Ch.8 SamplingDistributions
8.1 DistributionoftheSampleMean
1 Describethedistributionofthesamplemean:normalpopulation.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Determineμxandσxfromthegivenparametersofthepopulationandthesamplesize.Roundtheanswertothe
nearestthousandthwhereappropriate.
1) μ=54
,
σ=32
,
n=64
A) μx=54
,
σx=4B)μx
=54
,
σx
=32
C) μx=54
,
σx=0.5 D) μx
=6.75
,
σx
=4
2) μ=28
,
σ=8
,
n=22
A) μx=28
,
σx=1.706 B) μx
=28
,
σx
=0.364
C) μx=28
,
σx=8D)μx
=16.166
,
σx=1.706
Provideanappropriateresponse.
3) Whatarethevaluesofμxandσxforthesamplingdistributionofthesamplemeanshown?
350 390 430
A) μx=390
,
σx=40 B) μx
=40
,
σx
=390
C) μx=390
,
σx=120 D) μx
=390
,
σx
=80
4) Whatarethevaluesofμxandσxforthesamplingdistributionofthesamplemeanshown?
7.96 8 8.04
A) μx=8
,
σx=0.04 B) μx
=0.04
,
σx
=8
C) μx=8
,
σx=0.12 D) μx
=8
,
σx
=0.08
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5) Thesamplingdistributionofthesamplemeanisshown.Ifthesamplesizeisn=36
,
whatisthestandard
deviationofthepopulationfromwhichthesamplewasdrawn?Roundtothenearestthousandthwhere
appropriate.
420 470 520
A) σ=300 B) σ=1800 C) σ=8.333 D) σ=1.389
6) Thesamplingdistributionofthesamplemeanisshown.Ifthesamplesizeisn=16
,
whatisthestandard
deviationofthepopulationfromwhichthesamplewasdrawn?Roundtothenearestthousandthwhere
appropriate.
8.94 9 9.06
A) σ=0.24 B) σ=0.96 C) σ=0.015 D) σ=0.004
7) Supposeapopulationhasameanof7forsomecharacteristicofinterestandastandarddeviationof9.6.A
sampleisdrawnfromthispopulationofsize64.Whatisthestandarderrorofthemean?
A) 1.2 B) 0.7 C) 0.15 D) 3.3
8) Supposeapopulationhasameanweightof180poundsandastandarddeviationof25pounds.Asampleof
100itemsisdrawnfromthispopulation.Whatisthestandarderrorofthemean?
A) 2.5 B) 1.8 C) 18.0 D) 0.25
9) Thenumberofviolentcrimescommittedinadaypossessesadistributionwithameanof2.2crimesperday
andastandarddeviationof6crimesperday.Arandomsampleof100dayswasobserved,andthemean
numberofcrimesforthesamplewascalculated.Describethesamplingdistributionofthesamplemean.
A) approximatelynormalwithmean=2.2 andstandarddeviation=0.6
B) approximatelynormalwithmean=2.2 andstandarddeviation=6
C) shapeunknownwithmean=2.2andstandarddeviation=6
D) shapeunknownwithmean=2.2andstandarddeviation=0.6
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
10) Arandomsampleofsizenistobedrawnfromapopulationwithμ=600 andσ =300.Whatsizesample
wouldbenecessaryinordertoensureastandarderrorof20?
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MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
11) Theamountofcornchipsdispensedintoa16–ouncebagbythedispensingmachinehasbeenidentifiedas
possessinganormaldistributionwithameanof16.5ouncesandastandarddeviationof0.2ounce.Suppose
100bagsofchipswererandomlyselectedfromthisdispensingmachine.Findtheprobabilitythatthesample
meanweightofthese100bagsexceeded16.6ounces.
A) approximately0 B) 0.1915 C) 0.3085 D) 0.6915
12) Theaveragescoreofallgolfersforaparticularcoursehasameanof75 andastandarddeviationof4.5.
Suppose81golfersplayedthecoursetoday.Findtheprobabilitythattheaveragescoreofthe81golfers
exceeded76.
A) 0.0228 B) 0.1293 C) 0.4772 D) 0.3707
13) Oneyear,professionalsportsplayerssalariesaveraged$1.9 millionwithastandarddeviationof$0.7 million.
Supposeasampleof100majorleagueplayerswastaken.Findtheapproximateprobabilitythattheaverage
salaryofthe100playersexceeded$1.1million.
A) approximately1 B) approximately0 C) 0.2357 D) 0.7357
14) Furnacerepairbillsarenormallydistributedwithameanof267 dollarsandastandarddeviationof20 dollars.
If64oftheserepairbillsarerandomlyselected,findtheprobabilitythattheyhaveameancostbetween267
dollarsand269dollars.
A) 0.2881 B) 0.7881 C) 0.2119 D) 0.5517
15) Assumethattheheightsofmenarenormallydistributedwithameanof70.7 inchesandastandarddeviation
of2.8inches.If64menarerandomlyselected,findtheprobabilitythattheyhaveameanheightgreaterthan
71.7inches.
A) 0.0021 B) 0.0210 C) 0.9005 D) 0.9979
16) Assumethatbloodpressurereadingsarenormallydistributedwithameanof122andastandarddeviationof
6.4.If64peoplearerandomlyselected,findtheprobabilitythattheirmeanbloodpressurewillbelessthan124.
A) 0.9938 B) 0.0062 C) 0.9998 D) 0.8615
17) Accordingtothelawoflargenumbers,asmoreobservationsareaddedtothesample,thedifferencebetween
thesamplemeanandthepopulationmean
A) Tendstobecomesmaller B) Tendstobecomelarger
C) Remainsaboutthesame D) Isinverselyaffectedbythedataadded
18) Thestandarderrorofthemeanisgivenby
A) σ
nB) μ–x C) μ–x D) μ±σ
2 Describethedistributionofthesamplemean:nonnormalpopulation.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) Theamountofmoneycollectedbyasnackbaratalargeuniversityhasbeenrecordeddailyforthepastfive
years.Recordsindicatethatthemeandailyamountcollectedis$4000andthestandarddeviationis$450.The
distributionisskewedtotherightduetoseveralhighvolumedays(includingfootballgamedays).Suppose
that100dayswererandomlyselectedfromthefiveyearsandtheaverageamountcollectedfromthosedays
wasrecorded.Whichofthefollowingdescribesthesamplingdistributionofthesamplemean?
A) normallydistributedwithameanof$4000 andastandarddeviationof$45
B) normallydistributedwithameanof$4000 andastandarddeviationof$450
C) normallydistributedwithameanof$400 andastandarddeviationof$45
D) skewedtotherightwithameanof$4000 andastandarddeviationof$450
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2) Afarmerwasinterestedindetermininghowmanygrasshopperswereinhisfield.Heknowsthatthe
distributionofgrasshoppersmaynotbenormallydistributedinhisfieldduetogrowingconditions.Ashe
driveshistractordowneachrowhecountshowmanygrasshoppersheseesflyingaway.Afterseveralrowshe
figuresthemeannumberofflightstobe57withastandarddeviationof12.Whatistheprobabilityofthe
farmerwillcount60ormoreflightsonaverageinthenext40rowsdownwhichhedriveshistractor?
A) 0.0571 B) 0.4429 C) 0.9429 D) 0.5710
3) Afarmerwasinterestindetermininghowmanygrasshopperswereinhisfield.Heknowsthatthedistribution
ofgrasshoppersmaynotbenormallydistributedinhisfieldduetogrowingconditions.Ashedriveshis
tractordowneachrowhecountshowmanygrasshoppersheseesflyingaway.Afterseveralrowshefigures
themeannumberofflightstobe57withastandarddeviationof12.Whatistheprobabilityofthefarmerwill
count52orfewerflightsonaverageinthenext40rowsdownwhichhedriveshistractor?
A) 0.0041 B) 0.4959 C) 0.9959 D) 0.0410
4) Afarmerwasinterestedindetermininghowmanygrasshopperswereinhisfield.Heknowsthatthe
distributionofgrasshoppersmaynotbenormallydistributedinhisfieldduetogrowingconditions.Ashe
driveshistractordowneachrowhecountshowmanygrasshoppersheseesflyingaway.Afterseveralrowshe
figuresthemeannumberofflightstobe57withastandarddeviationof12.Whatistheprobabilityofthe
farmerwillcount52orfewerflightsor60ormoreflightsonaverageinthenext40rowsdownwhichhedrives
histractor?
A) 0.0612 B) 0.4959 C) 0.9388 D) 0.0530
5) Theownerofacomputerrepairshophasdeterminedthattheirdailyrevenuehasmean$7200andstandard
deviation$1200.Thedailyrevenuetotalsforthenext30dayswillbemonitored.Whatistheprobabilitythat
themeandailyrevenueforthenext30dayswillexceed$7500?
A) 0.0853 B) 0.9131 C) 0.0869 D) 0.9147
6) Theownerofacomputerrepairshophasdeterminedthattheirdailyrevenuehasmean$7200andstandard
deviation$1200.Thedailyrevenuetotalsforthenext30dayswillbemonitored.Whatistheprobabilitythat
themeandailyrevenueforthenext30dayswillexceed$7000?
A) 0.1814 B) 0.8186 C) 0.5675 D) 0.4325
7) Theownerofacomputerrepairshophasdeterminedthattheirdailyrevenuehasmean$7200andstandard
deviation$1200.Thedailyrevenuetotalsforthenext30dayswillbemonitored.Whatistheprobabilitythat
themeandailyrevenueforthenext30dayswillbebetween$7000and$7500?
A) 0.7333 B) 0.2667 C) 0.9147 D) 0.8186
8) Theagesoffiverandomlychosencarsinaparkinggaragearedeterminedtobe7,9,3,4,and6yearsold.Ifwe
considerthissampleof5ingroupsof3,howmanygroupscanbeformed?
A) 10 B) 5 C) 30 D) 60
9) Theagesoffiverandomlychosencarsinaparkinggaragearedeterminedtobe7,9,3,4,and6yearsold.Ifwe
considerthissampleof5ingroupsof3,whatistheprobabilityofthepopulationmeanfallingbetween5.5and
6.5years?
A) 0.5 B) 0.4 C) 0.55 D) 0.6
10) Theagesoffiverandomlychosencarsinaparkinggaragearedeterminedtobe7,9,3,4,and6yearsold.Ifwe
considerthissampleof5ingroupsof3,whatistheprobabilityofthepopulationmeanbeingmorethan6
years?
A) 0.4 B) 0.5 C) 0.1 D) 0.6
11)
J
ohnhassixbillsofpapermoneyinthefollowingdenominations:$1,$5,$10,$20,$50,and$100Ifheselects3
billsatatimehowmany,howmanygroupscanbeformed?
A) 20 B) 10 C) 30 D) 15
Page241
12)
J
ohnhassixbillsofpapermoneyinthefollowingdenominations:$1,$5,$10,$20,$50,and$100Ifheselects3
billsatatimewhatistheprobabilityofselectingagroupthathasanaveragevalueofatleast$26?
A) 0.55 B) 0.50 C) 0.60 D) 0.45
13)
J
ohnhassixbillsofpapermoneyinthefollowingdenominations:$1,$5,$10,$20,$50,and$100Ifheselects3
billsatatimewhatistheprobabilityofselectingagroupthathasanaveragevalueofbetween$40and$45?
A) 0.15 B) 0.05 C) 0.25 D) 0.20
14)
J
ohnhassixbillsofpapermoneyinthefollowingdenominations:$1,$5,$10,$20,$50,and$100Ifheselects3
billsatatimewhatistheprobabilityofselectingagroupthathasanaveragevalueofequaltoorlessthan$25?
A) 0.45 B) 0.86 C) 0.50 D) 0.35
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
15) Theamountoftimeittakesastudenttowalkfromherhometoclasshasaskewedrightdistributionwitha
meanof17minutesandastandarddeviationof2.2minutes.Ifdatawerecollectedfrom50randomlyselected
walks,describethesamplingdistributionofx,thesamplemeantime.
8.2 DistributionoftheSampleProportion
1 Describethesamplingdistributionofasampleproportion.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Describethesamplingdistributionofp
.
1) N=24,000
,
n=400
,
p=0.7
A) Approximatelynormal;μp=0.7
,
σp=0.023 B) Exactlynormal;μp=0.7
,
σp=0.023
C) Approximatelynormal;μp=0.7
,
σp=0.0935 D) Binomial;μp=280
,
σp=9.17
Provideanappropriateresponse.
2) ProfessorWhataGuysurveyedarandomsampleof420statisticsstudents.OneofthequestionswasʺWillyou
takeanothermathematicsclass?ʺTheresultsshowedthat252ofthestudentssaidyes.Whatisthesample
proportion,p
^ofstudentswhosaytheywilltakeanothermathclass?
A) 0.6 B) 0.42 C) 0.252 D) 0.775
3) Toassessattitudestowardsissuesthataffecttheresidentsofavillage,thevillagerandomlychose800families
toparticipateinasurveyoflifeattitudes.Thevillagereceived628completedsurveys.Whatisthesample
proportionofcompletedsurveys?
A) 0.785 B) 0.628 C) 1.274 D) 0.886
4) Anationalcatererdeterminedthat87%ofthepeoplewhosampledtheirfoodsaidthatitwasdelicious.A
randomsampleof144peopleisobtainedfromapopulationof5000.The144peopleareaskedtosamplethe
catererʹsfood.Willthedistributionofp
^,thesampleproportionsayingthatthefoodisdelicious,be
approximatelynormal?AnswerYesorNo.
A) Yes B) No
5) Anationalcatererdeterminedthat87%ofthepeoplewhosampledtheirfoodsaidthatitwasdelicious.A
randomsampleof144peopleisobtainedfromapopulationof5000.The144peopleareaskedtosamplethe
catererʹsfood.Ifp
^isthesampleproportionsayingthatthefoodisdelicious,whatisthemeanofthesampling
distributionofp
^?
A) 0.87 B) 1.25 C) 0.19 D) 0.42
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6) Anationalcatererdeterminedthat37%ofthepeoplewhosampledtheirfoodsaidthatitwasdelicious.A
randomsampleof144peopleisobtainedfromapopulationof5000.The144peopleareaskedtosamplethe
catererʹsfood.Ifp
^isthesampleproportionsayingthatthefoodisdelicious,whatisthestandarddeviationof
thesamplingdistributionofp
^?
A) 0.04 B) 0.002 C) 0.48 D) 0.23
7) Agreenhouseinatri–countyareahaskepttrackofitscustomersforthelastseveralyearsandhasdetermined
thatithasabout10,000regularcustomers.Ofthosecustomers,28%ofthemplantavegetablegardeninthe
spring.Thegreenhouseobtainsarandomsampleof800ofitscustomers.Isitsafetoassumethatthesampling
distributionofp
^,thesampleproportionofcustomersthatplantavegetablegarden,isapproximatelynormal?
AnswerYesorNo.
A) No B) Yes
8) Agreenhouseinatri–countyareahaskepttrackofitscustomersforthelastseveralyearsandhasdetermined
that28%ofitscustomersplantavegetablegardeninthespring.Thegreenhouseobtainsarandomsampleof
1000ofitscustomers.Whatisthemeanofthesamplingdistributionofp
^,thesampleproportionofcustomers
thatplantavegetablegardeninthespring?
A) 0.28 B) 0.72 C) 2800 D) 0.002
9) Ifapopulationproportionisbelievedtobe0.6,howmanyitemsmustbesampledtoensurethatthesampling
distributionofp
^willbeapproximatelynormal?AssumethatthesizeofthepopulationisN=10,000.
A) 42 B) 30 C) 13 D) 60
10) Asimplerandomsampleofsizen=1320 isobtainedfromapopulationwhosesizeisN=1,400,000 andwhose
populationproportionwithaspecifiedcharacteristicisp=0.46.Describethesamplingdistributionofp
^.
A) Approximatelynormal;μp=0.46
,
σp=0.014 B) Exactlynormal;μp=0.46
,
σp=0.014
C) Approximatelynormal;μp=0.46
,
σp=0.1332 D) Exactlynormal;μp=0.46
,
σp=0.1332
11) Accordingtoastudyconductedinonecity,39%ofadultsinthecityhavecreditcarddebtsofmorethan$2000.
Asimplerandomsampleofn=100adultsisobtainedfromthecity.Describethesamplingdistributionofp
^,
thesampleproportionofadultswhohavecreditcarddebtsofmorethan$2000.
A) Approximatelynormal;μp=0.39
,
σp=0.049 B) Exactlynormal;μp=0.39
,
σp=0.049
C) Approximatelynormal;μp=0.39
,
σp=0.0024 D) Binomial;μp=39
,
σp=4.88
2 Computeprobabilitiesofasampleproportion.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) Smithisaweldinspectoratashipyard.Heknowsfromkeepingtrackofgoodandsubstandardweldsthatfor
theafternoonshift5%ofallweldsdonewillbesubstandard.IfSmithchecks300ofthe7500weldscompleted
thatshift,whatistheprobabilitythathewillfindlessthan20substandardwelds?
A) 0.9066 B) 0.4066 C) 0.0934 D) 0.5934
2) Smithisaweldinspectoratashipyard.Heknowsfromkeepingtrackofgoodandsubstandardweldsthatfor
theafternoonshift5%ofallweldsdonewillbesubstandard.IfSmithchecks300ofthe7500weldscompleted
thatshift,whatistheprobabilitythathewillfindmorethan25substandardwelds?
A) 0.0040 B) 0.4960 C) 0.5040 D) 0.9960
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3) Smithisaweldinspectoratashipyard.Heknowsfromkeepingtrackofgoodandsubstandardweldsthatfor
theafternoonshift5%ofallweldsdonewillbesubstandard.IfSmithchecks300ofthe7500weldscompleted
thatshift,whatistheprobabilitythathewillfindbetween10and20substandardwelds?
A) 0.8132 B) 0.4066 C) 0.2033 D) 0.6377
4) Smithisaweldinspectoratashipyard.Heknowsfromkeepingtrackofgoodandsubstandardweldsthatfor
theafternoonshift5%ofallweldsdonewillbesubstandard.IfSmithchecks300ofthe7500weldscompleted
thatshift,woulditbeunusualforSmithtofind30ormoresubstandardwelds?
A) Yes B) No
5) TheNationalAssociationofRealtorsestimatesthat23%ofallhomespurchasedin2004wereconsidered
investmentproperties.Ifasampleof800homessoldin2004isobtainedwhatistheprobabilitythatatmost
200homesaregoingtobeusedasinvestmentproperty?
A) 0.9099 B) 0.4066 C) 0.0934 D) 0.5935
6) TheNationalAssociationofRealtorsestimatesthat23%ofallhomespurchasedin2004wereconsidered
investmentproperties.Ifasampleof800homessoldin2004isobtainedwhatistheprobabilitythatatleast175
homesaregoingtobeusedasinvestmentproperty?
A) 0.7764 B) 0.4821 C) 0.2236 D) 0.5189
7) TheNationalAssociationofRealtorsestimatesthat23%ofallhomespurchasedin2004wereconsidered
investmentproperties.Ifasampleof800homessoldin2004isobtainedanditwasnotedthat248homeswere
tobeusedasinvestmentproperty,wouldthisbeunusual?AnswerYesorNo.
A) Yes B) No
8) TheNationalAssociationofRealtorsestimatesthat23%ofallhomespurchasedin2004wereconsidered
investmentproperties.Ifasampleof800homessoldin2004isobtainedwhatistheprobabilitythatbetween
175and200homesaregoingtobeusedasinvestmentproperty?
A) 0.1335 B) 0.9099 C) 0.7764 D) 0.2236
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Ch.8 SamplingDistributions
AnswerKey
8.1 DistributionoftheSampleMean
1 Describethedistributionofthesamplemean:normalpopulation.
2 Describethedistributionofthesamplemean:nonnormalpopulation.
8.2 DistributionoftheSampleProportion
1 Describethesamplingdistributionofasampleproportion.
Page245
2 Computeprobabilitiesofasampleproportion.
Page246