(1.10)-FC5e-Ch06商务统计课件

BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc.Chap6-1Chapter6TheNormalDistributionBusinessStatistics:AFirstCourse

5thEditionPerspectiveDescriptiveversusInferentialStatisticsDescriptive:Chapter1-3Inferentialstatistics:TheremainderofthebookProbabilityDistributionsFrequencyversusprobabilitydistributionsDiscreteversuscontinuousprobabilitydistributionsChapter5:DiscreteprobabilitydistributionsChapter6:Aspecificcontinuousprobabilitydistribution:thenormaldistributionsBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-2BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-3LearningObjectivesInthischapter,youlearn:

TocomputeprobabilitiesfromthenormaldistributionTousethenormalprobabilityplottodeterminewhetherasetofdataisapproximatelynormallydistributedBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-4ContinuousProbabilityDistributionsAcontinuousrandomvariableisavariablethatcanassumeanyvalueonacontinuum(canassumeanuncountablenumberofvalues)thicknessofanitemtimerequiredtocompleteatasktemperatureofasolutionheight,ininchesThesecanpotentiallytakeonanyvaluedependingonlyontheabilitytopreciselyandaccuratelymeasureContinuousProbabilityDistributionsTherearemanytypesofcontinuousprobabilitydistributionsBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-5BellShaped

Probability

DistributionConstant

Probability

DistributionExponential

Probability

DistributionBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-6TheNormalDistributionBellShapedSymmetricalMean,MedianandModeareEqualOccursofteninbusinessMostcommonly,itoccursbecauseofrandomerrorsMean=Median=ModeXf(X)μσBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-7TheNormalDistributionLocationisdeterminedbythemean,μSpreadisdeterminedbythestandarddeviation,σ

Therandomvariablehasaninfinitetheoreticalrange:+

toMean=Median=ModeXf(X)μσBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-8TheNormalDistribution

DensityFunctionTheformulaforthenormalprobabilitydensityfunctionisWhere e=themathematicalconstantapproximatedby2.71828

π=themathematicalconstantapproximatedby3.14159

μ=thepopulationmean

σ=thepopulationstandarddeviation X=anyvalueofthecontinuousvariableBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-9Byvaryingtheparametersμandσ,weobtaindifferentnormaldistributionsManyNormalDistributionsBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-10TheNormalDistributionShapeXf(X)μσChanging

μshiftsthedistributionleftorright.Changingσincreasesordecreasesthespread.BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-11TheStandardizedNormalAnynormaldistribution(withanymeanandstandarddeviationcombination)canbetransformedintothestandardizednormal

distribution(Z)NeedtotransformXunitsintoZunitsThestandardizednormaldistribution(Z)hasameanof0andastandarddeviationof1BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-12TranslationtotheStandardizedNormalDistributionTranslatefromXtothestandardizednormal(the“Z”distribution)bysubtractingthemeanofXanddividingbyitsstandarddeviation:TheZdistributionalwayshasmean=0andstandarddeviation=1BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-13TheStandardizedNormalProbabilityDensityFunctionTheformulaforthestandardizednormalprobabilitydensityfunctionisWhere e=themathematicalconstantapproximatedby2.71828

π=themathematicalconstantapproximatedby3.14159 Z=anyvalueofthestandardizednormaldistributionBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-14TheStandardized

NormalDistributionAlsoknownasthe“Z”distributionMeanis0StandardDeviationis1Zf(Z)01ValuesabovethemeanhavepositiveZ-values,valuesbelowthemeanhavenegativeZ-valuesBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-15ExampleIfXisdistributednormallywithmeanof100andstandarddeviationof50,theZvalueforX=200

isThissaysthatX=200istwostandarddeviations(2incrementsof50units)abovethemeanof100.BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-16ComparingXandZunitsZ1002.00200XNotethattheshapeofthedistributionisthesame,onlythescalehaschanged.Wecanexpresstheprobleminoriginalunits(X)orinstandardizedunits(Z)(μ=100,σ=50)(μ=0,σ=1)ProbabilitiesBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-17Foradiscreterandomvariable,wecantalkaboutthepossibilityofasingleoccurrence,twooccurrences,etc.Incontrast,foracontinuousrandomvariable,wecanonlytalkabouttheprobabilityofarangeofvalues.BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-18FindingNormalProbabilities

abXf(X)PaXb()≤Probabilityismeasuredbytheareaunderthecurve≤PaXb()<<=(Notethattheprobabilityofanyindividualvalueiszero)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-19f(X)XμProbabilityas

AreaUndertheCurve0.50.5Thetotalareaunderthecurveis1.0,andthecurveissymmetric,sohalfisabovethemean,halfisbelowBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-20TheStandardizedNormalTable

TheCumulativeStandardizedNormaltableinthetextbook(AppendixtableE.2)givestheprobabilitylessthanadesiredvalueofZ(i.e.,fromnegativeinfinitytoZ)Z02.000.9772Example:

P(Z<2.00)=0.9772BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-21TheStandardizedNormalTable

ThevaluewithinthetablegivestheprobabilityfromZ=

uptothedesiredZvalue.97722.0P(Z<2.00)=0.9772

TherowshowsthevalueofZtothefirstdecimalpoint

Thecolumn

givesthevalueofZtotheseconddecimalpoint2.0...(continued)Z0.000.010.02…0.00.1BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-22GeneralProcedureforFindingNormalProbabilities

DrawthenormalcurvefortheproblemintermsofXTranslateX-valuestoZ-valuesUsetheStandardizedNormalTableTofindP(a<X<b)whenXisdistributednormally:BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-23FindingNormalProbabilitiesLetXrepresentthetimeittakestodownloadanimagefilefromtheinternet.SupposeXisnormalwithmean8.0andstandarddeviation5.0.FindP(X<8.6)X8.68.0BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-24LetXrepresentthetimeittakestodownloadanimagefilefromtheinternet.SupposeXisnormalwithmean8.0andstandarddeviation5.0.FindP(X<8.6)Z0.120X8.68μ=8

σ=10μ

=0σ=1(continued)FindingNormalProbabilitiesP(X<8.6)P(Z<0.12)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-25Z0.12Z.00.010.0.5000.5040.5080.5398.54380.2.5793.5832.58710.3.6179.6217.6255Solution:FindingP(Z<0.12).5478.020.1.5478StandardizedNormalProbabilityTable(Portion)0.00=P(Z<0.12)P(X<8.6)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-26

FindingNormal

UpperTailProbabilitiesSupposeXisnormalwithmean8.0andstandarddeviation5.0.NowFindP(X>8.6)X8.68.0BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-27NowFindP(X>8.6)…(continued)Z0.120Z0.120.547801.0001.0-0.5478=0.4522P(X>8.6)=P(Z>0.12)=1.0-P(Z≤0.12)=1.0-0.5478=0.4522

FindingNormal

UpperTailProbabilitiesBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-28FindingaNormalProbabilityBetweenTwoValuesSupposeXisnormalwithmean8.0andstandarddeviation5.0.FindP(8<X<8.6)P(8<X<8.6)=P(0<Z<0.12)Z0.120X8.68CalculateZ-values:BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-29Z0.12Solution:FindingP(0<Z<0.12)0.04780.00=P(0<Z<0.12)P(8<X<8.6)=P(Z<0.12)–P(Z≤0)=0.5478-.5000=0.04780.5000Z.00.010.0.5000.5040.5080.5398.54380.2.5793.5832.58710.3.6179.6217.6255.020.1.5478StandardizedNormalProbabilityTable(Portion)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-30SupposeXisnormalwithmean8.0andstandarddeviation5.0.NowFindP(7.4<X<8)X7.48.0ProbabilitiesintheLowerTailBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-31ProbabilitiesintheLowerTailNowFindP(7.4<X<8)…X7.48.0P(7.4<X<8)=P(-0.12<Z<0)=P(Z<0)–P(Z≤-0.12)=0.5000-0.4522=0.0478(continued)0.04780.4522Z-0.120TheNormaldistributionissymmetric,sothisprobabilityisthesameasP(0<Z<0.12)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-32EmpiricalRules

μ±1σenclosesabout68.26%ofX’s

f(X)Xμμ+1σμ-1σWhatcanwesayaboutthedistributionofvaluesaroundthemean?Foranynormaldistribution:σσ68.26%BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-33TheEmpiricalRule

μ±2σ

coversabout95%ofX’s

μ±3σ

coversabout99.7%ofX’sxμ2σ2σxμ3σ3σ95.44%99.73%(continued)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-34StepstofindtheXvalueforaknownprobability: 1.FindtheZvaluefortheknownprobability 2.ConverttoXunitsusingtheformula:GivenaNormalProbability

FindtheXValueBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-35FindingtheXvalueforaKnownProbabilityExample:LetXrepresentthetimeittakes(inseconds)todownloadanimagefilefromtheinternet.SupposeXisnormalwithmean8.0andstandarddeviation5.0FindXsuchthat20%ofdownloadtimesarelessthanX.X?8.00.2000Z?0(continued)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-36FindtheZvaluefor

20%intheLowerTail20%areainthelowertailisconsistentwithaZvalueof-0.84Z.03-0.9.1762.1736.2033-0.7.2327.2296.04-0.8.2005StandardizedNormalProbabilityTable(Portion).05.1711.1977.2266…………X?8.00.2000Z-0.8401.FindtheZvaluefortheknownprobabilityBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-37 2.ConverttoXunitsusingtheformula:FindingtheXvalue

So20%ofthevaluesfromadistributionwithmean8.0andstandarddeviation5.0arelessthan3.80BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-38EvaluatingNormalityNotallcontinuousdistributionsarenormalItisimportanttoevaluatehowwellthedatasetisapproximatedbyanormaldistribution.Normallydistributeddatashouldapproximatethetheoreticalnormaldistribution:Thenormaldistributionisbellshaped(symmetrical)wherethemeanisequaltothemedian.Theempiricalruleappliestothenormaldistribution.Theinterquartilerangeofanormaldistributionis1.33standarddeviations.BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-39EvaluatingNormalityComparingdatacharacteristicstotheoreticalpropertiesConstructchartsorgraphsForsmall-ormoderate-sizeddatasets,constructastem-and-leafdisplayoraboxplottocheckforsymmetryForlargedatasets,doesthehistogramorpolygonappearbell-shaped?ComputedescriptivesummarymeasuresDothemean,medianandmodehavesimilarvalues?Istheinterquartilerangeapproximately1.33σ?Istherangeapproximately6σ?(continued)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-40EvaluatingNormalityComparingdatacharacteristicstotheoreticalpropertiesObservethedistributionofthedatasetDoapproximately2/3oftheobservationsliewithinmean±1standarddeviation?Doapproximately80%oftheobservationsliewithinmean±1.28standarddeviations?Doapproximately95%oftheobservationsliewithinmean±2standarddeviations?EvaluatenormalprobabilityplotIsthenormalprobabilityplotapproximatelylinear(i.e.astraightline)withpositiveslope?(continued)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-41Constructing

ANormalProbabilityPlotNormalprobabilityplotArrangedataintoorderedarrayFindcorrespondingstandardizednormalquantilevalues(Z)Plotthepairsofpointswithobserveddatavalues(X)ontheverticalaxisandthestandardizednormalquantilevalues(Z)onthehorizontalaxisEvaluatetheplotforevidenceoflinearityBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-42Anormalprobabilityplotfordatafromanormaldistributionwillbeapproximatelylinear:306090-2-1012ZXTheNormalProbabilityPlot

InterpretationBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-43NormalProbabilityPlot

InterpretationLeft-SkewedRight-SkewedRectangular306090-2-1012ZX(continued)306090-2-1012ZX306090-2-1012ZXNonlinearplotsindicateadeviationfromnormalityBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-44EvaluatingNormality

AnExample:MutualFundsReturnsTheboxplotappearsreasonablysymmetric,withfourloweroutliersat-9.0,-8.0,-8.0,-6.5andoneupperoutlierat35.0.(Thenormaldistributionissymmetric.)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-45EvaluatingNormality

AnExample:MutualFundsReturnsDescriptiveStatistics(continued)Themean(12.5142)isslightlylessthanthemedian(13.1).(Inanormaldistributionthemeanandmedianareequal.)Theinterquartilerangeof9.2isapproximately1.46standarddeviations.(Inanormaldistributiontheinterquartilerangeis1.33standarddeviations.)Therangeof44isequalto6.99standarddeviations.(Inanormaldistributiontherangeis6standarddeviations.)72.2%oftheobservationsarewithin1standarddeviationofthemean.(Inanormaldistributionthispercentageis68.26%.87%oftheobservationsarewithin1.28standarddeviationsofthemean.(Inanormaldistributionpercentageis80%.)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-46EvaluatingNormality

AnExample:MutualFundsReturns(continued)Plotisapproximatelyastraightlineexceptforafewoutliersatthelowendandthehighend.BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-47EvaluatingNormality

AnExample:MutualFundsReturnsConclusionsThereturnsareslightlyleft-skewedThereturnshavemorevaluesconcentratedaroundthemeanthanexpectedTherangeislargerthanexpected(causedbyoneoutlierat35.0)NormalprobabilityplotisreasonablystraightlineOverall,thisdatasetdoesnotgreatlydifferfromthetheoreticalpropertiesofthenormaldistribution(continued)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-48ChapterSummaryPresentednormaldistributionFoundprobabilitiesforthenormaldistributionAppliednormaldistributiontoproblemsShowedhowtoassesswhetherornotadistributionisnormalBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap6-49ExerciseAnindustrialsewingmachineusesballbearingsthataretargetedtohaveadiameterof0.75inch.Thelowerandupperspecificationlimitsunderwhichtheballbearingscanoperateare0.74inchand0.76inch,respectively.Pastexperiencehasindicatedthattheactualdiameteroftheballbearingsisapproximatelynormallydistributed,withameanof0.753inchandastandarddeviationof0.004inch.Whatistheprobabilitythataballbearingisa.betweenthet