Ch.7 TheNormalProbabilityDistribution
7.1 PropertiesoftheNormalDistribution
1 Usetheuniformprobabilitydistribution.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) HightemperaturesinacertaincityforthemonthofAugustfollowauniformdistributionovertheinterval64°F
to92°F.WhatistheprobabilitythatarandomlyselectedAugustdayhasahightemperaturethatexceeded
69°F?
A) 0.8214 B) 0.1786 C) 0.4423 D) 0.0357
2) HightemperaturesinacertaincityforthemonthofAugustfollowauniformdistributionovertheinterval73°F
to103°F.Findthehightemperaturewhich90%oftheAugustdaysexceed.
A) 76°F B) 100°F C) 83°F D) 103°F
3) Thediameterofballbearingsproducedinamanufacturingprocesscanbeexplainedusingauniform
distributionovertheinterval7.5to9.5millimeters.Whatistheprobabilitythatarandomlyselectedball
bearinghasadiametergreaterthan8.6millimeters?
A) 0.45 B) 0.9053 C) 4 D) 0.5059
4) Supposexisauniformrandomvariablewithc=10 andd=60.Findtheprobabilitythatarandomlyselected
observationexceeds30.
A) 0.6 B) 0.4 C) 0.1 D) 0.9
5) Supposexisauniformrandomvariablewithc=30 andd=80.Findtheprobabilitythatarandomlyselected
observationisbetween33and75.
A) 0.84 B) 0.16 C) 0.5 D) 0.8
6) Amachineissettopumpcleanserintoaprocessattherateof5 gallonsperminute.Uponinspection,itis
learnedthatthemachineactuallypumpscleanserataratedescribedbytheuniformdistributionoverthe
interval4.5to7.5gallonsperminute.Findtheprobabilitythatbetween5.0gallonsand6.0gallonsarepumped
duringarandomlyselectedminute.
A) 0.33 B) 0.67 C) 0 D) 1
7) Supposeauniformrandomvariablecanbeusedtodescribetheoutcomeofanexperimentwiththeoutcomes
rangingfrom30to70.Whatistheprobabilitythatthisexperimentresultsinanoutcomelessthan40?
A) 0.25 B) 0.1 C) 0.31 D) 1
8) Arandomnumbergeneratorissettopgenerateintegerrandomnumbersbetween1and10inclusivefollowing
auniformdistribution.Whatistheprobabilityoftherandomnumbergeneratorgeneratinga7?
A) 0.7 B) 0 C) 0.07 D) 0.5
9) TrueorFalse:Inauniformprobabilitydistribution,anyrandomvariableisjustaslikelyasanyotherrandom
variabletooccur,providedtherandomvariablesbelongtothedistribution.
A) True B) False
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2 Graphanormalcurve.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) Compareagraphofthenormaldensityfunctionwithmeanof0andstandarddeviationof1withagraphofa
normaldensityfunctionwithmeanequalto4andstandarddeviationof1.Thegraphswould
A) Havethesameheightbutonewouldbeshifted4unitstotheright.
B) Havethesameheightbutonewouldbeshifter4unitstotheleft.
C) Havenohorizontaldisplacementbutonewouldbesteeperthattheother.
D) Havenohorizontaldisplacementbutonewouldbeflatterthantheother.
2) Compareagraphofthenormaldensityfunctionwithmeanof0andstandarddeviationof1withagraphofa
normaldensityfunctionwithmeanequalto0andstandarddeviationof0.5.Thegraphswould
A) Havenohorizontaldisplacementbutonewouldbesteeperthattheother.
B) Havenohorizontaldisplacementbutonewouldbeflatterthantheother.
C) Havethesameheightbutonewouldbeshifted4unitstotheright.
D) Havethesameheightbutonewouldbeshifter4unitstotheleft.
3) Drawanormalcurvewithμ=50andσ=16.Labelthemeanandtheinflectionpoints.
A)
34 50 66
B)
18 50 82
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C)
42 50 58
D)
34 50 66
3 Statethepropertiesofthenormalcurve.
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Provideanappropriateresponse.
1) Youareperformingastudyabouttheweightofpreschoolers.Apreviousstudyfoundtheweightstobe
normallydistributedwithameanof30andastandarddeviationof4.Yourandomlysample30preschool
childrenandfindtheirweightstobeasfollows.
25 25 26 26.5 27 27 27.5 28 28 28.5
29 29 30 30 30.5 31 31 32 32.5 32.5
33 33 34 34.5 35 35 37 37 38 38
a)Drawahistogramtodisplaythedata.Isitreasonabletoassumethattheweightsarenormally
distributed?Why?
b)Findthemeanandstandarddeviationofyoursample.
c)Isthereahighprobabilitythatthemeanandstandarddeviationofyoursampleareconsistent
withthosefoundinpreviousstudies?Explainyourreasoning.
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MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
2) Thegraphofanormalcurveisgiven.Usethegraphtoidentifythevalueofμandσ.
36 9 12 15 18 21
A) μ=12
,
σ=3B)μ=3
,
σ =12 C) μ=12
,
σ =9D)μ=9
,
σ =12
3) Thenormaldensitycurveissymmetricabout
A) Itsmean
B) Thehorizontalaxis
C) Aninflectionpoint
D) Apointlocatedonestandarddeviationfromthemean
4) Thehighestpointonthegraphofthenormaldensitycurveislocatedat
A) itsmean B) aninflectionpoint C) μ+σD) μ+3σ
5) Approximately____%oftheareaunderthenormalcurveisbetweenμ–3σandμ+3σ.
A) 99.7 B) 68 C) 95 D) 50
Determinewhetherthegraphcanrepresentanormalcurve.Ifitcannot,explainwhy.
6)
A) Thegraphcannotrepresentanormaldensityfunctionbecauseitincreasesasxbecomesverylargeor
verysmall.
B) Thegraphcannotrepresentanormaldensityfunctionbecauseittakesnegativevaluesforsomevaluesof
x.
C) Thegraphcannotrepresentanormaldensityfunctionbecausetheareaunderthegraphisgreaterthan1.
D) Thegraphcanrepresentanormaldensityfunction.
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7)
A) Thegraphcanrepresentanormaldensityfunction.
B) Thegraphcannotrepresentanormaldensityfunctionbecauseithasnoinflectionpoints.
C) Thegraphcannotrepresentanormaldensityfunctionbecauseasxincreaseswithoutbound,thegraph
takesnegativevalues.
D) Thegraphcannotrepresentanormaldensityfunctionbecausetheareaunderthegraphisgreaterthan1.
8)
A) Thegraphcannotrepresentanormaldensityfunctionbecausethegraphtakesnegativevaluesforsome
valuesofx.
B) Thegraphcannotrepresentanormaldensityfunctionbecausetheareaunderthegraphislessthan1.
C) Thegraphcannotrepresentanormaldensityfunctionbecauseitisnotsymmetric.
D) Thegraphcanrepresentanormaldensityfunction.
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9)
A) Thegraphcannotrepresentanormaldensityfunctionbecauseitdoesnotapproachthehorizontalaxisas
xincreasesordecreaseswithoutbound.
B) Thegraphcannotrepresentanormaldensityfunctionbecauseitisnotbellshaped.
C) Thegraphcanrepresentanormaldensityfunction.
D) Thegraphcannotrepresentanormaldensityfunctionbecauseithasnoinflectionpoints.
10)
A) Thegraphcanrepresentanormaldensityfunction.
B) Thegraphcannotrepresentanormaldensityfunctionbecauseithasnoinflectionpoints.
C) Thegraphcannotrepresentanormaldensityfunctionbecauseitsmaximumvalueistoosmall.
D) Thegraphcannotrepresentanormaldensityfunctionbecausetheareaunderthegraphislessthan1.
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11)
A) Thegraphcannotrepresentanormaldensityfunctionbecauseit isnotsymmetric.
B) Thegraphcannotrepresentanormaldensityfunctionbecauseasxincreaseswithoutbound,thegraph
takesnegativevalues.
C) Thegraphcannotrepresentanormaldensityfunctionbecauseit isbimodal.
D) Thegraphcanrepresentanormaldensityfunction.
12)
A) Thegraphcannotrepresentanormaldensityfunctionbecauseitisbimodal.
B) Thegraphcannotrepresentanormaldensityfunctionbecauseanormaldensitycurveshouldapproach
butnotreachthehorizontalaxisasxincreasesanddecreaseswithoutbound.
C) Thegraphcanrepresentanormaldensityfunction.
D) AandBarebothtrue.
4 Explaintheroleofareainthenormaldensityfunction.
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Provideanappropriateresponse.
1) Theanalyticscoresonastandardizedaptitudetestareknowntobenormallydistributedwithmeanμ=610
andstandarddeviationσ=115.
(a)Drawanormalcurvewiththeparameterslabeled.
(b)Shadetheregionthatrepresentstheproportionoftesttakerswhoscoredlessthan725.
(c)SupposetheareaunderthenormalcurvetotheleftofX=725is0.8413.Providetwointerpretationsofthis
result.
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2) Theweightof2–yearoldhyraxesisknowntobenormallydistributedwithmean
μ=2200gramsandstandarddeviationσ=365grams
(a)Drawanormalcurvewiththeparameterslabeled.
(b)Shadetheregionthatrepresentstheproportionofhyraxeswhoweighedmorethan2930grams.
(c)SupposetheareaunderthenormalcurvetotheleftofX=2930is0.0228.Providetwointerpretationsofthis
result.
3) Theaveragempg(milespergallon)ofanewmodelofmotorcycleisknowntobenormallydistributedwith
meanμ=27.4mpgandstandarddeviationσ
=2.9mpg.
(a)Drawanormalcurvewiththeparameterslabeled.
(b)Shadetheregionthatrepresentstheproportionofmpgsbetween29.3and23.7.
(c)SupposetheareaunderthenormalcurvetobetweenX=29.3andX=23.7is0.6419.Providetwo
interpretationsofthisresult.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
4) TrueorFalse:Theareaunderthenormalcurvedrawnwithregardtothepopulationparametersisthesameas
theproportionofthepopulationthathasthesecharacteristics.
A) True B) False
5) TrueorFalse:Theareaunderthenormalcurvedrawnwithregardtothepopulationparametersisthesameas
theprobabilitythatarandomlyselectedindividualofapopulationhasthesecharacteristics.
A) True B) False
6) TrueorFalse:Theproportionofthepopulationthathascertaincharacteristicsisthesameastheprobability
thatarandomlyselectedindividualofthepopulationhasthesesamecharacteristics.
A) True B) False
7.2 ApplicationsoftheNormalDistribution
1 Findandinterprettheareaunderanormalcurve.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) Findtheareaunderthestandardnormalcurvetotheleftofz=1.5.
A) 0.9332 B) 0.0668 C) 0.5199 D) 0.7612
2) Findtheareaunderthestandardnormalcurvetotheleftofz=1.25.
A) 0.8944 B) 0.1056 C) 0.2318 D) 0.7682
3) Findtheareaunderthestandardnormalcurvetotherightofz=1.
A) 0.1587 B) 0.8413 C) 0.1397 D) 0.5398
4) Findtheareaunderthestandardnormalcurvetotherightofz= –1.25.
A) 0.8944 B) 0.5843 C) 0.6978 D) 0.7193
5) Findtheareaunderthestandardnormalcurvebetweenz=0andz=3.
A) 0.4987 B) 0.9987 C) 0.0010 D) 0.4641
6) Findtheareaunderthestandardnormalcurvebetweenz=1andz=2.
A) 0.1359 B) 0.8413 C) 0.5398 D) 0.2139
7) Findtheareaunderthestandardnormalcurvebetweenz= –1.5andz=2.5.
A) 0.9270 B) 0.7182 C) 0.6312 D) 0.9831
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8) Findtheareaunderthestandardnormalcurvebetweenz=1.5andz=2.5.
A) 0.0606 B) 0.9938 C) 0.9332 D) 0.9816
9) Findtheareaunderthestandardnormalcurvebetweenz= –1.25andz=1.25.
A) 0.7888 B) 0.8817 C) 0.6412 D) 0.2112
10) Findthesumoftheareasunderthestandardnormalcurvetotheleftofz= –1.25andtotherightofz=1.25.
A) 0.2112 B) 0.7888 C) 0.1056 D) 0.3944
Determinetheareaunderthestandardnormalcurvethatliesbetween:
11) z=1andz=2
A) 0.1359 B) 0.8641 C) 0.0006 D) 0.0008
12) z=0.3andz=1.4
A) 0.3013 B) 0.6179 C) 0.9192 D) 0.3821
13) z=–0.7andz=0.7
A) 0.516 B) 0.242 C) 0.758 D) 0.5
14) z=–2andz=–0.1
A) 0.4374 B) 0.4602 C) 0.5398 D) 0.0228
Findtheindicatedprobability.
15) AssumethattherandomvariableXisnormallydistributed,withmeanμ=60 andstandarddeviationσ=16.
ComputetheprobabilityP(X<80).
A) 0.8944 B) 0.8849 C) 0.1056 D) 0.9015
16) AssumethattherandomvariableXisnormallydistributed,withmeanμ=100 andstandarddeviationσ=15.
ComputetheprobabilityP(X>112).
A) 0.2119 B) 0.1977 C) 0.7881 D) 0.2420
17) AssumethattherandomvariableXisnormallydistributed,withmeanμ=50 andstandarddeviationσ=16.
ComputetheprobabilityP(6<X<70).
A) 0.8914 B) 0.8819 C) 0.8944 D) 0.7888
Provideanappropriateresponse.
18) Aphysicalfitnessassociationisincludingthemileruninitssecondary–schoolfitnesstest.Thetimeforthis
eventforboysinsecondaryschoolisknowntopossessanormaldistributionwithameanof450secondsanda
standarddeviationof50seconds.Findtheprobabilitythatarandomlyselectedboyinsecondaryschoolcan
runthemileinlessthan335seconds.
A) 0.0107 B) 0.4893 C) 0.9893 D) 0.5107
19) Aphysicalfitnessassociationisincludingthemileruninitssecondary–schoolfitnesstest.Thetimeforthis
eventforboysinsecondaryschoolisknowntopossessanormaldistributionwithameanof440secondsanda
standarddeviationof60seconds.Findtheprobabilitythatarandomlyselectedboyinsecondaryschoolwill
takelongerthan302secondstorunthemile.
A) 0.9893 B) 0.4893 C) 0.0107 D) 0.5107
20) Supposeabreweryhasafillingmachinethatfills12ouncebottlesofbeer.Itisknownthattheamountofbeer
pouredbythisfillingmachinefollowsanormaldistributionwithameanof12.13ouncesandastandard
deviationof0.04ounce.Findtheprobabilitythatthebottlecontainsfewerthan12.03ouncesofbeer.
A) 0.0062 B) 0.4938 C) 0.9938 D) 0.5062
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21) Supposeabreweryhasafillingmachinethatfills12ouncebottlesofbeer.Itisknownthattheamountofbeer
pouredbythisfillingmachinefollowsanormaldistributionwithameanof13.14oncesandastandard
deviationof0.04ounce.Findtheprobabilitythatthebottlecontainsmorethan13.14ouncesofbeer.
A) 0.5 B) 1 C) 0 D) 0.4
22) Supposeabreweryhasafillingmachinethatfills12ouncebottlesofbeer.Itisknownthattheamountofbeer
pouredbythisfillingmachinefollowsanormaldistributionwithameanof12.29ouncesandastandard
deviationof0.04ounce.Findtheprobabilitythatthebottlecontainsbetween12.19and12.25ounces.
A) 0.1525 B) 0.8351 C) 0.1649 D) 0.8475
23) Thelengthoftimeittakescollegestudentstofindaparkingspotinthelibraryparkinglotfollowsanormal
distributionwithameanof3.0minutesandastandarddeviationof1minute.Findtheprobabilitythata
randomlyselectedcollegestudentwillfindaparkingspotinthelibraryparkinglotinlessthan2.5minutes.
A) 0.3085 B) 0.1915 C) 0.3551 D) 0.2674
24) Thelengthoftimeittakescollegestudentstofindaparkingspotinthelibraryparkinglotfollowsanormal
distributionwithameanof6.5minutesandastandarddeviationof1minute.Findtheprobabilitythata
randomlyselectedcollegestudentwilltakebetween5.0and7.5minutestofindaparkingspotinthelibrary
lot.
A) 0.7745 B) 0.4938 C) 0.0919 D) 0.2255
25) Theamountofsodaadispensingmachinepoursintoa12ouncecanofsodafollowsanormaldistributionwith
ameanof12.48ouncesandastandarddeviationof0.32ounce.Thecansonlyhold12.80ouncesofsoda.Every
canthathasmorethan12.80ouncesofsodapouredintoitcausesaspillandthecanneedstogothrougha
specialcleaningprocessbeforeitcanbesold.Whatistheprobabilityarandomlyselectedcanwillneedtogo
throughthisprocess?
A) 0.1587 B) 0.3413 C) 0.8413 D) 0.6587
26) Anewphonesystemwasinstalledlastyeartohelpreducetheexpenseofpersonalcallsthatwerebeingmade
byemployees.Beforethenewsystemwasinstalled,theamountbeingspentonpersonalcallsfolloweda
normaldistributionwithanaverageof$500permonthandastandarddeviationof$50permonth.Referto
suchexpensesasPCEʹs(personalcallexpenses).Usingthedistributionabove,whatistheprobabilitythata
randomlyselectedmonthhadaPCEofbetween$375.00and$590.00?
A) 0.9579 B) 0.0421 C) 0.9999 D) 0.0001
27) Anewphonesystemwasinstalledlastyeartohelpreducetheexpenseofpersonalcallsthatwerebeingmade
byemployees.Beforethenewsystemwasinstalled,theamountbeingspentonpersonalcallsfollowsanormal
distributionwithanaverageof$900permonthandastandarddeviationof$50permonth.Refertosuch
expensesasPCEʹs(personalcallexpenses).FindtheprobabilitythatarandomlyselectedmonthhadaPCEthat
fallsbelow$750.
A) 0.0013 B) 0.8333 C) 0.1667 D) 0.9987
28) Thetreadlifeofaparticularbrandoftireisarandomvariablebestdescribedbyanormaldistributionwitha
meanof60,000milesandastandarddeviationof2900miles.Whatistheprobabilityaparticulartireofthis
brandwilllastlongerthan57,100miles?
A) 0.8413 B) 0.1587 C) 0.2266 D) 0.7266
29) Thetreadlifeofaparticularbrandoftireisarandomvariablebestdescribedbyanormaldistributionwitha
meanof60,000milesandastandarddeviationof2900miles.Whatistheprobabilityacertaintireofthisbrand
willlastbetween53,910milesand54,780miles?
A) 0.0180 B) 0.9813 C) 0.4920 D) 0.4649
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SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
30) Afirmbelievestheinternalrateofreturnforitsproposedinvestmentcanbestbedescribedbyanormal
distributionwithmean26%andstandarddeviation3%.Whatistheprobabilitythattheinternalrateofreturn
fortheinvestmentwillbeatleast21.5%?
31) Afirmbelievestheinternalrateofreturnforitsproposedinvestmentcanbestbedescribedbyanormal
distributionwithmean36%andstandarddeviation3%.Whatistheprobabilitythattheinternalrateofreturn
fortheinvestmentexceeds42%?
32) Farmersoftensellfruitsandvegetablesatroadsidestandsduringthesummer.Onesuchroadsidestandhasa
dailydemandfortomatoesthatisapproximatelynormallydistributedwithameanequalto413tomatoesper
dayandastandarddeviationequalto30tomatoesperday.Ifthereare371tomatoesavailabletobesoldatthe
roadsidestandatthebeginningofaday,whatistheprobabilitythattheywillallbesold?
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
33) Givenadistributionthatfollowsastandardnormalcurve,whatdoesthegraphofthecurvedoaszincreases
inthepositivedirection?
A) Thegraphofthecurveapproacheszero.
B) Thegraphofthecurveapproaches1.
C) Thegraphofthecurveapproachesaninflectionpoint.
D) Thegraphofthecurveeventuallyintersectsthehorizontalaxis.
2 Findthevalueofanormalrandomvariable.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheindicatedz–score.
1) Findthez–scoreforwhichtheareaunderthestandardnormalcurvetoitsleftis0.96
A) 1.75 B) 1.82 C) 1.03 D) –1.38
2) Findthez–scoreforwhichtheareaunderthestandardnormalcurvetoitsleftis0.40
A) –0.25 B) 0.25 C) 0.57 D) –0.57
3) Findthez–scoreforwhichtheareaunderthestandardnormalcurvetoitsleftis0.09.
A) –1.34 B) –1.39 C) –1.26 D) –1.45
4) Findthez–scoreforwhichtheareaunderthestandardnormalcurvetoitsleftis0.04.
A) –1.75 B) –1.89 C) –1.48 D) –1.63
5) Findthez–scoreforwhichtheareaunderthestandardnormalcurvetoitsleftis0.70.
A) 0.53 B) 0.98 C) 0.81 D) 0.47
6) Findthez–scoreforwhichtheareaunderthestandardnormalcurvetoitsrightis0.07.
A) 1.48 B) 1.39 C) 1.26 D) 1.45
7) Findthez–scoreforwhichtheareaunderthestandardnormalcurvetoitsrightis0.70.
A) –0.53 B) –0.98 C) –0.81 D) –0.47
8) Findthez–scoreforwhichtheareaunderthestandardnormalcurvetoitsrightis0.09.
A) 1.34 B) 1.39 C) 1.26 D) 1.45
9) Findthez–scorehavingarea0.86toitsrightunderthestandardnormalcurve;thatis,findz0.86 .
A) –1.08 B) 1.08 C) 0.8051 D) 0.5557
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10) Forastandardnormalcurve,findthez–scorethatseparatesthebottom90%fromthetop10%.
A) 1.28 B) 0.28 C) 1.52 D) 2.81
11) Forastandardnormalcurve,findthez–scorethatseparatesthebottom30%fromthetop70%.
A) –0.53 B) –0.98 C) –0.47 D) –0.12
12) Forastandardnormalcurve,findthez–scorethatseparatesthebottom70%fromthetop30%.
A) 0.53 B) 0.98 C) 0.47 D) 0.12
13) Determinethetwoz–scoresthatseparatethemiddle87.4%ofthedistributionfromtheareainthetailsofthe
standardnormaldistribution.
A) –1.53
,
1.53 B) –1.39,1.39 C) –1.46,1.46 D) –1.45,1.45
14) Determinethetwoz–scoresthatseparatethemiddle96%ofthedistributionfromtheareainthetailsofthe
standardnormaldistribution.
A) –2.05and2.05 B) –1.75and1.75 C) 0and2.05 D) –2.33and2.33
15) Findthez–scoresforwhich90%ofthedistributionʹsarealiesbetween–zandz.
A) (–1.645,1.645) B) (–2.33,2.33) C) (–1.96,1.96) D) (–0.99,0.99)
16) Findthez–scoresforwhich98%ofthedistributionʹsarealiesbetween–zandz.
A) (–2.33,2.33) B) (–1.645,1.645) C) (–1.96,1.96) D) (–0.99,0.99)
Findthevalueofzα.
17) z0.05
A) 1.645 B) –1.645 C) 0.52 D) 1.75
Findtheindicatedpercentile.
18) AssumethattherandomvariableXisnormallydistributedwithmeanμ=50 andstandarddeviationσ=12.
Findthe24thpercentileforX.
A) 41.48 B) 58.52 C) 42.92 D) 38.6
Provideanappropriateresponse.
19) Aphysicalfitnessassociationisincludingthemileruninitssecondary–schoolfitnesstest.Thetimeforthis
eventforboysinsecondaryschoolisknowntopossessanormaldistributionwithameanof440secondsanda
standarddeviationof50seconds.Thefitnessassociationwantstorecognizethefastest10%oftheboyswith
certificatesofrecognition.Whattimewouldtheboysneedtobeatinordertoearnacertificateofrecognition
fromthefitnessassociation?
A) 376sec B) 504sec C) 522.25 sec D) 357.75 sec
20) Aphysicalfitnessassociationisincludingthemileruninitssecondary–schoolfitnesstest.Thetimeforthis
eventforboysinsecondaryschoolisknowntopossessanormaldistributionwithameanof460secondsanda
standarddeviationof60seconds.Betweenwhattimesdoweexpectmost(approximately95%)oftheboysto
runthemile?
A)
b
etween342.4and577.6sec B)
b
etween361.3 and558.736sec
C)
b
etween365and555sec D)
b
etween0and558.736sec
21) Theamountofcornchipsdispensedintoa16–ouncebagbythedispensingmachinehasbeenidentifiedas
possessinganormaldistributionwithameanof16.5ouncesandastandarddeviationof0.2ounce.Whatchip
amountrepresentsthe67thpercentileforthebagweightdistribution?Roundtothenearesthundredth.
A) 16.59oz B) 16.09 oz C) 16.63 oz D) 16.13 oz
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22) Supposeabreweryhasafillingmachinethatfills12–ouncebottlesofbeer.Itisknownthattheamountofbeer
pouredbythisfillingmachinefollowsanormaldistributionwithameanof12.47ouncesandastandard
deviationof0.04ounce.Thecompanyisinterestedinreducingtheamountofextrabeerthatispouredintothe
12ouncebottles.Thecompanyisseekingtoidentifythehighest1.5%ofthefillamountspouredbythis
machine.Forwhatfillamountaretheysearching?Roundtothenearestthousandth.
A) 12.557oz B) 11.913oz C) 12.383 oz D) 12.087oz
23) Thelengthoftimeittakescollegestudentstofindaparkingspotinthelibraryparkinglotfollowsanormal
distributionwithameanof5.5minutesandastandarddeviationof1minute.Findthecut–offtimewhich
75.8%ofthecollegestudentsexceedwhentryingtofindaparkingspotinthelibraryparkinglot.
A) 6.2min B) 6.0min C) 5.8 min D) 6.3 min
24) Theamountofsodaadispensingmachinepoursintoa12ouncecanofsodafollowsanormaldistributionwith
astandarddeviationof0.06ounce.Everycanthathasmorethan12.15ouncesofsodapouredintoitcausesa
spillandthecanneedstogothroughaspecialcleaningprocessbeforeitcanbesold.Whatisthemeanamount
ofsodathemachineshoulddispenseifthecompanywantstolimitthepercentagethatneedtobecleaned
becauseofspillageto3%?
A) 12.0372oz B) 12.2628 oz C) 12.0198 oz D) 12.2802 oz
25) Anewphonesystemwasinstalledlastyeartohelpreducetheexpenseofpersonalcallsthatwerebeingmade
byemployees.Beforethenewsystemwasinstalled,theamountbeingspentonpersonalcallsfollowsanormal
distributionwithanaverageof$800permonthandastandarddeviationof$50permonth.Refertosuch
expensesasPCEʹs(personalcallexpenses).Findthepointinthedistributionbelowwhich2.5%ofthePCEʹs
fell.
A) $702.00 B) $898.00 C) $20.00 D) $780.00
26) Abreweryhasabeerdispensingmachinethatdispensesbeerintothecompanyʹs12ouncebottles.The
distributionfortheamountofbeerdispensedbythemachinefollowsanormaldistributionwithastandard
deviationof0.10ounce.Thecompanycancontrolthemeanamountofbeerdispensedbythemachine.What
valueofthemeanshouldthecompanyuseifitwantstoguaranteethat98.5%ofthebottlescontainatleast12
ounces(theamountonthelabel)?Roundtothenearestthousandth.
A) 12.217oz B) 12.243 oz C) 12.002 oz D) 12.001 oz
27) Thetreadlifeofaparticularbrandoftireisarandomvariablebestdescribedbyanormaldistributionwitha
meanof60,000milesandastandarddeviationof3000miles.Whatwarrantyshouldthecompanyuseifthey
want96%ofthetirestooutlastthewarranty?
A) 54,750mi B) 65,250 mi C) 57,000 mi D) 63,000 mi
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
28) Theboardofexaminersthatadministerstherealestatebrokerʹsexaminationinacertainstatefoundthatthe
meanscoreonthetestwas493andthestandarddeviationwas72.Iftheboardwantstosetthepassingscore
sothatonlythebest10%ofallapplicantspass,whatisthepassingscore?Assumethatthescoresarenormally
distributed.
29) Theboardofexaminersthatadministerstherealestatebrokerʹsexaminationinacertainstatefoundthatthe
meanscoreonthetestwas426andthestandarddeviationwas72.Iftheboardwantstosetthepassingscore
sothatonlythebest80%ofallapplicantspass,whatisthepassingscore?Assumethatthescoresarenormally
distributed.
30) Farmersoftensellfruitsandvegetablesatroadsidestandsduringthesummer.Onesuchroadsidestandhasa
dailydemandfortomatoesthatisapproximatelynormallydistributedwithameanequalto125tomatoesper
dayandastandarddeviationequalto30tomatoesperday.Howmanytomatoesmustbeavailableonany
givendaysothatthereisonlya1.5%chancethatalltomatoeswillbesold?
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7.3 AssessingNormality
1 Usenormalprobabilityplotstoassessnormality.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Useanormalprobabilityplottoasseswhetherthesampledatacouldhavecomefromapopulationthatisnormally
distributed.
1) Determinewhetherthefollowingnormalprobabilityplotindicatesthatthesampledatacouldhavecomefrom
apopulationthatisnormallydistributed.
A) normallydistributed B) notnormallydistributed
2) Determinewhetherthefollowingnormalprobabilityplotindicatesthatthesampledatacouldhavecomefrom
apopulationthatisnormallydistributed.
A) normallydistributed B) notnormallydistributed
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3) Determinewhetherthefollowingnormalprobabilityplotindicatesthatthesampledatacouldhavecomefrom
apopulationthatisnormallydistributed.
A) notnormallydistributed B) normallydistributed
4) Determinewhetherthefollowingnormalprobabilityplotindicatesthatthesampledatacouldhavecomefrom
apopulationthatisnormallydistributed.
A) notnormallydistributed B) normallydistributed
5) Anindustrialpsychologistconductedanexperimentinwhich40employeesthatwereidentifiedasʺchronically
tardyʺbytheirmanagersweredividedintotwogroupsofsize20.Group1participatedinthenewʺItʹsGreatto
beAwake!ʺprogram,whileGroup2hadtheirpaypaydocked.Thefollowingdatarepresentthenumberof
minutesthatemployeesinGroup1werelateforworkafterparticipatingintheprogram.
124 132 147 151 147
163 148 183 138 141
153 129 138 153 158
159 159 160 131 153
A) normallydistributed B) notnormallydistributed
6) Thefollowingdatarepresentarandomsampleofthenumberofsharesofapharmaceuticalcompanyʹsstock
tradedfor20daysin2000.
4.71 9.63 11.13 8.25 14.57
6.02 10.42 9.67 12.83 13.98
4.97 9.27 12.32 7.48 28.27
8.44 10.87 5.01 7.97 11.61
A) notnormallydistributed B) normallydistributed
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Provideanappropriateresponse.
7) Anormalprobabilityplotisagraphthatplots_____________versus_____________.
A) observeddata,normalscores B) normalscore,observeddata
C) normaldata,observedscores D) observedscores,normaldata
8) Ifsampledataaretakenfromapopulationthatisnormallydistributed,anormalprobabilityplotofthe
observeddatavaluesversustheexpectedzscoreswill
A)
b
eapproximatelylinear. B) havenodiscernablepattern.
C) lookexponentialinnature. D) haveacorrelationcoefficientnear0.
7.4 TheNormalApproximationtotheBinomialProbabilityDistributio
n
1 Approximatebinomialprobabilitiesusingthenormaldistribution.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
ComputeP(x)usingthebinomialprobabilityformula.Thendeterminewhetherthenormaldistributioncanbeusedas
anapproximationforthebinomialdistribution.Ifso,approximateP(x)andcomparetheresulttotheexactprobability.
1) n=80,p=0.7
,
x=62(Roundthestandarddeviationtothreedecimalplacestoworktheproblem.)
A) Exact:0.0343;Approximate:0.0342 B) Exact:0.0343;Approximate:0.0351
C) Exact:0.0352;Approximate:0.0351 D) Exact:0.0343;Approximate:0.0333
Provideanappropriateresponse.
2) Astudentanswersall48questionsonamultiple–choicetestbyguessing.Eachquestionhasfourpossible
answers,onlyoneofwhichiscorrect.Findtheprobabilitythatthestudentgetsexactly15correctanswers.Use
thenormaldistributiontoapproximatethebinomialdistribution.
A) 0.0823 B) 0.8577 C) 0.7967 D) 0.0606
3) Iftheprobabilityofanewbornkittenbeingfemaleis0.5,findtheprobabilitythatin100births,55ormorewill
befemale.Usethenormaldistributiontoapproximatethebinomialdistribution.
A) 0.1841 B) 0.7967 C) 0.8159 D) 0.0606
4) Alocalconcertcenterreportsthatithasbeenexperiencinga15%rateofno–showsonadvancedreservations.
Among150advancedreservations,findtheprobabilitythattherewillbefewerthan20no –shows.Roundthe
standarddeviationtothreedecimalplacestoworktheproblem.
A) 0.2451 B) 0.7549 C) 0.7967 D) 0.3187
5) Findtheprobabilitythatin200tossesofafairsix–sideddie,athreewillbeobtainedatleast40times.
A) 0.1190 B) 0.8810 C) 0.0871 D) 0.3875
6) Findtheprobabilitythatin200tossesofafairsix–sideddie,athreewillbeobtainedatmost40times.
A) 0.9147 B) 0.1190 C) 0.8810 D) 0.0853
7) Asalespersonfoundthattherewasa1%chanceofasalefromherphonesolicitations.Findtheprobabilityof
getting5ormoresalesfor1000telephonecalls.Roundthestandarddeviationtothreedecimalplacestowork
theproblem.
A) 0.9599 B) 0.0401 C) 0.8810 D) 0.0871
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
8) Theauthorofaneconomicsbookhastroubledecidingwhethertousethewordsʺheʺorʺsheʺinthebookʹs
examples.Tosolvetheproblem,theauthorflipsacoineachtimetheproblemarises.Ifaheadshows,the
authorusesʺheʺandifatailshows,theauthorusesʺsheʺ.Ifthisproblemoccurs100timesinthebook,whatis
theprobabilitythatʺsheʺwillbeused58times?
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9) Inarecentsurvey,84%ofthecommunityfavoredbuildingmoreparksintheirneighborhood.Yourandomly
select19citizensandaskeachifheorshethinksthecommunityneedsmoreparks.Decidewhetheryoucan
usethenormaldistributiontoapproximatethebinomialdistribution.Ifso,findthemeanandstandard
deviation.Ifnot,explainwhy.
10) Arecentsurveyfoundthat73%ofalladultsover50owncellphones.Yourandomlyselect42adultsover50,
andaskifheorsheownsacellphone.Decidewhetheryoucanusethenormaldistributiontoapproximatethe
binomialdistribution.Ifso,findthemeanandstandarddeviation,Ifnot,explainwhy.Roundtothenearest
hundredthwhennecessary.
11) Accordingtogovernmentdata,theprobabilitythananadultneverhadthefluis15%.Yourandomlyselect75
adultsandaskifheorsheeverhadtheflu.Decidewhetheryoucanusethenormaldistributionto
approximatethebinomialdistribution,Ifso,findthemeanandstandarddeviation,Ifnot,explainwhy.Round
tothenearesthundredthwhennecessary.
12) Forthefollowingconditions,determineifitisappropriatetousethenormaldistributiontoapproximatea
binomialdistributionwithn=6andp=0.2.
13) Forthefollowingconditions,determineifitisappropriatetousethenormaldistributiontoapproximatea
binomialdistributionwithn=38andp=0.6.
14) ThefailurerateinaGermanclassis30%.Inaclassof50students,findtheprobabilitythatexactlyfivestudents
willfail.Usethenormaldistributiontoapproximatethebinomialdistribution.Roundthestandarddeviationto
threedecimalplacestoworktheproblem.
15) Alocalrentalcaragencyhas100cars.Therentalrateforthewintermonthsis60%.Findtheprobabilitythatin
agivenwintermonthatleast70carswillberented.Usethenormaldistributiontoapproximatethebinomial
distribution.Roundthestandarddeviationtothreedecimalplacestoworktheproblem.
16) Alocalrentalcaragencyhas200cars.Therentalrateforthewintermonthsis60%.Findtheprobabilitythatin
agivenwintermonthfewerthan140carswillberented.Usethenormaldistributiontoapproximatethe
binomialdistribution.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
17) TrueorFalse:Inordertouseanormalapproximationtothebinomialprobabilitydistribution,np(1–p)≥10.
A) True B) False
18) Assumingthatallconditionsaremettoapproximateabinomialprobabilitydistributionwiththestandard
normaldistribution,thentocomputeP(x≥19)fromthebinomialdistributionwemustcompute
asthenormalapproximation.
A) P(x≥18.5) B) P(x≤18.5) C) P(x≥19.1) D) P(x≤18.9)
19) Assumingthatallconditionsaremettoapproximateabinomialprobabilitydistributionwiththestandard
normaldistribution,thentocomputeP(x≤23)fromthebinomialdistributionwemustcompute
asthenormalapproximation.
A) P(x≤23.5) B) P(x≥23.5) C) P(x≤22.9) D) P(x≤23.1)
20) Assumingthatallconditionsaremettoapproximateabinomialprobabilitydistributionwiththestandard
normaldistribution,thentocomputeP(12≤x≤15)fromthebinomialdistributionwemustcompute
asthenormalapproximation.
A) P(11.5<x<15.5) B) P(12.5<x<14.5)
C) P(x>15.5)andP(x<11.5) D) P(x>12.5)andP(x<14.5)
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Ch.7 TheNormalProbabilityDistribution
AnswerKey
7.1 PropertiesoftheNormalDistribution
1 Usetheuniformprobabilitydistribution.
2 Graphanormalcurve.
3 Statethepropertiesofthenormalcurve.
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4 Explaintheroleofareainthenormaldensityfunction.
7.2 ApplicationsoftheNormalDistribution
1 Findandinterprettheareaunderanormalcurve.
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2 Findthevalueofanormalrandomvariable.
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7.3 AssessingNormality
1 Usenormalprobabilityplotstoassessnormality.
7.4 TheNormalApproximationtotheBinomialProbabilityDistributio
n
1 Approximatebinomialprobabilitiesusingthenormaldistribution.
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