Download Section 7-3 SAMPLE MEANS POINT ESTIMATE

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Section 7-3
Estimating a Population Mean
SAMPLE MEANS
1. Formanypopulations,thedistributionof
samplemeans ̅ tendstobemore
consistent(withlessvariation)thanthe
distributionsofothersamplestatistics.
2. Forallpopulations,thesamplemean ̅ is
anunbiasedestimatorofthepopulation
meanµ,meaningthatthedistributionof
samplemeanstendstocenteraboutthe
valueofthepopulationmeanµ.
POINT ESTIMATE
PointEstimate: Thesamplemean ̅ isthe
bestpointestimate (orsinglevalueestimate)
ofthepopulationmean .
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COMMENT
Itisrarethatwewanttoestimatethe
unknownvalueofapopulationmeanbutwe
somehowknowthevalueofthepopulation
standarddeviation .Therealisticsituationis
that isnotknown.(Webeginthissectionby
consideringthismorerealisticscenario.)
When isnotknown,weconstructthe
confidenceintervalbyusingtheStudent
distributioninsteadofthestandardnormal
distribution.
ASSUMPTIONS FOR CONFIDENCE
INTERVAL OF MEAN WITH
σ NOT KNOWN
1. Thesampleisasimplerandomsample.
2. Eitherorbothofthefollowingconditions
aresatisfied:
• Thepopulationisnormally
distributed
• n >30
THE STUDENT t DISTRIBUTION
Ifapopulationhasanormaldistribution,then
thedistributionof
̅
isaStudentt distribution forallsamplesofsize
.TheStudent distributionisoftenreferredto
asthe distribution.
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DEGREES OF FREEDOM
Findingacriticalvalue / requiresavaluefor
thedegreesoffreedom (ordf).Ingeneral,
thenumberofdegreesoffreedomfora
collectionofsampledataisthenumberof
samplevaluesthatvaryaftercertainrestraints
havebeenimposedonthedatavalues.Forthe
methodsofthissection,thenumberofdegrees
offreedomisthesamplesizeminus1;thatis,
degreesoffreedom
1
FINDING THE CRITICAL
VALUE
Acriticalvalue / canbefoundusingTableA‐3which
isfoundonpage586,insidethebackcover,andonthe
FormulasandTablescard.Ifthetabledoesnotinclude
thenumberofdegreesoffreedomthatyouneed,you
could
• usetheclosestvalue
• beconservativeandusingthenextlowernumber
ofdegreesoffreedom
• interpolate.Forexample,ifyouhave55degrees
offreedom,youcouldfindthemeanofthe
criticalvaluesfor50and60.
Tokeepthingssimple,wewillusetheclosestvalue.
MARGIN OF ERROR ESTIMATE
OF µ (WITH σ NOT KNOWN)
/
·
where(1
)istheconfidenceleveland
has
1 degreesoffreedom.
/
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CONFIDENCE INTERVAL
ESTIMATE OF THE POPULATION
MEAN μ (WITH σ NOT KNOWN)
̅
where
̅
/
·
CONSTRUCTING A CONFIDENCE
INTERVAL FOR μ (σ NOT KNOWN)
1.
2.
Verifythatthetworequiredassumptionsaremet.
With unknown(asisusuallythecase),use
1
degreesoffreedomandrefertoTableA‐3tofindthe
criticalvalue / thatcorrespondstothedesired
confidenceinterval.(Fortheconfidencelevel,refer
to“AreainTwoTails.”)
3.
Evaluatethemarginoferror
4.
Findthevaluesof ̅
and ̅
.Substitutethese
inthegeneralformatoftheconfidenceinterval: ̅
̅
.
Roundtheresultusingthesameround‐offruleon
thefollowingslide.
5.
/
·
ROUND-OFF RULE FOR
CONFIDENCE INTERVALS USED
TO ESTIMATE μ
1. Whenusingtheoriginalsetofdata to
constructtheconfidenceinterval,round
theconfidenceintervallimitstoonemore
decimalplace thanisusedfortheoriginal
dataset.
2. Whentheoriginalsetofdataisunknown
andonlythesummarystatistics , ̅ ,
areused,roundtheconfidenceinterval
limitstothesamenumberofplaces as
usedforthesamplemean.
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FINDING A CONFIDENCE
INTERVAL FOR µ WITH TI-83/84
1.
2.
3.
4.
5.
6.
7.
8.
9.
Select STAT.
Arrow right to TESTS.
Select 8:TInterval….
Select input (Inpt) type: Data or Stats. (Most of
the time we will use Stats.)
Enter the sample mean, x.
Enter the sample standard deviation, Sx.
Enter the size of the sample, n.
Enter the confidence level (C-Level).
Arrow down to Calculate and press ENTER.
PROPERTIES OF THE
STUDENT t DISTRIBUTION
1. TheStudent distributionisdifferentfor
differentsamplesizes(seeFigurebelow
forthecases
3 and
12).
PROPERTIES OF THE
STUDENT t DISTRIBUTION (CONTINUED)
2.
3.
4.
5.
TheStudent distributionhasthesamegeneral
symmetricbellshapeasthenormaldistribution
butitreflectsthegreatervariability(withwider
distributions)thatisexpectedwithsmallsamples.
TheStudentt distributionhasameanof
0
(justasthestandardnormaldistributionhasa
meanof
0).
ThestandarddeviationoftheStudentt
distributionvarieswiththesamplesizeandis
greaterthan1(unlikethestandardnormal
distribution,whichhasa
1).
Asthesamplesize getslarger,theStudent
distributiongetsclosertothenormaldistribution.
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ESTIMATING A MEAN WHEN
σ IS KNOWN
Requirements:
1. Thesampleisasimplerandomsample.
2. Eitherorbothoftheseconditionsaresatisfied:The
populationisnormallydistributedor
30.
ConfidenceInterval:
̅
̅
wherethemarginoferror isfoundfromthefollowing:
/
⋅
Note:Thecriticalvalue / isfoundfromTableA‐2(the
standardnormaldistribution).
FINDING A CONFIDENCE
INTERVAL FOR µ WITH TI-83/84
1.
2.
3.
4.
5.
6.
7.
8.
9.
SelectSTAT.
ArrowrighttoTESTS.
Select7:ZInterval….
Selectinput(Inpt)type:Data orStats.(Mostof
thetimewewilluseStats.)
Enterthestandarddeviation,σ.
Enterthesamplemean, .
Enterthesizeofthesample,n.
Entertheconfidencelevel(C‐Level).
ArrowdowntoCalculate andpressENTER.
SAMPLE SIZE FOR
ESTIMATING µ
⁄
·
where zα/2 = criticalz scorebasedondesired
confidencelevel
E = desiredmarginoferror
σ = populationstandarddeviation
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ROUND-OFF RULE FOR SAMPLE
SIZE n
Whenfindingthesamplesize ,iftheuseof
theformulaonthepreviousslidedoesnot
resultinawholenumber,alwaysincrease the
valueof tothenextlarger wholenumber.
FINDING THE SAMPLE SIZE
WHEN σ IS UNKNOWN
1. Usetherangeruleofthumb(seeSection3‐3)
toestimatethestandarddeviationasfollows:
range/4.
2. Startthesamplecollectionprocesswithout
knowing and,usingthefirstseveralvalues,
calculatethesamplestandarddeviations and
useitinplaceofσ.Theestimatedvalueof
canthenbeimprovedasmoresampledataare
obtained,andtherequiredsamplesizecanbe
adjustedasyoucollectmoresampledata.
3. Estimatethevalueofσ byusingtheresultsof
someotherearlierstudy.
CHOOSING THE APPROPRIATE
DISTRIBUTION
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CHOOSING BETWEEN z AND t
Conditions
Method
σ notknownandnormallydistributed
population
or
σ notknownand
30
UseStudent
distribution
σ knownandnormallydistributed
population
or
σ knownand
30
Usenormal( )
distribution
Populationisnotnormallydistributed
and
30.
Useanonparametric
methodor
bootstrapping
FINDING A POINT ESTIMATE AND E
FROM A CONFIDENCE INTERVAL
Pointestimateofµ:
̅
upperconfidencelimit
lowerconfidencelimit
2
Marginoferror:
upperconfidencelimit
lowerconfidencelimit
2
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