ch07 - the normal distribution - rick jerz homepage · · 2018-04-30goals • understand the...

TheNormalProbabilityDistribution

Chapter7Dr.RichardJerz

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Goals

• Understandthedifferencebetweendiscreteandcontinuous distributions.

• Computethemean,standarddeviation,andprobabilitiesforauniformdistribution.

• Listthecharacteristicsofthenormalprobabilitydistribution.

• Defineandcalculatezvalues.• Determinetheprobabilityanobservationisabove orbelow apoint,orisbetween twopointsonanormalprobabilitydistribution.

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Adistribution

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ADiscreteProbabilityDistribution

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ResultsofTossingTwoDice

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Probabilitiesinvolve…

• Areasofinterestunderaprobabilitydistribution

• “Areaunderthecurve”• Totalarea=1• Wecanmakethewidthofeachbar=1

• Area=lengthxwidth• Exactlythesameasaddinguptheheights

• Example:.167x1=.167

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BinomialDistribution:n,x,π

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ExcelModel

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A“Continuous”ProbabilityDistribution(weights)

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ANY“continuous”probabilitydistribution

• Centraltendency• Dispersion• Probabilityquestions

• “Areaundercurve”

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SimpleContinuousDistributionTheUniformDistribution

• Theuniformprobabilitydistributionisperhapsthesimplestdistributionforacontinuousrandomvariable.

• Thisdistributionisrectangularinshapeandisdefinedbyminimumandmaximumvalues.

• Totalarea=1(LxH=1)• (b-a)*(1/(b-a))=1


TheUniformDistribution:MeanandStandardDeviation


Example:UniformDistribution

SouthwestArizonaStateUniversityprovidesbusservicetostudentswhiletheyareoncampus.AbusarrivesattheNorthMainStreetandCollegeDrivestopevery30minutesbetween6A.M.and11P.M.duringweekdays.Studentsarriveatthebusstopatrandomtimes.Thetimethatastudentwaitsisuniformlydistributedfrom0to30minutes.

• Drawagraphofthisdistribution.• Howlongwillastudent“typically”havetowaitforabus?In

otherwordswhatisthemeanwaitingtime?Whatisthestandarddeviationofthewaitingtimes?

• Whatistheprobabilityastudentwillwaitmorethan25minutes?

• Whatistheprobabilityastudentwillwaitbetween10and20minutes?


TheUniformDistributionExample

• Draw a graph of this distribution.

• Area = 1, L=30-0, H=1/(30-0) = .0333


TheUniformDistribution–Example

• Howlongwillastudent“typically”havetowaitforabus?Inotherwordswhatisthemeanwaitingtime?Whatisthestandarddeviationofthewaitingtimes?



• Whatistheprobabilityastudentwillwaitmorethan25minutes?



• Whatistheprobabilityastudentwillwaitbetween10and20minutes?


TheNormalProbabilityDistribution


TheNormalDistribution–Graphically


CharacteristicsofaNormalProbabilityDistribution

• Itisbell-shaped andhasasinglepeakatthecenterofthedistribution.

• Thearithmeticmean,median,andmode areequal• Thetotalareaunderthecurveis1.00;halftheareaunderthenormalcurveistotherightofthiscenterpointandtheotherhalftotheleftofit.

• Itissymmetrical aboutthemean.• Itisasymptotic:ThecurvegetscloserandclosertotheX-axisbutneveractuallytouchesit.Toputitanotherway,thetailsofthecurveextendindefinitelyinbothdirections.

• Thelocation ofanormaldistributionisdeterminedbythemean,µ, thedispersionorspreadofthedistributionisdeterminedbythestandarddeviation,σ .


Whatdataisnormallydistributed?

• Examscores• Height• Weight• Amountofcoffeedispensed

• Manysituations!


“Bell”shape


TheNormalDistribution–Families


TheEmpiricalRule

• About68percentoftheareaunderthenormalcurveiswithinonestandarddeviationofthemean.

• About95percentiswithintwostandarddeviationsofthemean.

• Practicallyalliswithinthreestandarddeviationsofthemean.


TheEmpiricalRule- Example

Aspartofitsquality assuranceprogram, theAutoliteBatteryCompanyconducts testsonbattery life.ForaparticularD-cellalkalinebattery, themeanlifeis19hours.Theusefullifeofthebatteryfollowsanormal distributionwithastandard deviation of1.2hours.

Answerthefollowingquestions.• About68percentofthe

batteries failedbetweenwhattwovalues?

• About95percentofthebatteries failedbetweenwhattwovalues?

• Virtually allofthebatteriesfailedbetweenwhattwovalues?


NormalDistributionandStandard NormalDistribution


TheStandardNormalProbabilityDistribution

• Thestandardnormalprobabilitydistributionisanormalprobabilitydistributionwithameanof0andastandarddeviationof1.

• Itisalsocalledthezdistribution.• Az-valueisthedistancebetweenaselectedvalue,designatedX,andthepopulationmeanµ,dividedbythepopulationstandarddeviation,σ.

• Theformulais:


AreasUndertheNormalCurve


Usingtablefrombackoftextbook


TheNormalDistribution–Example

Theweeklyincomesofshiftforemenintheglassindustryfollowthenormalprobabilitydistributionwithameanof$1,000andastandarddeviationof$100.Whatisthezvaluefortheincome,let’scallitX,ofaforemanwhoearns$1,100perweek?Foraforemanwhoearns$900perweek?


NormalDistribution– FindingProbabilities

• Inanearlierexamplewereportedthatthemeanweeklyincomeofashiftforeman intheglassindustry isnormallydistributedwithameanof$1,000andastandarddeviationof$100.

• Whatisthelikelihoodofselectingaforemanwhoseweeklyincomeisbetween$1,000and$1,100?

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NormalDistributionFindingProbabilities


FindingAreasforZUsingExcel

The Excel function=NORMDIST(x,Mean,Standard_dev,Cumu)=NORMDIST(1100,1000,100,true)generates area (probability) fromZ=1 and below


FindingProbabilities(Example2)

• Refertotheinformationregarding theweeklyincomeofshiftforemenintheglassindustry.Thedistribution ofweeklyincomes followsthenormal probabilitydistribution withameanof$1,000andastandarddeviationof$100.

• Whatistheprobability ofselectingashiftforemanintheglass industrywhoseincomeis:

• Between$790and$1,000?

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FindingProbabilities(Example3)

Refertotheinformationregardingtheweeklyincomeofshiftforemenintheglassindustry.Thedistributionofweeklyincomesfollowsthenormalprobabilitydistributionwithameanof$1,000andastandarddeviationof$100.

Whatistheprobabilityofselectingashiftforemanintheglassindustrywhoseincomeis:

Lessthan$790?


ch07 - the normal distribution - rick jerz homepage · · 2018-04-30goals • understand the...

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