Chapter 05 Find the area of the indicated region under the standard

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Ch.5 NormalProbabilityDistributions

5.1 IntroductiontoNormalDistributionsandtheStandardNormalDistributio

1 FindAreasUndertheStandardNormalCurve

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Provideanappropriateresponse.

1) Findtheareaoftheindicatedregionunderthestandardnormalcurve.

A) 0.0968 B) 0.9032 C) 0.4032 D) 0.0823

2) Findtheareaoftheindicatedregionunderthestandardnormalcurve.

A) 0.9032 B) 0.0968 C) 0.0823 D) 0.9177

3) Findtheareaoftheindicatedregionunderthestandardnormalcurve.

A) 0.6562 B) 1.309 C) 0.3438 D) 0.309

4) Findtheareaoftheindicatedregionunderthestandardnormalcurve.

A) 0.1504 B) 0.1292 C) 0.8489 D) 0.0212

5) Findtheareaunderthestandardnormalcurvetotheleftofz=1.5.

A) 0.9332 B) 0.0668 C) 0.5199 D) 0.7612

6) Findtheareaunderthestandardnormalcurvetotheleftofz=1.25.

A) 0.8944 B) 0.1056 C) 0.2318 D) 0.7682

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7) Findtheareaunderthestandardnormalcurvetotherightofz=1.

A) 0.1587 B) 0.8413 C) 0.1397 D) 0.5398

8) Findtheareaunderthestandardnormalcurvetotherightofz= –1.25.

A) 0.8944 B) 0.5843 C) 0.6978 D) 0.7193

9) Findtheareaunderthestandardnormalcurvebetweenz=0andz=3.

A) 0.4987 B) 0.9987 C) 0.0010 D) 0.4641

10) Findtheareaunderthestandardnormalcurvebetweenz=1andz=2.

A) 0.1359 B) 0.8413 C) 0.5398 D) 0.2139

11) Findtheareaunderthestandardnormalcurvebetweenz= –1.5andz=2.5.

A) 0.9270 B) 0.7182 C) 0.6312 D) 0.9831

12) Findtheareaunderthestandardnormalcurvebetweenz=1.5andz=2.5.

A) 0.0606 B) 0.9938 C) 0.9332 D) 0.9816

13) Findtheareaunderthestandardnormalcurvebetweenz= –1.25andz=1.25.

A) 0.7888 B) 0.8817 C) 0.6412 D) 0.2112

14) Findthesumoftheareasunderthestandardnormalcurvetotheleftofz= –1.25andtotherightofz=1.25.

A) 0.2112 B) 0.7888 C) 0.1056 D) 0.3944

2 DetermineifDistributionisNormal

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Provideanappropriateresponse.

1) Determinewhetherthegraphcouldrepresentavariablewithanormaldistribution.Explainyourreasoning.

A) Yes,thegraphfulfillsthepropertiesofthenormaldistribution

B) No,thegraphisskewedleft.

C) No,thegraphcrossesthex–axis.

D) No,thegraphisnotsymmetric.

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2) Determinewhetherthegraphcouldrepresentavariablewithanormaldistribution.Explainyourreasoning.

A) No,thegraphcrossesthex–axis.

B) No,thegraphisskewedright.

C) No,thegraphisnotsymmetric.

D) Yes,thegraphfulfillsthepropertiesofthenormaldistribution.

3) Determinewhetherthegraphcouldrepresentavariablewithanormaldistribution.Explainyourreasoning.

A) No,thegraphisskewedright.

B) No,thegraphcrossesthex–axis.

C) No,thegraphisskewedleft.

D) Yes,thegraphfulfillsthepropertiesofthenormaldistribution

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

4) Youareperformingastudyabouttheweightofpreschoolers.Apreviousstudyfoundtheweightstobe

normallydistributedwithameanof30andastandarddeviationof4.Yourandomlysample30preschool

childrenandfindtheirweightstobeasfollows.

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)Drawahistogramtodisplaythesedata.Usesevenclasses.Dotheweightsappeartobenormally

distributed?Explain.

b)Findthemeanandstandarddeviationofyoursample.

c)Isthereahighprobabilitythatthemeanandstandarddeviationofyoursampleareconsistent

withthosefoundinpreviousstudies?Explainyourreasoning.

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3 Computez–Score

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Provideanappropriateresponse.

1) TheSATisanexamusedbycollegesanduniversitiestoevaluateundergraduateapplicants.Thetestscoresare

normallydistributed.Inarecentyear,themeantestscorewas1475andthestandarddeviationwas308.The

testscoresoffourstudentsselectedatrandomare1930,1340,2150,and1450.

a)Findthez–scoresthatcorrespondtoeachvalue

b)Determinewhetheranyofthevaluesareunusual.

2) TheACTisanexamusedbycollegesanduniversitiestoevaluateundergraduateapplicants.Thetestscoresare

normallydistributed.Inarecentyear,themeantestscorewas22.2andthestandarddeviationwas4.9.Thetest

scoresoffourstudentsselectedatrandomare16,24,9,and35.

a)Findthez–scoresthatcorrespondtoeachvalue.

b)Determinewhetheranyofthevaluesareunusual.

4 FindProbabilitiesUsingtheStandardNormalCurve

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheprobabilityofzoccurringintheindicatedregion.

01.82

A) 0.9656 B) 0.0344 C) 0.9772 D) 0.4656

–0.59 0

A) 0.2776 B) 0.7224 C) 0.2224 D) 0.1894

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–1.33 0

A) 0.9082 B) 0.0918 C) 0.9332 D) 0.0668

01.75

A) 0.0401 B) 0.0668 C) 0.9599 D) 0.0228

–203

A) 0.9772 B) 0.0228 C) 0.9544 D) 0.0456

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0 1.50

A) 0.4332 B) 0.5668 C) 0.0668 D) 0.9332

5 FindProbabilitiesUsingtheStandardNormalDistribution

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Provideanappropriateresponse.

1) UsethestandardnormaldistributiontofindP(0<z<2.25).

A) 0.4878 B) 0.5122 C) 0.8817 D) 0.7888

2) UsethestandardnormaldistributiontofindP(–2.25<z<0).

A) 0.4878 B) 0.5122 C) 0.6831 D) 0.0122

3) UsethestandardnormaldistributiontofindP(–2.25<z<1.25).

A) 0.8822 B) 0.0122 C) 0.4878 D) 0.8944

4) UsethestandardnormaldistributiontofindP(–2.50<z<1.50).

A) 0.9270 B) 0.8822 C) 0.6167 D) 0.5496

5) UsethestandardnormaldistributiontofindP(z< –2.33orz>2.33).

A) 0.0198 B) 0.9802 C) 0.7888 D) 0.0606

6) Forthestandardnormalcurve,findthez–scorethatcorrespondstothethirdquartile.

A) 0.67 B) –0.67 C) 0.77 D) –0.23

7) Forthestandardnormalcurve,findthez–scorethatcorrespondstothefirstquartile.

A) –0.67 B) 0.67 C) 0.77 D) –0.23

8) Forthestandardnormalcurve,findthez–scorethatcorrespondstothefirstdecile.

A) –1.28 B) 1.28 C) –2.33 D) 0.16

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5.2 NormalDistributions:FindingProbabilities

1 FindProbabilitiesforNormallyDistributedVariables

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Provideanappropriateresponse.

1) AssumethattherandomvariableXisnormallydistributed,withmeanμ=60 andstandarddeviationσ=12.

ComputetheprobabilityP(X<75).

A) 0.8944 B) 0.8849 C) 0.1056 D) 0.9015

2) AssumethattherandomvariableXisnormallydistributed,withmeanμ=80 andstandarddeviationσ=15.

ComputetheprobabilityP(X>92).

A) 0.2119 B) 0.1977 C) 0.7881 D) 0.2420

3) AssumethattherandomvariableXisnormallydistributed,withmeanμ=60 andstandarddeviationσ=8.

ComputetheprobabilityP(38<X<70).

A) 0.8914 B) 0.8819 C) 0.8944 D) 0.7888

Provideanappropriateresponse.UsetheStandardNormalTabletofindtheprobability.

4) IQtestscoresarenormallydistributedwithameanof100andastandarddeviationof15.AnindividualʹsIQ

scoreisfoundtobe110.Findthez–scorecorrespondingtothisvalue.

A) 0.67 B) –0.67 C) 1.33 D) –1.33

5) IQtestscoresarenormallydistributedwithameanof100andastandarddeviationof15.AnindividualʹsIQ

scoreisfoundtobe90.Findthez–scorecorrespondingtothisvalue.

A) –0.67 B) 0.67 C) 1.33 D) –1.33

6) IQtestscoresarenormallydistributedwithameanof100andastandarddeviationof15.AnindividualʹsIQ

scoreisfoundtobe120.Findthez–scorecorrespondingtothisvalue.

A) 1.33 B) –0.67 C) 0.67 D) –1.33

7) IQtestscoresarenormallydistributedwithameanof100andastandarddeviationof15.FindtheIQscore

thatcorrespondstoaz–scoreof1.96.

A) 129.4 B) 115.6 C) 122.4 D) 132.1

8) IQtestscoresarenormallydistributedwithameanof105 andastandarddeviationof11.AnindividualʹsIQ

scoreisfoundtobe119.Findthez–scorecorrespondingtothisvalue.

A) 1.27 B) -1.27 C) 0.79 D) -0.79

9) IQtestscoresarenormallydistributedwithameanof104 andastandarddeviationof15.AnindividualʹsIQ

scoreisfoundtobe111.Findthez–scorecorrespondingtothisvalue.

A) 0.47 B) -0.47 C) 2.14 D) -2.14

10) Thelengthsofpregnanciesofhumansarenormallydistributedwithameanof268daysandastandard

deviationof15days.Findtheprobabilityofapregnancylastingmorethan300days.

A) 0.0166 B) 0.9834 C) 0.2375 D) 0.3189

11) Thelengthsofpregnanciesofhumansarenormallydistributedwithameanof268daysandastandard

deviationof15days.Findtheprobabilityofapregnancylastinglessthan250days.

A) 0.1151 B) 0.1591 C) 0.0606 D) 0.0066

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12) Thedistributionofcholesterollevelsinteenageboysisapproximatelynormalwithμ=170andσ =30 (Source:

U.S.NationalCenterforHealthStatistics).Levelsabove200warrantattention.Findtheprobabilitythata

teenageboyhasacholesterollevelgreaterthan200.

A) 0.1587 B) 0.8413 C) 0.3419 D) 0.2138

13) Thedistributionofcholesterollevelsinteenageboysisapproximatelynormalwithμ=170andσ =30 (Source:

U.S.NationalCenterforHealthStatistics).Levelsabove200warrantattention.Findtheprobabilitythata

teenageboyhasacholesterollevelgreaterthan225.

A) 0.0336 B) 0.0606 C) 0.0718 D) 0.0012

14) Anairlineknowsfromexperiencethatthedistributionofthenumberofsuitcasesthatgetlosteachweekona

certainrouteisapproximatelynormalwithμ=15.5andσ=3.6.Whatistheprobabilitythatduringagiven

weektheairlinewillloselessthan20suitcases?

A) 0.8944 B) 0.3944 C) 0.1056 D) 0.4040

15) Anairlineknowsfromexperiencethatthedistributionofthenumberofsuitcasesthatgetlosteachweekona

certainrouteisapproximatelynormalwithμ=15.5andσ=3.6.Whatistheprobabilitythatduringagiven

weektheairlinewilllosemorethan20suitcases?

A) 0.1056 B) 0.3944 C) 0.4040 D) 0.8944

16) Anairlineknowsfromexperiencethatthedistributionofthenumberofsuitcasesthatgetlosteachweekona

certainrouteisapproximatelynormalwithμ=15.5andσ=3.6.Whatistheprobabilitythatduringagiven

weektheairlinewilllosebetween10and20suitcases?

A) 0.8314 B) 0.3944 C) 0.1056 D) 0.4040

17) AssumethatthesalariesofelementaryschoolteachersintheUnitedStatesarenormallydistributedwitha

meanof$32,000andastandarddeviationof$3000.Ifateacherisselectedatrandom,findtheprobabilitythat

heorshemakesmorethan$36,000.

A) 0.0918 B) 0.9082 C) 0.1056 D) 0.4040

18) AssumethatthesalariesofelementaryschoolteachersintheUnitedStatesarenormallydistributedwitha

meanof$32,000andastandarddeviationof$3000.Ifateacherisselectedatrandom,findtheprobabilitythat

heorshemakeslessthan$28,000.

A) 0.0918 B) 0.9981 C) 0.2113 D) 0.9827

19) Assumethattheheightsofwomenarenormallydistributedwithameanof63.6inchesandastandard

deviationof2.5inches.Thecheerleadersforalocalprofessionalbasketballteammustbebetween65.5and68.0

inches.Ifawomanisrandomlyselected,whatistheprobabilitythatherheightisbetween65.5and68.0

inches?

A) 0.1844 B) 0.9608 C) 0.7881 D) 0.3112

20) Thelengthsofpregnanciesofhumansarenormallydistributedwithameanof268daysandastandard

deviationof15days.Ababyisprematureifitisbornthreeweeksearly.Whatpercentofbabiesareborn

prematurely?

A) 8.08% B) 6.81% C) 9.21% D) 10.31%

21) Thedistributionofcholesterollevelsinteenageboysisapproximatelynormalwithμ=170andσ =30 (Source:

U.S.NationalCenterforHealthStatistics).Levelsabove200warrantattention.Whatpercentofteenageboys

havelevelsbetween170and225?

A) 3.36% B) 6.06% C) 46.64% D) 56.13%

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22) Assumethatbloodpressurereadingsarenormallydistributedwithμ=120 andσ=8.Abloodpressure

readingof145ormoremayrequiremedicalattention.Whatpercentofpeoplehaveabloodpressurereading

greaterthan145?

A) 0.09% B) 99.91% C) 6.06% D) 11.09%

23) AssumethattheheightsofAmericanmenarenormallydistributedwithameanof69.0inchesandastandard

deviationof2.8inches.TheU.S.MarineCorpsrequiresthatmenhaveheightsbetween64and78inches.Find

thepercentofmenmeetingtheseheightrequirements.

A) 96.26% B) 3.67% C) 99.93% D) 31.12%

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

24) Assumethattheheightsofwomenarenormallydistributedwithameanof63.6inchesandastandard

deviationof2.5inches.TheU.S.Armyrequiresthattheheightsofwomenbebetween58and80inches.Ifa

womanisrandomlyselected,whatistheprobabilitythatherheightisbetween58and80inches?

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Provideanappropriateresponse.

25) ThegraphdepictsIQscoresofadults,andthosescoresarenormallydistributedwithameanof100anda

standarddeviationof15.Findtheprobabilitythatamemberselectedatrandomisfromtheshadedareaofthe

graph.

A) 0.7486 B) 0.4400 C) 0.6293 D) 0.8051

26) ThegraphdepictsIQscoresofadults,andthosescoresarenormallydistributedwithameanof100anda

standarddeviationof15.Findtheprobabilitythatamemberselectedatrandomisfromtheshadedareaofthe

graph.

A) 0.7938 B) 0.7745 C) 0.7303 D) 0.7619

2 InterpretNormalDistributions

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Provideanappropriateresponse.UsetheStandardNormalTabletofindtheprobability.

1) Thelengthsofpregnanciesarenormallydistributedwithameanof268daysandastandarddeviationof15

days.Outof50pregnancies,howmanywouldyouexpecttolastlessthan250days?

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2) Thedistributionofcholesterollevelsinteenageboysisapproximatelynormalwithμ=170andσ =30. Levels

above200warrantattention.If95teenageboysareexamined,howmanywouldyouexpecttohavecholesterol

levelsgreaterthan225?

3) Anairlineknowsfromexperiencethatthedistributionofthenumberofsuitcasesthatgetlosteachweekona

certainrouteisapproximatelynormalwithμ=15.5andσ=3.6.Inoneyear,howmanyweekswouldyou

expecttheairlinetolosebetween10and20suitcases?

4) Assumethattheheightsofwomenarenormallydistributedwithameanof63.6inchesandastandard

deviationof2.5inches.TheU.S.Armyrequiresthattheheightsofwomenbebetween58and80inches.If200

womenwanttoenlistintheU.S.Army,howmanywouldyouexpecttomeettheheightrequirements?

5) Assumethattheheightsofmenarenormallydistributedwithameanof69.0inchesandastandarddeviation

of2.8inches.TheU.S.MarineCorpsrequiresthattheheightsofmenbebetween64and78inches.If500men

wanttoenlistintheU.S.MarineCorps,howmanywouldyounotexpecttomeettheheightrequirements?

5.3 NormalDistributions:FindingValues

1 Findaz–scoreGiventheAreaUndertheNormalCurve

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Provideanappropriateresponse.

1) Findthez–scorethatcorrespondstothegivenareaunderthestandardnormalcurve.

2) Findthez–scorethatcorrespondstothegivenareaunderthestandardnormalcurve.

3) Findthez–scorethatcorrespondstothegivenareaunderthestandardnormalcurve.

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4) Findthez–scorethatcorrespondstothegivenareaunderthestandardnormalcurve.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

5) Findthez–scoresforwhich90%ofthedistributionʹsarealiesbetween–zandz.

A) (–1.645,1.645) B) (–2.33,2.33) C) (–1.96,1.96) D) (–0.99,0.99)

6) Findthez–scoresforwhich98%ofthedistributionʹsarealiesbetween–zandz.

A) (–2.33,2.33) B) (–1.645,1.645) C) (–1.96,1.96) D) (–0.99,0.99)

7) Findthez–scoreforwhich70%ofthedistributionʹsarealiestoitsright.

A) –0.53 B) –0.98 C) –0.81 D) –0.47

8) Findthez–scorethatisgreaterthanthemeanandforwhich70%ofthedistributionʹsarealiestoitsleft.

A) 0.53 B) 0.98 C) 0.81 D) 0.47

9) Useastandardnormaltabletofindthez–scorethatcorrespondstothecumulativeareaof0.7019.

A) 0.53 B) -0.53 C) 0.835 D) -0.835

10) Findthez–scorethathas84.85%ofthedistributionʹsareatoitsright.

A) -1.03 B) 1.03 C) -0.39 D) 0.39

11) Findthez–scoreforwhich99%ofthedistributionʹsarealiesbetween–zandz.

A) (–2.575

2.575) B) (–1.645

1.645) C) (–2.33

2.33) D) (–1.28

1.28)

2 Findaz–scoreGivenaPercentile

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Provideanappropriateresponse.

1) Forthestandardnormalcurve,findthez–scorethatcorrespondstothe90thpercentile.

A) 1.28 B) 0.28 C) 1.52 D) 2.81

2) Forthestandardnormalcurve,findthez–scorethatcorrespondstothe30thpercentile.

A) –0.53 B) –0.98 C) –0.47 D) –0.12

3) Forthestandardnormalcurve,findthez–scorethatcorrespondstothe7thdecile.

A) 0.53 B) 0.98 C) 0.47 D) 0.12

4) Useastandardnormaltabletofindthez–scorethatcorrespondstothe80thpercentile.

A) 0.84 B) 1.405 C) 2.41 D) 0.2

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3 UseNormalDistributionstoAnswerQuestions

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Provideanappropriateresponse.

1) IQtestscoresarenormallydistributedwithameanof100andastandarddeviationof15.Findthex–scorethat

correspondstoaz–scoreof2.33.

A) 134.95 B) 125.95 C) 139.55 D) 142.35

2) IQtestscoresarenormallydistributedwithameanof100andastandarddeviationof15.Findthex–scorethat

correspondstoaz–scoreof–1.645.

A) 75.3 B) 79.1 C) 82.3 D) 91.0

3) Thescoresonamathematicsexamhaveameanof70 andastandarddeviationof5.Findthex–valuethat

correspondstothez–score2.33.

A) 81.7 B) 58.4 C) 75.0 D) 72.3

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

4) Amathematicsprofessorgivestwodifferentteststotwosectionsofhiscollegealgebracourses.Thefirstclass

hasameanof56withastandarddeviationof9whilethesecondclasshasameanof75withastandard

deviationof15.Astudentfromthefirstclassscoresa62onthetestwhileastudentfromthesecondclass

scoresan83onthetest.Comparethescores.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

5) Comparethescores:ascoreof75onatestwithameanof65andastandarddeviationof8andascoreof75on

atestwithameanof70andastandarddeviationof4.

A) Thetwoscoresarestatisticallythesame.

B) Ascoreof75withameanof65andastandarddeviationof8isbetter.

C) Ascoreof75withameanof70andastandarddeviationof4isbetter.

D) Youcannotdeterminewhichscoreisbetterfromthegiveninformation.

6) Comparethescores:ascoreof88onatestwithameanof79andascoreof78onatestwithameanof70.

A) Youcannotdeterminewhichscoreisbetterfromthegiveninformation.

B) Thetwoscoresarestatisticallythesame.

C) Ascoreof75withameanof65andastandarddeviationof8isbetter.

D) Ascoreof75withameanof70andastandarddeviationof4isbetter.

7) Comparethescores:ascoreof220onatestwithameanof200andastandarddeviationof21andascoreof90

onatestwithameanof80andastandarddeviationof8.

A) Ascoreof90withameanof80andastandarddeviationof8isbetter.

B) Ascoreof220withameanof200andastandarddeviationof21isbetter.

C) Thetwoscoresarestatisticallythesame.

D) Youcannotdeterminewhichscoreisbetterfromthegiveninformation.

8) Twohighschoolstudentstookequivalentlanguagetests,oneinGermanandoneinFrench.Thestudenttaking

theGermantest,forwhichthemeanwas66andthestandarddeviationwas8,scoredan82,whilethestudent

takingtheFrenchtest,forwhichthemeanwas27andthestandarddeviationwas5,scoreda35.Comparethe

scores.

A) Ascoreof82withameanof66andastandarddeviationof8isbetter.

B) Ascoreof35withameanof27andastandarddeviationof5isbetter.

C) Thetwoscoresarestatisticallythesame.

D) Youcannotdeterminewhichscoreisbetterfromthegiveninformation.

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9) SATscoreshaveameanof1026andastandarddeviationof209.ACTscoreshaveameanof20.8anda

standarddeviationof4.8.Astudenttakesbothtestswhileajuniorandscores1130ontheSATand25onthe

ACT.Comparethescores.

A) Ascoreof25ontheACTtestwasbetter.

B) Ascoreof1130ontheSATtestwasbetter.

C) Thetwoscoresarestatisticallythesame.

D) Youcannotdeterminewhichscoreisbetterfromthegiveninformation.

10) SATscoreshaveameanof1026andastandarddeviationof209.ACTscoreshaveameanof20.8anda

standarddeviationof4.8.Astudenttakesbothtestswhileajuniorandscores860ontheSATand16onthe

ACT.Comparethescores.

A) Ascoreof860ontheSATtestwasbetter.

B) Ascoreof16ontheACTtestwasbetter.

C) Thetwoscoresarestatisticallythesame.

D) Youcannotdeterminewhichscoreisbetterfromthegiveninformation.

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

11) Assumethatbloodpressurereadingsarenormallydistributedwithμ=114 andσ=8.Aresearcherwishesto

selectpeopleforastudybutwantstoexcludethetopandbottom10percent.Whatwouldbetheupperand

lowerreadingstoqualifypeopletoparticipateinthestudy?

12) AssumethatthesalariesofelementaryschoolteachersintheUnitedStatesarenormallydistributedwitha

meanof$29,000andastandarddeviationof$2000.Whatisthecutoffsalaryforteachersinthetop10%?

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

13) AssumethatthesalariesofelementaryschoolteachersintheUnitedStatesarenormallydistributedwitha

meanof$37,000andastandarddeviationof$5000.Whatisthecutoffsalaryforteachersinthebottom10%?

A) $30,600 B) $43,400 C) $28,775 D) $45,225

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

14) Thetimesforcompletingonecircuitofabicyclecoursearenormallydistributedwithameanof62.9 minutes

andastandarddeviationof6.1minutes.Anassociationwantstosponsoraracebutwillcutthebottom25%of

riders.Inatrialrun,whatshouldbethecutofftime?

15) Assumethattheheightsofmenarenormallydistributedwithameanof69.4 inchesandastandarddeviation

of1.3inches.Ifthetop5percentandbottom5percentareexcludedforanexperiment,whatarethecutoff

heightstobeeligibleforthisexperiment?Roundyouranswerstoonedecimalplace.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

16) Assumethattheheightsofwomenarenormallydistributedwithameanof62.2 inchesandastandard

deviationof2.3inches.FindQ3,thethirdquartilethatseparatesthebottom75%fromthetop25%.

A) 63.8 B) 60.6 C) 64.8 D) 65.1

17) Thebodytemperaturesofadultsarenormallydistributedwithameanof98.6 ° Fandastandarddeviationof

0.95°F.Whattemperaturerepresentsthe95thpercentile?

A) 100.16°F B) 97.04° F C) 99.82° F D) 100.46° F

18) Inacertainnormaldistribution,findthestandarddeviationσ whenμ =50and10.56%ofthearealiestothe

rightof55.

A) 4 B) 2 C) 3 D) 5

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19) Inacertainnormaldistribution,findthemeanμ whenσ =5and5.48%ofthearealiestotheleftof78.

A) 86 B) 70 C) 94 D) 62

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

20) Inacertainnormaldistribution,6.3%ofthearealiestotheleftof36and6.3%ofthearealiestotherightof42.

Findthemeanμandthestandarddeviationσ.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

21) Atirecompanyfindsthelifespanforonebrandofitstiresisnormallydistributedwithameanof46,800 miles

andastandarddeviationof4000miles.Ifthemanufactureriswillingtoreplacenomorethan10%ofthetires,

whatshouldbetheapproximatenumberofmilesforawarranty?

A) 41,680 B) 51,920 C) 40,220 D) 53,380

5.4 SamplingDistributionsandtheCentralLimitTheorem

1 InterpretSamplingDistributions

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Provideanappropriateresponse.

1) Thedistributionofroomandboardexpensesperyearatafour–yearcollegeisnormallydistributedwitha

meanof$5850andstandarddeviationof$1125.Randomsamplesofsize20aredrawnfromthispopulation

andthemeanofeachsampleisdetermined.Whichofthefollowingmeanexpenseswouldbeconsidered

unusual?

A) $5180 B) $6350 C) $6180 D) noneofthese

2 FindProbabilities

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Provideanappropriateresponse.

1) Thelengthsofpregnanciesarenormallydistributedwithameanof264 daysandastandarddeviationof15

days.If36womenarerandomlyselected,findtheprobabilitythattheyhaveameanpregnancybetween264

daysand266days.

A) 0.2881 B) 0.7881 C) 0.2119 D) 0.5517

2) Thelengthsofpregnanciesarenormallydistributedwithameanof268daysandastandarddeviationof15

days.If64womenarerandomlyselected,findtheprobabilitythattheyhaveameanpregnancybetween266

daysand268days.

A) 0.3577 B) 0.7881 C) 0.2881 D) 0.5517

3) Assumethattheheightsofwomenarenormallydistributedwithameanof63.6inchesandastandard

deviationof2.5inches.If100womenarerandomlyselected,findtheprobabilitythattheyhaveameanheight

greaterthan63.0inches.

A) 0.9918 B) 0.0082 C) 0.2881 D) 0.8989

4) Assumethattheheightsofwomenarenormallydistributedwithameanof63.6inchesandastandard

deviationof2.5inches.If75womenarerandomlyselected,findtheprobabilitythattheyhaveameanheight

between63and65inches.

A) 0.9811 B) 0.3071 C) 0.0188 D) 0.2119

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5) Assumethattheheightsofmenarenormallydistributedwithameanof69.5 inchesandastandarddeviation

of2.1inches.If36menarerandomlyselected,findtheprobabilitythattheyhaveameanheightgreaterthan

70.5inches.

A) 0.0021 B) 0.8188 C) 0.9005 D) 9.9671

6) Assumethattheheightsofmenarenormallydistributedwithameanof69.0inchesandastandarddeviation

of2.8inches.If64menarerandomlyselected,findtheprobabilitythattheyhaveameanheightbetween68

and70inches.

A) 0.9958 B) 0.9979 C) 0.0021 D) 0.9015

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

7) Thebodytemperaturesofadultsarenormallydistributedwithameanof98.6 ° Fandastandarddeviationof

0.60°F.If25adultsarerandomlyselected,findtheprobabilitythattheirmeanbodytemperatureislessthan

99°F.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

8) Thebodytemperaturesofadultsarenormallydistributedwithameanof98.6 ° Fandastandarddeviationof

0.60°F.If36adultsarerandomlyselected,findtheprobabilitythattheirmeanbodytemperatureisgreater

than98.4°F.

A) 0.9772 B) 0.0228 C) 0.8188 D) 0.9360

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

9) AssumethatthesalariesofelementaryschoolteachersintheUnitedStatesarenormallydistributedwitha

meanof$32,000andastandarddeviationof$3000.If100teachersarerandomlyselected,findtheprobability

thattheirmeansalaryislessthan$32,500.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

10) AssumethatthesalariesofelementaryschoolteachersintheUnitedStatesarenormallydistributedwitha

meanof$32,000andastandarddeviationof$3000.If100teachersarerandomlyselected,findtheprobability

thattheirmeansalaryisgreaterthan$32,500.

A) 0.0475 B) 0.9525 C) 0.3312 D) 0.1312

11) Assumethatbloodpressurereadingsarenormallydistributedwithameanof115andastandarddeviationof

8.If100peoplearerandomlyselected,findtheprobabilitythattheirmeanbloodpressurewillbelessthan117.

A) 0.9938 B) 0.0062 C) 0.8819 D) 0.8615

12) Assumethatbloodpressurereadingsarenormallydistributedwithameanof120andastandarddeviationof

8.If100peoplearerandomlyselected,findtheprobabilitythattheirmeanbloodpressurewillbegreaterthan

122.

A) 0.0062 B) 0.9938 C) 0.8819 D) 0.8615

13) Theaveragenumberofpoundsofredmeatapersonconsumeseachyearis196withastandarddeviationof22

pounds(Source:AmericanDieteticAssociation).Ifasampleof50individualsisrandomlyselected,findthe

probabilitythatthemeanofthesamplewillbelessthan200pounds.

A) 0.9015 B) 0.0985 C) 0.7613 D) 0.8815

14) Theaveragenumberofpoundsofredmeatapersonconsumeseachyearis196withastandarddeviationof22

pounds(Source:AmericanDieteticAssociation).Ifasampleof50individualsisrandomlyselected,findthe

probabilitythatthemeanofthesamplewillbegreaterthan200pounds.

A) 0.0985 B) 0.7613 C) 0.8815 D) 0.9015

Page124

15) Acoffeemachinedispensesnormallydistributedamountsofcoffeewithameanof12ouncesandastandard

deviationof0.2ounce.Ifasampleof9cupsisselected,findtheprobabilitythatthemeanofthesamplewillbe

lessthan12.1ounces.Findtheprobabilityifthesampleisjust1cup.

A) 0.9332;0.6915 B) 0.9332;0.1915 C) 0.4332;0.1915 D) 0.4332;0.6915

16) Acoffeemachinedispensesnormallydistributedamountsofcoffeewithameanof12ouncesandastandard

deviationof0.2ounce.Ifasampleof9cupsisselected,findtheprobabilitythatthemeanofthesamplewillbe

greaterthan12.1ounces.Findtheprobabilityifthesampleisjust1cup.

A) 0.0668;0.3085 B) 0.0668;0.8085 C) 0.5668;0.8085 D) 0.5668;0.3085

3 UseCentralLimitTheoremtoFindMean/StdErrorofMean

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

UsetheCentralLimitTheoremtofindthemeanandstandarderrorofthemeanoftheindicatedsamplingdistribution.

1) Theamountsoftimeemployeesofatelecommunicationscompanyhaveworkedforthecompanyarenormally

distributedwithameanof5.4yearsandastandarddeviationof2.1years.Randomsamplesofsize12are

drawnfromthepopulationandthemeanofeachsampleisdetermined.

A) 5.4years,0.61years B) 5.4 years,0.18 years

C) 1.56years,0.61years D) 1.56 years,2.1 years

2) Themonthlyrentsforstudioapartmentsinacertaincityhaveameanof$920 andastandarddeviationof$190.

Randomsamplesofsize30aredrawnfromthepopulationandthemeanofeachsampleisdetermined.

A) $920

$34.69 B) $920

$6.33 C) $167.97

$34.69 D) $167.97

$190

Provideanappropriateresponse.

3) Scoresonatesthaveameanof72andastandarddeviationof12.Randomsamplesofsize64aredrawnfrom

thispopulationandthemeanofeachsampleisdetermined.UsetheCentralLimitTheoremtofindthemean

andstandarderrorofthemeanofthesamplingdistribution.Thensketchagraphofthesamplingdistribution.

A) 72,1.5

67.5 69 70.5 72 73.5 75 76.5

B) 72,12

36 48 60 72 84 96 108

C) 72,9

45 54 63 72 81 90 99

D) 9,1.5

4.5 6 7.5 9 10.5 12 13.5

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4) Theweightsofpeopleinacertainpopulationarenormallydistributedwithameanof157lbandastandard

deviationof23lb.Findthemeanandstandarderrorofthemeanforthissamplingdistributionwhenusing

randomsamplesofsize8.

A) 157

8.13 B) 157

23 C) 157

8 D) 157

2.88

5) Themeanannualincomeforadultwomeninonecityis$28,520andthestandarddeviationoftheincomesis

$5600.Thedistributionofincomesisskewedtotheright.Findthemeanandstandarderrorofthemeanfor

thissamplingdistributionwhenusingrandomsamplesofsize66.Roundyouranswerstothenearestdollar.

A) $28,520,$689 B) $28,520,$5600 C) $28,520,$66 D) $28,520,$85

6) Whathappenstothemeanandstandarddeviationofthedistributionofsamplemeansasthesizeofthe

sampledecreases?

A) Themeanofthesamplemeansstaysconstantandthestandarderrorincreases.

B) Themeanofthesamplemeansstaysconstantandthestandarderrordecreases.

C) Themeanofthesamplemeansdecreasesandthestandarderrorincreases.

D) Themeanofthesamplemeansincreasesandthestandarderrorstaysconstant.

7) Ifthesamplesizeismultipliedby4,whathappenstothestandarddeviationofthedistributionofsample

means?

A) Thestandarderrorishalved.

B) Thestandarderrorisdecreasedbyafactorof4.

C) Thestandarderrorisdoubled.

D) Thestandarderrorisincreasedbyafactorof4.

4 DetermineiftheFiniteCorrectionFactorShouldbeUsed

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Provideanappropriateresponse.

1) Astudyof800homeownersinacertainareashowedthattheaveragevalueofthehomesis$182,000andthe

standarddeviationis$15,000.If50homesareforsale,findtheprobabilitythatthemeanvalueofthesehomes

islessthan$185,000.Remember:checktoseeifthefinitecorrectionfactorapplies.

2) Astudyof800homeownersinacertainareashowedthattheaveragevalueofthehomesis$182,000andthe

standarddeviationis$15,000.If50homesareforsale,findtheprobabilitythatthemeanvalueofthesehomes

isgreaterthan$185,000.Remember:checktoseeifthefinitecorrectionfactorapplies.

5Concepts

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Provideanappropriateresponse.

1) Asodamachinedispensesnormallydistributedamountsofsodawithameanof20ouncesandastandard

deviationof0.2ounce.Areyoumorelikelytorandomlyselectonebottlewithmorethan20.3ouncesorare

youmorelikelytoselectasampleofeightbottleswithameanamountofmorethan20.3ounces?Explain.

2) Asodamachinedispensesnormallydistributedamountsofsodawithameanof20ouncesandastandard

deviationof0.2ounce.Areyoumorelikelytorandomlyselectonebottlewithanamountbetween19.8ounces

and20.2ouncesorareyoumorelikelytoselectasampleofeightbottleswithameanamountbetween19.8

ouncesand20.2ounces?Explain.

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5.5 NormalApproximationstoBinomialDistributions

1 DecideiftheNormalDistributionCanbeUsedtoApproximatetheBinomialDistribution

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Provideanappropriateresponse.

1) Inarecentsurvey,83%ofthecommunityfavoredbuildingapolicesubstationintheirneighborhood.You

randomlyselect16citizensandaskeachifheorshethinksthecommunityneedsapolicesubstation.

Decidewhetheryoucanusethenormaldistributiontoapproximatethebinomialdistribution.Ifso,findthe

meanandstandarddeviation.Ifnot,explainwhy.

2) Arecentsurveyfoundthat72%ofalladultsover50wearglassesfordriving.Yourandomlyselect33 adults

over50,andaskifheorshewearsglasses.Decidewhetheryoucanusethenormaldistributiontoapproximate

thebinomialdistribution.Ifso,findthemeanandstandarddeviation,Ifnot,explainwhy.

3) Accordingtogovernmentdata,theprobabilitythananadultwasnevermarriedis19%.Yourandomlyselect49

adultsandaskifheorshewasevermarried.Decidewhetheryoucanusethenormaldistributionto

approximatethebinomialdistribution,Ifso,findthemeanandstandarddeviation,Ifnot,explainwhy.

4) Decideifitisappropriatetousethenormaldistributiontoapproximatetherandomvariablexforabinomial

experimentwithsamplesizeofn=5andprobabilityofsuccessp=0.3.

5) Decideifitisappropriatetousethenormaldistributiontoapproximatetherandomvariablexforabinomial

experimentwithsamplesizeofn=48andprobabilityofsuccessp=0.6.

6) Decideifitisappropriatetousethenormaldistributiontoapproximatetherandomvariablexforabinomial

experimentwithsamplesizeofn=25andprobabilityofsuccessp=0.2

2 FindtheCorrectionforContinuity

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Provideanappropriateresponse.

1) Tenpercentofthepopulationisleft–handed.Aclassof100studentsisselected.Convertthebinomial

probabilityP(x>12)toanormalprobabilitybyusingthecorrectionforcontinuity.

A) P(x>12.5) B) P(x<11.5) C) P(x≤12.5) D) P(x≥11.5)

2) Tenpercentofthepopulationisleft–handed.Aclassof1800 studentsisselected.Convertthebinomial

probabilityP(x≥15)toanormalprobabilitybyusingthecorrectionforcontinuity.

A) P(x≥14.5) B) P(x<14.5) C) P(x≤15.5) D) P(x≥15.5)

3) Tenpercentofthepopulationisleft–handed.Aclassof100studentsisselected.Convertthebinomial

probabilityP(x<12)toanormalprobabilitybyusingthecorrectionforcontinuity.

A) P(x<11.5) B) P(x>12.5) C) P(x≤12.5) D) P(x≥11.5)

4) Tenpercentofthepopulationisleft–handed.Aclassof5350 studentsisselected.Convertthebinomial

probabilityP(x≤22)toanormalprobabilitybyusingthecorrectionforcontinuity.

A) P(x≤22.5) B) P(x<21.5) C) P(x>22.5) D) P(x≥21.5)

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3 FindaProbabilityUsingtheAppropriateDistribution

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Provideanappropriateresponse.

1) Thefailurerateinastatisticsclassis10%.Inaclassof40 students,findtheprobabilitythatexactlyfivestudents

willfail.Usethenormaldistributiontoapproximatethebinomialdistribution.

2) Alocalmotelhas100rooms.Theoccupancyrateforthewintermonthsis60%.Findtheprobabilitythatina

givenwintermonthatleast70roomswillberented.Usethenormaldistributiontoapproximatethebinomial

distribution.

3) Alocalmotelhas50rooms.Theoccupancyrateforthewintermonthsis60%.Findtheprobabilitythatina

givenwintermonthfewerthan35roomswillberented.Usethenormaldistributiontoapproximatethe

binomialdistribution.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

4) Astudentanswersall48questionsonamultiple–choicetestbyguessing.Eachquestionhasfourpossible

answers,onlyoneofwhichiscorrect.Findtheprobabilitythatthestudentgetsexactly15correctanswers.Use

thenormaldistributiontoapproximatethebinomialdistribution.

A) 0.0823 B) 0.8577 C) 0.7967 D) 0.0606

5) Iftheprobabilityofanewbornchildbeingfemaleis0.5,findtheprobabilitythatin100births,55ormorewill

befemale.Usethenormaldistributiontoapproximatethebinomialdistribution.

A) 0.1841 B) 0.7967 C) 0.8159 D) 0.0606

6) Anairlinereportsthatithasbeenexperiencinga15%rateofno–showsonadvancedreservations.Among150

advancedreservations,findtheprobabilitythattherewillbefewerthan20no–shows.

A) 0.2451 B) 0.7549 C) 0.7967 D) 0.3187

7) Findtheprobabilitythatin200tossesofafairsix–sideddie,afivewillbeobtainedatleast40times.

A) 0.1210 B) 0.8810 C) 0.0871 D) 0.3875

8) Findtheprobabilitythatin200tossesofafairsix–sideddie,afivewillbeobtainedatmost40times.

A) 0.9131 B) 0.1190 C) 0.8810 D) 0.0853

9) Atelemarketerfoundthattherewasa1%chanceofasalefromhisphonesolicitations.Findtheprobabilityof

getting5ormoresalesfor1000telephonecalls.

A) 0.9599 B) 0.0401 C) 0.8810 D) 0.0871

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

10) Theauthorofastatisticsbookhastroubledecidingwhethertousethewordsʺheʺorʺsheʺinthebookʹs

examples.Tosolvetheproblem,theauthorflipsacoineachtimetheproblemarises.Ifaheadshows,the

authorusesʺheʺandifatailshows,theauthorusesʺsheʺ.Ifthisproblemoccurs100timesinthebook,whatis

theprobabilitythatʺsheʺwillbeused58times?

4Concepts

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Provideanappropriateresponse.

1) MatchthebinomialprobabilityP(x<23)withthecorrectstatement.

A) P(therearefewerthan23successes) B) P(thereareatmost23successes)

C) P(therearemorethan23successes) D) P(thereareatleast23successes)

Page128

2) MatchthebinomialprobabilityP(x<40)withthecorrectstatement.

A) P(therearefewerthan40successes) B) P(thereareatmost40successes)

C) P(thereareatleast 40successes) D) P(therearemorethan40successes)

3) MatchthebinomialprobabilityP(x≤23)withthecorrectstatement.

A) P(thereareatmost23successes) B) P(thereareatleast23successes)

C) P(therearemorethan23successes) D) P(therearefewerthan23successes)

4) MatchthebinomialprobabilityP(x>23)withthecorrectstatement.

A) P(therearemorethan23successes) B) P(thereareatleast23successes)

C) P(thereareatmost23successes) D) P(therearefewerthan23successes)

5) Tenpercentofthepopulationisleft–handed.Inaclassof100students,writethebinomialprobabilityforthe

statementʺThereareatleast12left–handedstudentsintheclass.ʺ

A) P(x≥12) B) P(x<12) C) P(x>12) D) P(x≤12)

6) Tenpercentofthepopulationisleft–handed.Inaclassof100students,writethebinomialprobabilityforthe

statementʺTherearemorethan12left–handedstudentsintheclass.ʺ

A) P(x>12) B) P(x<12) C) P(x≤12) D) P(x=12)

7) Tenpercentofthepopulationisleft–handed.Inaclassof100students,writethebinomialprobabilityforthe

statementʺThereareatmost12left–handedstudentsintheclass.ʺ

A) P(x≤12) B) P(x<12) C) P(x=12) D) P(x≥12)

8) Tenpercentofthepopulationisleft–handed.Inaclassof56 students,writethebinomialprobabilityforthe

statementʺThereareatleast10left–handedstudentsintheclass.ʺ

A) P(x≥10) B) P(x≤10) C) P(x>10) D) P(x<10)

9) Tenpercentofthepopulationisleft–handed.Inaclassof100students,writethebinomialprobabilityforthe

statementʺThereareexactly12left–handedstudentsintheclass.ʺ

A) P(x=12) B) P(x≥12) C) P(x≤12) D) P(x>12)

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Ch.5 NormalProbabilityDistributions

AnswerKey

5.1 IntroductiontoNormalDistributionsandtheStandardNormalDistributio

1 FindAreasUndertheStandardNormalCurve

2 DetermineifDistributionisNormal

3 Computez–Score

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4 FindProbabilitiesUsingtheStandardNormalCurve

5.2 NormalDistributions:FindingProbabilities

1 FindProbabilitiesforNormallyDistributedVariables

2InterpretNormalDistributions

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5.3 NormalDistributions:FindingValues

1 Findaz–scoreGiventheAreaUndertheNormalCurve

2 Findaz–scoreGivenaPercentile

5.4 SamplingDistributionsandtheCentralLimitTheorem

1 InterpretSamplingDistributions

2 FindProbabilities

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3 UseCentralLimitTheoremtoFindMean/StdErrorofMean

4 DetermineiftheFiniteCorrectionFactorShouldbeUsed

5Concepts

5.5 NormalApproximationstoBinomialDistributions

1 DecideiftheNormalDistributionCanbeUsedtoApproximatetheBinomialDistribution

2 FindtheCorrectionforContinuity

3 FindaProbabilityUsingtheAppropriateDistribution

4Concepts

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