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1、会计学1统计学入门统计学入门正态分布正态分布Chap 6-2Chap 6-2Learning ObjectivesIn this chapter,you learn:n nTo compute probabilities from the normal distributionTo compute probabilities from the normal distributionn nHow to use the normal distribution to solve business problemsHow to use the normal distribution to solve
2、business problemsn nTo use the normal probability plot to determine whether a set To use the normal probability plot to determine whether a set of data is approximately normally distributedof data is approximately normally distributed第1页/共36页Chap 6-3Chap 6-3Continuous Probability Distributionsn nA c
3、ontinuous random variable is a variable that can assume any value on a continuum(can assume an uncountable number of values)n nthickness of an itemthickness of an itemn ntime required to complete a tasktime required to complete a taskn ntemperature of a solutiontemperature of a solutionn nheight,in
4、inchesheight,in inchesn nThese can potentially take on any value depending only on the ability to precisely and accurately measure第2页/共36页Chap 6-4Chap 6-4The Normal Distributionn n Bell ShapedBell Shaped n n SymmetricalSymmetrical n n Mean,Median and ModeMean,Median and Mode are Equal are EqualLocat
5、ion is determined by the Location is determined by the mean,mean,Spread is determined by the Spread is determined by the standard deviation,standard deviation,The random variable has an infinite The random variable has an infinite theoretical range:theoretical range:+to to Mean=Median=ModeXf(X)第3页/共
6、36页Chap 6-5Chap 6-5The Normal DistributionDensity FunctionnThe formula for the normal probability density function isWheree=the mathematical constant approximated by 2.71828=the mathematical constant approximated by 3.14159=the population mean=the population standard deviationX=any value of the cont
7、inuous variable第4页/共36页Chap 6-6Chap 6-6By varying the parameters and,we obtain different normal distributionsMany Normal Distributions第5页/共36页Chap 6-7Chap 6-7The Normal Distribution ShapeXf(X)Changing shifts the distribution left or right.Changing increases or decreases the spread.第6页/共36页Chap 6-8Ch
8、ap 6-8The Standardized Normaln nAny normal distribution(with any mean and standard deviation combination)can be transformed into the standardized normal distribution(Z)n nNeed to transform X units into Z unitsn nThe standardized normal distribution(Z)has a mean of 0 and a standard deviation of 1第7页/
9、共36页Chap 6-9Chap 6-9Translation to the Standardized Translation to the Standardized Normal DistributionNormal Distributionn nTranslate from X to the standardized normal(the“Z”distribution)by subtracting the mean of X and dividing by its standard deviation:The Z distribution always has mean=0 and sta
10、ndard deviation=1第8页/共36页Chap 6-10Chap 6-10The Standardized Normal Distributionn nAlso known as the“Z”distributionAlso known as the“Z”distributionn nMean is 0Mean is 0n nStandard Deviation is 1Standard Deviation is 1Zf(Z)01Values above the mean have positive Z-values,values below the mean have negat
11、ive Z-values第9页/共36页Chap 6-11Chap 6-11Examplen nIf X is distributed normally with mean of$100 and standard deviation of$50,the Z value for X=$200 isn nThis says that X=$200 is two standard deviations(2 increments of$50 units)above the mean of$100.第10页/共36页Chap 6-12Chap 6-12Comparing X and Z unitsZ$1
12、00 2.00$200$XNote that the shape of the distribution is the same,only the scale has changed.We can express the problem in the original units(X in dollars)or in standardized units(Z)(=$100,=$50)(=0,=1)第11页/共36页Chap 6-13Chap 6-13Finding Normal Probabilities abXf(X)P aXb()Probability is measured by the
13、 area under the curveP aXb()=(Note that the probability of any individual value is zero)第12页/共36页Chap 6-14Chap 6-14f(X)XProbability as Area Under the Curve0.50.5The total area under the curve is 1.0,and the curve is symmetric,so half is above the mean,half is below第13页/共36页Chap 6-15Chap 6-15The Stan
14、dardized Normal TableThe Standardized Normal Tablen n The Cumulative Standardized Normal table in the textbook(Appendix table E.2)gives the probability less than a desired value of Z(i.e.,from negative infinity to Z)Z02.000.9772Example:P(Z 2.00)=0.9772第14页/共36页Chap 6-16Chap 6-16The Standardized Norm
15、al Table(Table The Standardized Normal Table(Table E2)E2)The value within the The value within the table gives the table gives the probability probability from Z=from Z=up to the up to the desired Z-valuedesired Z-value.97722.0P(Z 2.00)=0.9772 The row shows the value of Z to the first decimal point
16、The column gives the value of Z to the second decimal point2.0.(continued)Z 0.00 0.01 0.02 0.00.1第15页/共36页Chap 6-17Chap 6-17General Procedure for Finding Normal Probabilitiesn n Draw the normal curve for the problem in terms of Xn n Translate X-values to Z-valuesn n Use the Standardized Normal Table
17、(Table E2)To find Prob(a X b)=P(a X b)when X is distributed normally:第16页/共36页Chap 6-18Chap 6-18Finding Normal Probabilitiesn nLet X represent the time it takes(in seconds)to Let X represent the time it takes(in seconds)to download an image file from the internet.download an image file from the inte
18、rnet.n nSuppose X is normal with a mean of 18.0 seconds Suppose X is normal with a mean of 18.0 seconds and a standard deviation of 5.0 seconds.and a standard deviation of 5.0 seconds.Find P(X Find P(X 18.6)18.6)18.6X18.0第17页/共36页Chap 6-19Chap 6-19n nLet X represent the time it takes,in seconds to d
19、ownload an image file from the Let X represent the time it takes,in seconds to download an image file from the internet.internet.n nSuppose X is normal with a mean of 18.0 seconds and a standard deviation of Suppose X is normal with a mean of 18.0 seconds and a standard deviation of 5.0 seconds.5.0
20、seconds.Find P(X 18.6)Find P(X 18.6)Z0.12 0X18.6 18=18 =5=0=1(continued)Finding Normal ProbabilitiesP(X 18.6)P(Z 0.12)第18页/共36页Chap 6-20Chap 6-20Z0.12Z.00.010.0.5000.5040.5080.5398.54380.2.5793.5832.58710.3.6179.6217.6255Solution:Finding P(Z 0.12)0.5478.020.1.5478Standardized Normal Probability Tabl
21、e(Portion)0.00=P(Z 0.12)P(X 18.6)Now Find P(X 18.6)X18.618.0第20页/共36页Chap 6-22Chap 6-22n nNow Find P(X 18.6)Now Find P(X 18.6)(continued)Z0.12 0Z0.120.5478 01.0001.0-0.5478=0.4522P(X 18.6)=P(Z 0.12)=1.0-P(Z .12)=1-.5478=.4522Finding NormalFinding NormalUpper Tail Probabilities(Table E2)Upper Tail Pr
22、obabilities(Table E2)第21页/共36页Chap 6-23Chap 6-23Finding a Normal Probability Finding a Normal Probability Between Two ValuesBetween Two Valuesn nSuppose X is normal with mean 18.0 and standard deviation 5.0.Find P(18 X 18.6)P(18 X 18.6)=P(0 Z 0.12)Z0.12 0X18.6 18Calculate Z-values:第22页/共36页Chap 6-24
23、Chap 6-24Z0.12Solution:Finding P(0 Z 0.12)0.04780.00=P(0 Z 0.12)P(18 X 18.6)=P(Z 0.12)P(Z 0)=0.5478-0.5000=0.04780.5000Z.00.010.0.5000.5040.5080.5398.54380.2.5793.5832.58710.3.6179.6217.6255.020.1.5478Standardized Normal Probability Table(Portion)第23页/共36页Chap 6-25Chap 6-25n nSuppose X is normal wit
24、h mean 18.0 and Suppose X is normal with mean 18.0 and standard deviation 5.0.standard deviation 5.0.n nNow Find P(17.4 X 18)Now Find P(17.4 X 18)X17.418.0Probabilities in the Lower Tail 第24页/共36页Chap 6-26Chap 6-26Probabilities in the Lower Tail Now Find P(17.4 X 18)Now Find P(17.4 X 18)X17.4 18.0 P
25、(17.4 X 18)=P(-0.12 Z 0)=P(Z 0)P(Z -0.12)=0.5000-0.4522=0.0478(continued)0.04780.4522Z-0.12 0The Normal distribution is symmetric,so this probability is the same as P(0 Z 0.12)第25页/共36页Chap 6-27Chap 6-27n nSteps to find the X value for a known probability:Steps to find the X value for a known probab
26、ility:1.Find the Z-value for the known probability2.Convert to X units using the formula:Given a Normal ProbabilityFind the X Value第26页/共36页Chap 6-28Chap 6-28Finding the X value for a Known ProbabilityExample:n nLet X represent the time it takes(in seconds)to download an Let X represent the time it
27、takes(in seconds)to download an image file from the internet.image file from the internet.n nSuppose X is normal with mean 18.0 and standard deviation Suppose X is normal with mean 18.0 and standard deviation 5.05.0n nFind X such that 20%of download times are less than X.Find X such that 20%of downl
28、oad times are less than X.X?18.00.2000Z?0(continued)第27页/共36页Chap 6-29Chap 6-29Find the Z-value for 20%in the Lower Tailn n20%area in the lower tail is consistent with a Z-value of-0.84Z.03-0.9.1762.1736.2033-0.7.2327.2296.04-0.8.2005Standardized Normal Probability Table(Portion).05.1711.1977.2266X?
29、18.00.2000Z-0.84 01.Find the Z-value for the known probability第28页/共36页Chap 6-30Chap 6-302.Convert to X units using the formula:Finding the X valueSo 20%of the values from a distribution with mean 18.0 and standard deviation 5.0 are less than 13.80第29页/共36页Chap 6-31Chap 6-31Evaluating Normalityn nNo
30、t all continuous distributions are normalNot all continuous distributions are normaln nIt is important to evaluate how well the data set is It is important to evaluate how well the data set is approximated by a normal distribution.approximated by a normal distribution.n nNormally distributed data sh
31、ould approximate the Normally distributed data should approximate the theoretical normal distribution:theoretical normal distribution:n nThe normal distribution is bell shaped(symmetrical)where The normal distribution is bell shaped(symmetrical)where the mean is equal to the median.the mean is equal
32、 to the median.n nThe empirical rule applies to the normal distribution.The empirical rule applies to the normal distribution.n nThe interquartile range of a normal distribution is 1.33 The interquartile range of a normal distribution is 1.33 standard deviations.standard deviations.第30页/共36页Chap 6-3
33、2Chap 6-32Evaluating NormalityComparing data characteristics to theoretical propertiesn nConstruct Construct charts or graphscharts or graphsn nFor small-or moderate-sized data sets,construct a stem-and-leaf display For small-or moderate-sized data sets,construct a stem-and-leaf display or a boxplot
34、 to check for symmetryor a boxplot to check for symmetryn nFor large data sets,does the histogram or polygon appear bell-shaped?For large data sets,does the histogram or polygon appear bell-shaped?n nCompute Compute descriptive summary measuresdescriptive summary measuresn nDo the mean,median and mo
35、de have similar values?Do the mean,median and mode have similar values?n nIs the interquartile range approximately 1.33 Is the interquartile range approximately 1.33?n nIs the range approximately 6 Is the range approximately 6?(continued)第31页/共36页Chap 6-33Chap 6-33Evaluating NormalityComparing data
36、characteristics to theoretical propertiesn n Observe the distribution Observe the distribution of the data set of the data setn nDo approximately 2/3 of the observations lie within mean Do approximately 2/3 of the observations lie within mean 1 1 standard deviation?standard deviation?n nDo approxima
37、tely 80%of the observations lie within mean Do approximately 80%of the observations lie within mean 1.28 1.28 standard deviations?standard deviations?n nDo approximately 95%of the observations lie within mean Do approximately 95%of the observations lie within mean 2 2 standard deviations?standard de
38、viations?n n Evaluate Evaluate normal probability plotnormal probability plotn nIs the normal probability plot approximately linear(i.e.a straight line)Is the normal probability plot approximately linear(i.e.a straight line)with positive slope?with positive slope?(continued)第32页/共36页Chap 6-34Chap 6-
39、34Constructing A Quantile-Quantile Normal Probability Constructing A Quantile-Quantile Normal Probability PlotPlotn nNormal probability plotn nArrange data into ordered arrayArrange data into ordered arrayn nFind corresponding standardized normal quantile Find corresponding standardized normal quant
40、ile values(Z)values(Z)n nPlot the pairs of points with observed data values Plot the pairs of points with observed data values(X)on the vertical axis and the standardized normal(X)on the vertical axis and the standardized normal quantile values(Z)on the horizontal axisquantile values(Z)on the horizo
41、ntal axisn nEvaluate the plot for evidence of linearityEvaluate the plot for evidence of linearity第33页/共36页Chap 6-35Chap 6-35A quantile-quantile normal probability plot for data from a normal distribution will be approximately linear:306090-2-1012ZXThe Quantile-Quantile Normal Probability Plot The Q
42、uantile-Quantile Normal Probability Plot InterpretationInterpretation第34页/共36页Chap 6-36Chap 6-36Quantile-Quantile Normal Probability Plot Quantile-Quantile Normal Probability Plot InterpretationInterpretationLeft-SkewedRight-SkewedRectangular306090-2-1 012ZX(continued)306090-2-1 012ZX306090-2-1 012ZXNonlinear plots indicate a deviation from normality第35页/共36页