
t
0
&
&
t
Step5
:
Comparethecriticalvaluewiththeteststatistic.
Step5
:
If rejectthenullhypothesis.
P
-value
6
Two-Tailed Left-Tailed Right-Tailed
If or
t
0
6-t
t
0
6-t
If reject If reject
rejectthenullhypothesis.
thenullhypothesis.
thenullhypothesis.
Step6
:
Statetheconclusion.
Theproceduresjustpresentedare
robust
,whichmeansthatminordepar-
turesfromnormalitywillnotadverselyaffecttheresultsofthetest.Ifthedata
haveoutliers,however,theprocedureshouldnotbeused.
Wewillverifytheassumptionthatthedifferenceddatacomefromapopula-
tionthatisnormallydistributedbyconstructingnormalprobabilityplots.Weuse
boxplotstodeterminewhetherthereareoutliers.Ifthenormalprobabilityplot
indicatesthatthedifferenceddataarenotnormallydistributedortheboxplot
revealsoutliers,nonparametrictestsshouldbeperformed,whicharenotdis-
cussedinthistext.
EXAMPLE2
TestingHypothesesRegardingMatched-PairsData
Problem
:
ProfessorAndyNeillmeasuredthetime(inseconds)requiredtocatch
afallingmeterstickfor12randomlyselectedstudents’dominanthandandnon-
dominanthand.ProfessorNeillwantstoknowifthereactiontimeinanindividual’s
dominanthandislessthanthereactiontimeinhisorhernondominanthand.Con-
ductthetestatthe levelofsignificance.Thedataobtainedarepresented
inTable1.
=
0.05
Table1
Student DominantHand, NondominantHand,
1
0 .177
0.179
2
0 .210
0.202
3
0 .186
0.208
4
0 .189
0.184
5
0 .198
0.215
6
0 .194
0.193
7
0 .160
0.194
8
0 .163
0.160
9
0 .166
0.209
10
0.152
0.164
11
0.190
0.210
12
0.172
0.197
Source
:ProfessorAndyNeill,JolietJuniorCollege
                                               Section11.1 InferenceaboutTwoMeans:DependentSamples
511
Approach
:
Thisisamatched-pairsdesignbecausethevariableismeasured
onthesamesubjectforboththedominantandnondominanthand,thetreat-
mentinthisexperiment.Wecomputethedifferencebetweenthedominanttime
andthenondominanttime.So,forthefirststudentwecompute forthe
secondstudentwecompute andsoon.Ifthereactiontimeinthedom-
inanthandislessthanthereactiontimeinthenondominanthand,wewould
expectthevaluesof tobenegative.Beforeweperformthehypothesis
test,wemustverifythatthedifferencescomefromapopulationthatisapprox-
imatelynormallydistributedwithnooutliersbecausethesamplesizeissmall.
Wewillconstructanormalprobabilityplotandboxplotofthedifferenceddata
toverifytheserequirements.WethenproceedtofollowSteps1through6.
Solution
:
Wecomputethedifferencesas ofdominant
handfor
i
thstudentminustimeofnondominanthandfor
i
thstudent.Weexpect
thesedifferencestobenegative,sowewishtodetermineif Table2
displaysthedifferences.
=
time
6
0.
Table2
StudentDominantHand, NondominantHand, Difference,
1
0 .177
0.179
0.177
-
0.179
=-
0.002
2
0 .210
0.202
0.210
-
0.202
=
0.008
3
0 .186
0.208
-
0.022
4
0 .189
0.184
0.005
5
0 .198
0.215
-
0.017
6
0 .194
0.193
0.001
7
0 .160
0.194
-
0.034
8
0 .163
0.160
0.003
9
0 .166
0.209
-
0.043
CAUTION
Thewaythatwedefinethe
differencedeterminesthedirectionof
thealternativehypothesisinone-
tailedtests.InExample1,weexpect
sothedifference is
expectedtobenegative.Therefore,
thealternativehypothesisis
andwehavealeft-tailed
test.However,ifwecomputedthe
differencesas we’dexpect
thedifferencestobepositive,andwe
havearight-tailedtest!
10 0.152
0.164
-
0.012
11 0.190
0.210
-
0.020
12 0.172
0.197
-
0.025
i
=-
0.158
Wecomputethemeanandstandarddeviationofthedifferencesandobtain
roundedtofourdecimalplacesand roundedtofourdec-
imalplaces.Wemustverifythatthedatacomefromapopulationthatisapproxi-
matelynormalwithnooutliers.Figure1showsthenormalprobabilityplotand
boxplotofthedifferenceddata.
1
:
m
6
0,
d=-
0.0132
=
0.0164
Figure1
"
0.04
"
0.03
"
0.01
Difference
"
0.02 0.00
0.01
Thenormalprobabilityplotisroughlylinearandtheboxplotdoesnotshow
anyoutliers.Wecanproceedwiththehypothesistest.
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