SAMPLE MEANS 8/4/2014

8/4/2014
SAMPLE MEANS
Section 7-3
Estimating a Population Mean
1. Formanypopulations,thedistributionof
samplemeans ̅ tendstobemore
consistent(withlessvariation)thanthe
distributionsofothersamplestatistics.
2. Forallpopulations,thesamplemean ̅ is
anunbiasedestimatorofthepopulation
meanµ,meaningthatthedistributionof
samplemeanstendstocenteraboutthe
valueofthepopulationmeanµ.
POINT ESTIMATE
COMMENT
PointEstimate: Thesamplemean ̅ isthe
bestpointestimate (orsinglevalueestimate)
ofthepopulationmean .
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
THE STUDENT t DISTRIBUTION
1. Thesampleisasimplerandomsample.
2. Eitherorbothofthefollowingconditions
aresatisfied:
• Thepopulationisnormally
distributed
• n >30
Ifapopulationhasanormaldistribution,then
thedistributionof
̅
isaStudentt distribution forallsamplesofsize
.TheStudent distributionisoftenreferredto
asthe distribution.
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FINDING THE CRITICAL
VALUE
DEGREES OF FREEDOM
Findingacriticalvalue / requiresavaluefor
thedegreesoffreedom (ordf).Ingeneral,
thenumberofdegreesoffreedomfora
collectionofsampledataisthenumberof
samplevaluesthatvaryaftercertainrestraints
havebeenimposedonthedatavalues.Forthe
methodsofthissection,thenumberofdegrees
offreedomisthesamplesizeminus1;thatis,
degreesoffreedom
1
MARGIN OF ERROR ESTIMATE
OF µ (WITH σ NOT KNOWN)
/
CONFIDENCE INTERVAL
ESTIMATE OF THE POPULATION
MEAN μ (WITH σ NOT KNOWN)
∙
̅
where(1
)istheconfidenceleveland
has
1 degreesoffreedom.
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.
/
̅
/
CONSTRUCTING A CONFIDENCE
INTERVAL FOR μ (σ NOT KNOWN)
5.
Acriticalvalue / canbefoundusingTableA‐3which
isfoundonpage586,insidethebackcover,andonthe
FormulasandTablescard.Ifthetabledoesnotinclude
thenumberofdegreesoffreedomthatyouneed,you
could
• usetheclosestvalue
• beconservativeandusingthenextlowernumber
ofdegreesoffreedom
• interpolate.Forexample,ifyouhave55degrees
offreedom,youcouldfindthemeanofthe
criticalvaluesfor50and60.
Tokeepthingssimple,wewillusetheclosestvalue.
∙
where
/
∙
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 (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).
Asthesamplesizen getslarger,theStudent
distributiongetsclosertothenormaldistribution.
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.
PROPERTIES OF THE
STUDENT t DISTRIBUTION
1. TheStudent distributionisdifferentfor
differentsamplesizes(seeFigurebelow
forthecases
3 and
12).
ESTIMATING A MEAN WHEN
σ IS KNOWN
Requirements:
1. Thesampleisasimplerandomsample.
2. Eitherorbothoftheseconditionsaresatisfied:The
populationisnormallydistributedor
30.
ConfidenceInterval:
̅
̅
wherethemarginoferror isfoundfromthefollowing:
/
⋅
Note:Thecriticalvalue / isfoundfromTableA‐2(the
standardnormaldistribution).
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.
CHOOSING THE APPROPRIATE
DISTRIBUTION
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 BETWEEN z AND t
Conditions
Method
σ notknownandnormallydistributed
population
or
σ notknownand
30
UseStudent
distribution
σ knownandnormallydistributed
population
or
σ knownand
30
Populationisnotnormallydistributed
and
30.
Usenormal( )
distribution
Useanonparametric
methodor
bootstrapping
FINDING A POINT ESTIMATE AND E
FROM A CONFIDENCE INTERVAL
Pointestimateofµ:
̅
upperconfidencelimit
lowerconfidencelimit
2
Marginoferror:
upperconfidencelimit
lowerconfidencelimit
2
4