Download - Section 7.3 Completed - Washington-Liberty
Section 7.3 Completed
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Thesamplingdistributionofisofparticularinterest:
IfisthemeanofanSRSofsize ndrawnfromalargepopulationwithmeanμandstandarddeviationσ,then
themeanofthesamplingdistributionofisμandits
standarddeviationis
ThismeansthatthesamplingdistributionofhastheSAMEmeanasthepopulationandhasaSMALLERstandarddeviationthanthepopulation(unlessofcoursethesamplesizeis1...)
FOREXAMPLE:
Juanmakesameasurementinachemistrylaboratoryandrecordstheresultinhislabreport.Thestandarddeviationofstudents'labmeasurementsisσ=10milligrams.Juanrepeatsthemeasurement3timesandrecordsthemeanofhis3measurements.
a)WhatisthestandarddeviationofJuan'smeanresult?(Thatis,ifJuankeptonmaking3measurementsandaveragingthem,whatwouldbethestandarddeviationofallhiss?)
b)HowmanytimesmustJuanrepeatthemeasurementtoreducethestandarddeviationofto5?
c)Explaintosomeonewhoknowsnostatisticstheadvantageofreportingtheaverageofseveralmeasurementsratherthantheresultofasinglemeasurement.
Section 7.3 Completed
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ThisgivesusinformationonthecenterandspreadforEVERYpossiblesamplingdistributionofx-bar,but...
Whatabouttheshape?
IfthedistributionofthePOPULATIONisNormal,thenthesamplingdistributionofalsohasaNormaldistribution
IfindividualobservationshavetheN(μ,σ)distribution,
thenthesamplingdistributionoffromsamplesof
sizenhastheN(μ,)distribution.
Section 7.3 Completed
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Thescoresof12th-gradestudentsontheNationalAssessmentofEducationalProgressyear2000mathematicstesthaveadistributionthatisapproximatelyNormalwithmeanμ=300pointsandstandarddeviationσ=35points.
a)Chooseone12thgraderatrandom.Whatistheprobabilitythathisorherscoreishigherthan325?Higherthan350?
b)NowchooseanSRSoffour12th-gradersandcalculatetheirmeanscore.Ifyoudidthismanytimes,whatwouldbethemeanandstandarddeviationofallthe-values?
c)WhatistheprobabilitythatthemeanscoreforyourSRSishigherthan325?Higherthan350?
TheweightsofGrannySmithapplesfromalargeorchardareNormallydistributedwithameanof380gandastandarddeviationof28g.
a)Asingleappleisselectedatrandomfromthisorchard.Whatistheprobabilitythatitweighsmorethan400g?
b)Threeapplesareselectedatrandomfromthisorchard.Whatistheprobabilitythattheirmeanweightisgreaterthan400g?
c)Explainwhytheprobabilitiesin(a)and(b)arenotequal.
Section 7.3 Completed
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TheweightsofnewbornchildrenintheUnitedStatesvaryaccordingtotheNormaldistributionwithmean7.5poundsandstandarddeviation1.25pounds.Thegovernmentclassi]iesanewbornashavinglowbirthweightiftheweightislessthan5.5pounds.
a)Whatistheprobabilitythatababychosenatrandomweighslessthan5.5poundsatbirth?
b)Youchoosethreebabiesatrandomandcomputetheirmeanweight.Whatistheprobabilitythattheiraveragebirthweightislessthan5.5pounds?
*****CENTRALLIMITTHEOREM(CLT)*****
(OneoftheMOSTimportanttheoremsinStatistics-KNOWANDUNDERSTANDTHIS!!!!!!)
DrawanSRSofsizenfromANYpopulationwithmeanμ
andstandarddeviationσ.WhennisLARGE,the
samplingdistributionofthesamplemeanis
approximatelyNORMAL!!!withmeanμandstandard
deviation.
Section 7.3 Completed
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FOREXAMPLE:
KEEPINMIND:
1.YouMUSThaveasamplethatislargeenoughtousetheCentralLimitTheorem(≥30isusuallygood-wewillspendmoretimeonthislater...).
2.NormaldistributionsareGREATforcalculatingprobabilities(likewehaveseenmanytimesbefore)sotheCLTisaVERYpowerfultool
http://onlinestatbook.com/stat_sim/sampling_dist/index.html
FOREXAMPLE:
Aninsurancecompanyknowsthatthemeanlossfrom]irefortheentirepopulationofhomeownersisμ=$250andthestandarddeviationoflossisσ=$1000.Thedistributionoflossesisstronglyskewedtotheright:manypolicieshave$0loss,butafewhaveverylargelosses.Ifthecompanysells10,000policies,whatistheapproximateprobabilitythattheaveragelosswillbegreaterthan$275?
Whatiftheyhadonlysold20policies?
Section 7.3 Completed
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ThescoresofstudentsontheACTcollegeentranceexaminationin2001hadameanscoreof21withstandarddeviationof4.7.Thedistributionofscoresisslightlyskewedleft.
a)Whatistheapproximateprobabilitythatasinglestudentrandomlychosenfromallthosetakingthetestscores23orhigher?
b)NowtakeanSRSof50studentswhotookthetest.Whatistheapproximateprobabilitythatthemeanscoreofthesestudentsis23orhigher?
7.3Homework:p.454#s43-49all,51-65odd,66,67,68