section 7.3 completed - washington-liberty

Section 7.3 Completed

1

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

2

ThisgivesusinformationonthecenterandspreadforEVERYpossiblesamplingdistributionofx-bar,but...

Whatabouttheshape?

IfthedistributionofthePOPULATIONisNormal,thenthesamplingdistributionofalsohasaNormaldistribution

IfindividualobservationshavetheN(μ,σ)distribution,

thenthesamplingdistributionoffromsamplesof

sizenhastheN(μ,)distribution.

Section 7.3 Completed

3

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

4

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

5

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

6

ThescoresofstudentsontheACTcollegeentranceexaminationin2001hadameanscoreof21withstandarddeviationof4.7.Thedistributionofscoresisslightlyskewedleft.

a)Whatistheapproximateprobabilitythatasinglestudentrandomlychosenfromallthosetakingthetestscores23orhigher?

b)NowtakeanSRSof50studentswhotookthetest.Whatistheapproximateprobabilitythatthemeanscoreofthesestudentsis23orhigher?

7.3Homework:p.454#s43-49all,51-65odd,66,67,68