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[ Pobierz całość w formacie PDF ]Inference:From
SamplestoPopulation
PART
CHAPTER8
SamplingDistributions
CHAPTER9
EstimatingtheValueofa
ParameterUsingConfidence
Intervals
CHAPTER10
HypothesisTestsRegardinga
Parameter
CHAPTER11
InferenceonTwoSamples
CHAPTER12
AdditionalInferential
Procedures
InChapter1,wepresentedthefollowingprocessof
statistics:
Iftheinformation(data)collectedisfromapopula-
tion,wecanusethesummariesobtainedinStep3to
drawconclusionsaboutthepopulationbeingstudiedand
thestatisticalprocessisover.
However,itisoftendifficultorimpossibletogainac-
cesstopopulations,sotheinformationobtainedinStep2
isoftensampledata.Thesampledataareusedtomake
inferencesaboutthepopulation.Forexample,wemight
computeasamplemeanfromtheinformationcollected
inStep2andusethisinformationtodrawconclusionsre-
gardingthepopulationmean.Thelastpartofthistext
discusseshowsampledataareusedtodrawconclusions
aboutpopulations.
Step1
:
Identifyaresearchobjective.
Step2
:
Collecttheinformationneededtoanswerthe
questionsposedinStep1.
Step3
:
Organizeandsummarizetheinformation.
Step4
:
Drawconclusionsfromtheinformation.
ThemethodsforconductingSteps1and2werediscussed
inChapter1.ThemethodsforconductingStep3were
discussedinChapters2through4.Wetookabreakfrom
thestatisticalprocessinChapters5through7sothatwe
coulddevelopskillsthatallowustotackleStep4.
373
SamplingDistributions
CHAPTER
Outline
8.1
DistributionoftheSampleMean
8.2
DistributionoftheSampleProportion
"
ChapterReview
"
CaseStudy:SamplingDistributionofthe
Median(OnCD)
DECISIONS
TheAmericanTimeUseSurveyisasurveyofadultAmericanscon-
ductedbytheBureauofLaborStatistics.Thepurposeofthesurvey
istolearnhowAmericansallocatetheirtimeduringaday.Asare-
porterfortheschoolnewspaper,youwishtofileareportthatcom-
paresthetypicalstudentatyourschooltotherestofAmericans.
SeetheDecisionsprojectonpage388.
PuttingItAllTogether
InChapters6and7,welearnedaboutrandomvariables
andtheirprobabilitydistributions.Arandomvariableisa
numericalmeasureoftheoutcometoaprobability
experiment.Aprobabilitydistributionprovidesawayto
assignprobabilitiestotherandomvariable.Fordiscrete
randomvariables,wediscussedthebinomialprobability
distribution.Weassignedprobabilitiesusingaformula.
Forcontinuousrandomvariables,wediscussedthenor-
malprobabilitydistribution.Tocomputeprobabilitiesfor
anormalrandomvariable,wefoundtheareaundera
normaldensitycurve.
Inthischapter,wecontinueourdiscussionofproba-
bilitydistributionswherestatistics,suchas willbe
therandomvariable.Statisticsarerandomvariables
becausethevalueofastatisticvariesfromsampleto
sample.Forthisreason,statisticshaveprobabilitydistri-
butionsassociatedwiththem.Forexample,thereisa
probabilitydistributionforthesamplemean,sample
variance,andsoon.Weuseprobabilitydistributionsto
makeprobabilitystatementsregardingthestatistic.So
thischapterdiscussestheshape,center,andspreadof
statisticssuchas
x
374
Section8.1DistributionoftheSampleMean
375
8.1
DistributionoftheSampleMean
PreparingforThisSection
Beforegettingstarted,reviewthefollowing:
•
Simplerandomsampling(Section1.2,pp.16–19)
•
Themean(Section3.1,pp.107–110)
•
Thestandarddeviation(Section3.2,pp.129–130)
•
Applicationsofthenormaldistribution(Section7.3,
pp.345–349)
Objectives
Understandtheconceptofasamplingdistribution
Describethedistributionofthesamplemeanfor
samplesobtainedfromnormalpopulations
Describethedistributionofthesamplemeanforsam-
plesobtainedfromapopulationthatisnotnormal
SupposethatthegovernmentwantedtoestimatethemeanincomeofallU.S.
households.Oneapproachthegovernmentcouldtakeistoliterallysurveyeach
householdintheUnitedStatestodeterminethepopulationmean, This
wouldbeaveryexpensiveandtime-consumingsurvey!
Asecondapproachthatthegovernmentcould(anddoes)takeistosurvey
arandomsampleofU.S.householdsandusetheresultsofthesurveytoesti-
matethemeanhouseholdincome.ThisisdonethroughtheAmericanCommu-
nitySurvey.Thesurveyisadministeredtoapproximately250,000randomly
selectedhouseholdseachmonth.Amongthemanyquestionsonthesurvey,re-
spondentsareaskedtoreporttheincomeofeachindividualinthehousehold.
Fromthisinformation,thefederalgovernmentobtainsasamplemeanhouse-
holdincomeforU.S.households.Forexample,in2003themeanannualhouse-
holdincomeintheUnitedStateswasestimatedtobe The
governmentmightinferfromthisresultthatthemeanannualhouseholdin-
comeof
all
U.S.householdsin2003was Thistypeofstatementis
anexampleof
statisticalinference
usinginformationfromasampletodraw
conclusionsaboutapopulation.
ThehouseholdsthatwereadministeredtheAmericanCommunitySur-
veyweredeterminedbychance(randomsampling).Asecondrandomsample
ofhouseholdswouldlikelyleadtoadifferentsamplemeansuchas
andathirdrandomsampleofhouseholdswouldlikelyleadtoa
thirddistinctsamplemeansuchas Becausethehouseholdsare
selectedbychance,thesamplemeanofhouseholdincomeisalsodetermined
bychance.Weconcludefromthisthatthereisvariabilityinourestimates.
Thisvariabilityleadstouncertaintyastowhetherourestimatesarecorrect.
Therefore,weneedawaytoassessthereliabilityofinferencesmadeabouta
populationbasedonsampledata.
Themeasureofreliabilityisactuallyastatementofprobability.Probability
describeshowlikelyanoutcomeistooccur.Thegoalofthischapteristolearn
thedistributionofstatisticssuchasthesamplemeansothatourestimatesare
accompaniedbystatementsthatindicatethelikelihoodthatourmethodsare
accurate.
=
$58,036.
=
$58,036.
=
$58,132,
=
$58,095.
UnderstandtheConceptofaSampling
Distribution
Ingeneral,thesamplingdistributionofastatisticisaprobabilitydistributionfor
allpossiblevaluesofthestatisticcomputedfromasampleofsize
n
.The
samplingdistributionofthesamplemean
istheprobabilitydistributionofall
possiblevaluesoftherandomvariablecomputedfromasampleofsize
n
from
apopulationwithmeanandstandarddeviation
s
376
Chapter8SamplingDistributions
Theideabehindobtainingthesamplingdistributionofthemeanisasfollows:
Step1
:
Obtainasimplerandomsampleofsize
n
InOtherWords
Ifthenumberofindividualsina
populationisapositiveinteger,wesay
thepopulationisfinite.Otherwise,the
populationisinfinite.
Step2
:
Computethesamplemean.
Step3
:
Assumingthatwearesamplingfromafinitepopulation,repeat
Steps1and2untilallsimplerandomsamplesofsize
n
havebeenobtained.
Note:
Onceaparticularsampleisobtained,itcannotbeobtainedasecond
time.
Wepresentanexampletoillustratetheideabehindasamplingdistribution.
EXAMPLE1
ASamplingDistribution
Problem
:
Onesemester,ProfessorGoehlhadasmallstatisticsclassofseven
students.Heaskedthemtheagesoftheircarsandobtainedthefollowingdata:
2,4,6,8,4,3,7
Constructasamplingdistributionofthemeanforsamplesofsize What
istheprobabilityofobtainingasamplemeanbetween4and6years,inclusive;
thatis,whatis
Approach
:
WefollowSteps1to3listedabovetoconstructtheprobability
distribution.
Solution
:
Therearesevenindividualsinthepopulation.Weareselecting
themtwoatatimewithoutreplacement.Therefore,thereare samples
ofsize Welistthese21samplesalongwiththesamplemeansinTable1.
=
2.
=
21
=
2.
Table1
Sample SampleMean Sample SampleMean Sample SampleMean
2,4
3 4,8
6 6,7
6 .5
2,6
4 4,4
4 8,4
6
2,8
5 4,3
3 .5 8,3
5 .5
2,4
3 4,7
5 .5 8,7
7 .5
2,3
2 .5 6,8
7 4,3
3 .5
2,7
4 .5 6,4
5 4,7
5 .5
4,6
5 6,3
4 .5 3,7
5
Table2displaysthesamplingdistributionofthesamplemean,
x
Table2
SampleMean Frequency Probability SampleMean Frequency Probability
1
21
3
21
2.5 1
5 .5 3
2
21
2
21
3
2
6
2
2
21
1
21
3.5 2
6 .5 1
2
21
1
21
4
2
7
1
2
21
1
21
4.5 2
7 .5 1
4
21
5
4
Section8.1DistributionoftheSampleMean
377
FromTable2wecancompute
2
21
+
2
21
+
4
21
+
3
21
+
21
=
13
21
=
0.619
Ifwetook10simplerandomsamplesofsize2fromthispopulation,about6of
themwouldresultinsamplemeansbetween4and6years,inclusive.
Thesamplemeanwiththehighestprobabilityis Thisshouldnotbe
surprisingsincethepopulationmeanofthedatainExample1is
roundedtoonedecimalplace.Figure1isaprobabilityhistogramofthesam-
plingdistributionforthesamplemeangiveninTable2.
=
5.
=
4.9,
Figure1
ProbabilityDistributionoftheSampleMean
0.2
0.15
0.1
0.05
2.5 3
3 .5 4
4 .5 5
5 .5 6
6 .5 7
7 .5
NowWorkProblem31.
SampleMean
In-ClassActivity:SamplingDistributions
Randomlyselectsixstudentsfromtheclasstotreatasapopulation.Chooseaquan-
titativevariable(suchaspulserate,age,ornumberofsiblings)touseforthisactivi-
ty,andgatherthedataappropriately.Compute forthepopulation.Dividethe
classintofourgroupsandhaveonegrouplistallsamplesofsize another
grouplistallsamplesofsize andothergroupslistallsamplesofsize
and Eachgroupshoulddothefollowing:
(a)
Computethesamplemeanofeachsample.
(b)
Formtheprobabilitydistributionforthesamplemean.
(c)
Drawaprobabilityhistogramoftheprobabilitydistribution.
(d)
Verifythat
Comparethespreadineachprobabilitydistributionbasedontheprobabilityhis-
togram.Whatdoesthisresultimplyaboutthestandarddeviationofthesample
mean?
=
2,
=
3,
=
5.
DescribetheDistributionoftheSampleMean
forSamplesObtainedfromNormal
Populations
ThepointofExample1istohelpyourealizethatstatisticssuchasarerandom
variablesandthereforehaveprobabilitydistributionsassociatedwiththem.In
practice,asinglerandomsampleofsize
n
isobtainedfromapopulation.The
probabilitydistributionofthesamplestatistic(orsamplingdistribution)isde-
terminedfromstatisticaltheory.Wewillusesimulationtohelpjustifytheresult
thatstatisticaltheoryprovides.Weconsidertwopossibilities.Inthefirstcase,we
samplefromapopulationthatisknowntobenormallydistributed.Inthesec-
ondcase,wesamplefromadistributionthatisnotnormallydistributed.
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