Chapter 21 The Accuracy of Percentages Confidence Intervals Example
Therearenearly25,000registeredvotersinthecounty.Use
asimplerandomsampleofsize400toestimatethe
percentageofregisteredvoterswhofavorthesaleof
municipalbondsfortheconstructionoftwonewelementary
schools.
1.Isthisachanceprocess?Whatistheboxmodel?
2.Itturnsoutthat55%ofthevotersinyoursamplefavor
thesaleofbonds.Whatisyourestimatefortheentire
population?
3.Howaccurateisyourestimate?Couldanothersample
yieldadifferentresult?
4.IfyouknewtheboxSD,couldyoufindtheSEforthe
percentageof1'sdrawn?
5.CanyouestimatetheboxSD?
6.UseyourestimateoftheBoxSDtofind(approximately)
theSEforthepercentageof1'sdrawn?
7.Builda95%confidenceintervalforthetruepopulation
percentageinfavorofbonds.
8.Builda68%confidenceintervalforthetruepopulation
percentageinfavorofbonds.
9.Ifeachstudentintheclassweretosample400voters
andconstructa68%confidenceinterval,abouthowmany
oftheseintervalswouldactuallycontainthetruepopulation
percentageinfavorofbonds?
TheBootstrap
Whenwedonotknowwhatisinthebox,weestimatetheSD
oftheboxbyusingtheSDofthesample.
ConfidenceIntervals
A95%confidenceintervalforthepopulationpercentageis
givenby
Samplepercentage±2(SEfor%)
Theconfidenceintervalisvalidifthenumberofdrawsis
largeenough.
Foraconfidenceintervalwithadifferentconfidencelevel,
normaltablesareusedtofindthemultiplier.
An80%confidenceintervalis
Samplepercentage±1.3(SEfor%)
A90%confidenceintervalis
Samplepercentage±1.65(SEfor%)
Example.Ahealthinspectortakesarandomsampleof300
10‐year‐oldsinacity,andfindsthat73%ofthemhavehad
chicken‐pox.Finda90%confidenceintervalforthe
percentageof10‐year‐oldsinthecitywhohavehadchicken‐
pox.
Example.SenatorSmithwantstoknowifsheshouldseek
re‐election.Fromapopulationof1,000,000voters,asimple
randomsampleofsize2500istaken.Inthesample,53%of
thevotersfavorSenatorSmith.Asherstatisticalconsultant,
whatshouldyoutellher?