商务统计学课件-统计06-学生版

BusinessChapterBusinessTheNormalDistribution Chap6-LearningInthischapter,youTocomputeprobabilitiesfromthenormaldistributionTousethenormalprobabilityplottodeterminewhetherasetofdataisapproxima ynormallydistributed利用正 Chap6-ContinuousProbabilityAcontinuousrandomvariableisavariablethatcanassumeanyvalueonacontinuum(canassumeanuncountablenumberofvalues) thicknessofantimerequiredtocompleteatemperatureofaheight,inThesecanpotentiallytakeonanyvaluedependingonlyontheabilitytopreciselyandaccura Chap6-TheNormalDistributionBellShaped钟 对Mean,MedianandLocationisdeterminedbymean,μ位置由均值Spreadisdeterminedby

σXσμstandarddeviation,σ离散程度由Therandomvariablehasaninfinitetheoreticalrange:to理论上分布空间无

== Chap6-TheNormalDistributionDensityFunctionTheformulaforthenormalprobabilitydensityfunctionf(X)f(X)1e1(Xμ)22Wheree=themathematicalconstantapproximatedby2.71828π=themathematicalconstantapproximatedby3.14159μ=thepopulationmeanσ=thepopulationstandardX=anyvalueofthecontinuous Chap6-ManyNormalByvaryingtheparametersμandσ,weobtaindifferentnormaldistributions Chap6-TheNormalDistribution

ChangingμshiftsX出现的概率distributionleftσ

Changingσincreasesordecreasesthe Chap6-TheStandardizedAnynormaldistribution(withanymeanandstandarddeviationcombination)canbetransformedintothestandardizednormalNeedtotransformXunitsintoZZ：标准正态 Thestandardizednormaldistribution(Z)hasameanof0andastandarddeviationof1 Chap6-TranslationtotheStandardizedNormalDistributionTranslatefromXtothestandardizednormal(the“Z”distribution)bysubtractingthemeanofXanddividingbyitsstandarddeviation:ZXZXσTheTheZdistributionalwayshasmean=0andstandarddeviation=1 Chap6-TheStandardizedNormalDistribution标准正态分布Alsoknownasthe“Z”MeanisStandardDeviationis11 ValuesabovethemeanhavepositiveZ-valuesbelowthemeanhavenegativeZ- Chap6-IfXisdistributednormallywithmeanof$100andstandarddeviationof$50,theZvalueforX=$200isZZXμ$200$100σThissaysthatX=$200istwostandarddeviations(2incrementsof$50units)abovethemeanof$100. Chap6-ComparingXandZ

$X(μ=$100,σ=

(μ=0,σ=Notethattheshapeofthedistributionisthesame,onlythescalehaschanged.Wecanexpresstheproblemintheoriginalunits(Xindollars)orinstandardizedunits(Z) Chap6-FindingNormalProbabilityismeasuredbytheareaunderthecurve概率可以用曲线下的面

P(a≤X≤b=P(a<X<b(Notethattheprobabilityofanyindividualvalueiszero) Chap6-ProbabilityasAreaUndertheCurveThetotalareaunderthecurveis1.0,andthecurveissymmetric,sohalfisabovethemean,halfisbelowμP(μμP(μX)P(Xμ)XP(P(X) Chap6-TheStandardizedNormalTheCumulativeStandardizedNormaltableinthetextbook(AppendixtableE.2)givestheprobabilitylessthanadesiredvalueofZ(i.e.,fromnegativeinfinitytoZ)P(Z<2.00)= Chap6-TheStandardizedNormalTherowshowsthevalueofZtothefirstdecimal

ThecolumngivesthevalueZtotheseconddecimal.... 0.02ThevaluewithinthetablegivestheP(ZP(Z<2.00)=

probabilityfromZ=uptothedesiredZ- Chap6-GeneralProcedureforFindingNormalProbabilitiesTofindP(a<X<b)whenXisdistributednormally:DrawthenormalcurvefortheproblemintermsofXTranslateX-valuestoZ-UsetheStandardizedNormal Chap6-FindingNormalLetXrepresentthetimeittakes(inseconds)todownloadanimagefilefromtheinternet. 的速SupposeXisnormalwithamean均值of18.0secondsandastandarddeviationof5.0seconds.FindP(X<18.6)

FindingNormalLetXrepresentthetimeittakes,insecondstodownloadanimagefilefromtheinternet.SupposeXisnormalwithameanof18.0secondsandastandarddeviationof5.0seconds.FindP(X<18.6)ZZXμ18.618.0σμ=σμ=σ=μ=σ=P(X<

0 P(Z< Chap6-Solution:FindingP(Z<P(X<P(X<=P(Z<Z Chap6-FindingNormalUpperTailProbabilitiesSupposeXisnormalwithmean18.0andstandarddeviation5.0.NowFindP(X>X Chap6-FindingNormalUpperTailProbabilitiesNowFindP(X>

P(XP(X>18.6)=P(Z>0.12)=1.0-P(Z≤=1.0-0.547801.001.0-=Z0

Chap6-FindingaNormalProbabilityBetweenTwoValuesSupposeXisnormalwithmean18.0andstandarddeviation5.0.FindP(18<X<18.6)CalculateZ-ZZXμ1818σ5ZXZXμ18.6180.12σ50 P(18P(18<X<=P(0<Z< Chap6-Solution:FindingP(0<Z<P(18<XP(18<X<=P(0<Z<=P(Z<0.12)–P(Z≤=0.5478-0.5000=ZZ

Chap6-ProbabilitiesintheLowerSupposeXisnormalwithmean18.0andstandarddeviation5.0.NowFindP(17.4<X<X Chap6-ProbabilitiesintheLowerNowFindP(17.4<X<P(17.4<X<=P(-0.12<Z<=P(Z<0)–P(Z≤-=0.5000-0.4522

TheNormaldistributionissymmetric,sothisprobabilityisthesameTheNormaldistributionissymmetric,sothisprobabilityisthesameasP(0<Z<0.12)-0.12 Chap6-GivenaNormalProbabilityFindtheXValueStepstofindtheXvalueforaknownFindtheZ-valuefortheknownConverttoXunitsusingtheXXμ Chap6-FindingtheXvalueforaKnownProbability布 Chap6-FindtheZ-valuefor20%intheLowerTail1.1.FindtheZ-valuefortheknownStandardizedNormalProbabilityTable(Portion)Z…-…

20%areainthelowertailisconsistentwithaZ-valueof-0.84-…-… .2005-….2327.2296 - Chap6-FindingtheX2.2.ConverttoXunitsusingtheXXμ18.0SoSo20%ofthevaluesfromadistributionwithmean18.0andstandarddeviation5.0arelessthan对于均值为18，标准差为5的正态分布 Chap6-Notallcontinuousdistributionsarenormal不是所Itisimportanttoevaluatehowwellthedatasetisapproximatedbyanormaldistribution.在多大程度上近正态分布 Chap6-Normallydistributeddatashouldapproximatethetheoreticalnormaldistribution:正态分布的数据应该Thenormaldistributionisbellshaped(symmetrical称wherethemeanisequaltothemedian.均值=Theempiricalruleappliestothenormaldistribution.Theinterquartilerangeofanormaldistributionis Chap6-Comparingd haracteristicstotheoreticalConstructchartsorForsmall-ormoderate-sizeddatasets,constructastem-and-leafdisplayoraboxplottocheckfor Forlargedatasets,doesthehistogramorappearbell-shaped? Chap6-Comparingd haracteristicstotheoreticalComputedescriptivesummarymeasures描述性的Dothemean,medianandmodehavesimilarIstheinterquartilerange y1.33四分位间距是否大致等于1.33Istherange y6全距是否大致等于6 Chap6-EvaluatingComparing haracteristicstoObservethedistributionofthedatasetDoapproxima y2/3oftheobservationsliewithinmean±1standarddeviation?Doapproxima y80%oftheobservationsliewithinmean±1.28standarddeviations?Doapproxima y95%oftheobservationsliewithinmean±2standarddeviations? Chap6-Evaluating

Comparingd haracteristicstotheoreticalEvaluatenormalprobabilityplotIsthenormalprobabilityplot ylinear(i.e.aline)withpositive Chap6-ConstructingA NormalProbabilityPlot 图1Arrangedataintoorderedarray2.Findcorrespondingstandardizednormal valuesZ)找到相应位置的标准化的正态四分位值（Z）最小值的Z值对应的累计面积3.Plotthepairsofpointswithobserveddatavalues(X)ontheverticalaxisandthestandardizednormallevalues(Z)onthehorizontal4EvaluatetheplotforevidenceoflinearityAleprobabilityAleprobabilityplotfordatafromnormaldistributionwillbe ylinear:X- Chap6-X-X- - 2X- - 2X- - 2Nonlinearplotsindicateadeviationfromnormality Chap6-EvaluatingAnExample:BondFundsTheboxplotisskewedtotheright.(ThenormaldistributionTheboxplotisskewedtotheright.(Thenormaldistributionissymmetric.) Chap6-EvaluatingAnExample:BondFundsThemean(7.1641)isgreaterthanthemedianThemean(7.1641)isgreaterthanthemedian6.4).(Inanormaldistributionthemeanandmedianareequal.)Theinterquartilerange四分位间距of7.4 y1.21standarddeviations.(In normalnormaldistributiontheinterquartilerange1.33standardTherangeof40.8isequalto6.70standarddeviationsInanormaldistributiontherangeis6standarddeviations.)73.91%oftheobservationsarewithin1standarddeviationofthemean.(Inanormaldistributionthispercentageis68.26%.85.33%oftheobservationsarewithin1.28standarddeviationsofthemean.(Inanormaldistributionthispercentageis80%.) Chap6-EvaluatingAnExample:BondFunds96.20%ofthereturnsarewithin2standarddeviationsofthemean.(Inanormaldistribution,95.44%96.20%ofthereturnsarewithin2standarddeviationsofthemean.(Inanormaldistribution,95.44%ofthevaluesliewithin2s