商務(wù)統(tǒng)計學英文版教學課件第6章

1、The Normal Distribution Chapter 6The Normal Distribution ChapteObjectivesIn this chapter, you learn: To compute probabilities from the normal distributionHow to use the normal distribution to solve business problemsTo use the normal probability plot to determine whether a set of data is approximatel

2、y normally distributedObjectivesIn this chapter, youContinuous Probability DistributionsA continuous variable is a variable that can assume any value on a continuum (can assume an uncountable number of values)thickness of an itemtime required to complete a tasktemperature of a solutionheight, in inc

3、hesThese can potentially take on any value depending only on the ability to precisely and accurately measureContinuous Probability DistribThe Normal Distribution Bell Shaped Symmetrical Mean, Median and Mode are EqualLocation is determined by the mean, Spread is determined by the standard deviation,

4、 The random variable has an infinite theoretical range: + to Mean = Median = ModeXf(X)The Normal Distribution Bell The Normal DistributionDensity FunctionThe formula for the normal probability density function isWheree = the mathematical constant approximated by 2.71828 = the mathematical constant a

5、pproximated by 3.14159 = the population mean = the population standard deviationX = any value of the continuous variableThe Normal DistributionDensitBy varying the parameters and , we obtain different normal distributionsABCA and B have the same mean but different standard deviations.B and C have di

6、fferent means and different standard deviations.By varying the parameters anThe Normal Distribution ShapeXf(X)Changing shifts the distribution left or right.Changing increases or decreases the spread.The Normal Distribution ShapeXThe Standardized NormalAny normal distribution (with any mean and stan

7、dard deviation combination) can be transformed into the standardized normal distribution (Z)To compute normal probabilities need to transform X units into Z unitsThe standardized normal distribution (Z) has a mean of 0 and a standard deviation of 1The Standardized NormalAny norTranslation to the Sta

8、ndardized Normal DistributionTranslate 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 standard deviation = 1Translation to the StandardizeThe Standardized Normal Probability Densi

9、ty FunctionThe formula for the standardized normal probability density function isWheree = the mathematical constant approximated by 2.71828 = the mathematical constant approximated by 3.14159Z = any value of the standardized normal distributionThe Standardized Normal ProbabThe Standardized Normal D

10、istributionAlso known as the “Z” distributionMean is 0Standard Deviation is 1Zf(Z)01Values above the mean have positive Z-values. Values below the mean have negative Z-values.The Standardized Normal DistrExampleIf X is distributed normally with mean of $100 and standard deviation of $50, the Z value

11、 for X = $200 isThis says that X = $200 is two standard deviations (2 increments of $50 units) above the mean of $100.ExampleIf X is distributed nComparing X and Z unitsNote that the shape of the distribution is the same, only the scale has changed. We can express the problem in the original units (

12、X in dollars) or in standardized units (Z)Z$100 2.00$200$X( = $100, = $50)( = 0, = 1)Comparing X and Z unitsNotFinding Normal Probabilities Probability is measured by the area under the curveabXf(X)PaXb()PaXb()=(Note that the probability of any individual value is zero)Finding Normal Probabilities P

13、robability as Area Under the CurveThe total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is belowf(X)X0.50.5Probability as Area Under theThe Standardized Normal TableThe Cumulative Standardized Normal table in the textbook (Appendix table E.2) gives the pr

14、obability less than a desired value of Z (i.e., from negative infinity to Z)Z02.000.9772Example: P(Z 2.00) = 0.9772The Standardized Normal TableTThe Standardized Normal Table The value within the table gives the probability from Z = up to the desired Z value.97722.0P(Z 2.00) = 0.9772 The row shows t

15、he value of Z to the first decimal point The column gives the value of Z to the second decimal point2.0.(continued) Z 0.00 0.01 0.02 0.00.1The Standardized Normal Table General Procedure for Finding Normal Probabilities Draw the normal curve for the problem in terms of X Translate X-values to Z-valu

16、es Use the Standardized Normal TableTo find P(a X b) when X is distributed normally:General Procedure for Finding Finding Normal ProbabilitiesLet X represent the time it takes (in seconds) to download an image file from the internet.Suppose X is normal with a mean of18.0 seconds and a standard devia

17、tion of 5.0 seconds. Find P(X 18.6)18.6X18.0Finding Normal ProbabilitiesLeLet X represent the time it takes, in seconds to download an image file from the internet.Suppose X is normal with a mean of 18.0 seconds and a standard deviation of 5.0 seconds. Find P(X 18.6)Z0.12 0X18.6 18 = 18 = 5 = 0 = 1(

18、continued)Finding Normal ProbabilitiesP(X 18.6)P(Z 0.12)Let X represent the time it taZ0.12Solution: Finding P(Z 0.12)0.5478Standardized Normal Probability Table (Portion)0.00= P(Z 0.12)P(X 18.6)Z.00.010.0.5000.5040.5080.5398.54380.2.5793.5832.58710.3.6179.6217.6255.020.1.5478Z0.12Solution: Finding

19、P(Z 18.6)X18.618.0Finding NormalUpper Tail ProNow Find P(X 18.6)(continued)Z0.12 0Z0.120.5478 01.0001.0 - 0.5478 = 0.4522 P(X 18.6) = P(Z 0.12) = 1.0 - P(Z 0.12) = 1.0 - 0.5478 = 0.4522Finding NormalUpper Tail ProbabilitiesNow Find P(X 18.6)(continueFinding a Normal Probability Between Two ValuesSup

20、pose 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:Finding a Normal Probability BZ0.12Solution: 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.00.010.0.500

21、0.5040.5080.5398.54380.2.5793.5832.58710.3.6179.6217.6255.020.1.5478Standardized Normal Probability Table (Portion)Z0.12Solution: Finding P(0 ZSuppose X is normal with mean 18.0 and standard deviation 5.0. Now Find P(17.4 X 18)X17.418.0Probabilities in the Lower Tail Suppose X is normal with meaProb

22、abilities in the Lower Tail Now Find P(17.4 X 18)X17.418.0 P(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)Probabilities in the Lower TaiEmpirical Rule 1 enclose

23、s about 68.26% of Xsf(X)X+1-1What can we say about the distribution of values around the mean? For any normal distribution:68.26%Empirical Rule 1 enclosThe Empirical Rule 2 covers about 95.44% of Xs 3 covers about 99.73% of Xsx22x3395.44%99.73%(continued)The Empirical Rule 2 covSteps to find the X v

24、alue for a known probability:1. Find the Z value for the known probability2. Convert to X units using the formula:Given a Normal ProbabilityFind the X ValueSteps to find the X value for Finding the X value for a Known ProbabilityExample:Let X represent the time it takes (in seconds) to download an i

25、mage file from the internet.Suppose X is normal with mean 18.0 and standard deviation 5.0Find X such that 20% of download times are less than X.X?18.00.2000Z? 0(continued)Finding the X value for a KnowFind the Z value for 20% in the Lower Tail20% area in the lower tail is consistent with a Z value o

26、f -0.84Z.03-0.9.1762.1736.2033-0.7.2327.2296.04-0.8.2005Standardized Normal Probability Table (Portion).05.1711.1977.2266X?18.00.2000Z-0.84 01. Find the Z value for the known probabilityFind the Z value for 20% in t2. Convert to X units using the formula:Finding the X valueSo 20% of the values from

27、a distribution with mean 18.0 and standard deviation 5.0 are less than 13.802. Convert to X units using Both Minitab & Excel Can Be Used To Find Normal ProbabilitiesFind P(X 9) where X is normal with a mean of 7and a standard deviation of 2 ExcelMinitabBoth Minitab & Excel Can Be UsEvaluating Normal

28、ityNot all continuous distributions are normalIt is important to evaluate how well the data set is approximated by a normal distribution.Normally distributed data should approximate the theoretical normal distribution:The normal distribution is bell shaped (symmetrical) where the mean is equal to th

29、e median.The empirical rule applies to the normal distribution.The interquartile range of a normal distribution is 1.33 standard deviations.Evaluating NormalityNot all coEvaluating NormalityComparing data characteristics to theoretical propertiesConstruct charts or graphsFor small- or moderate-sized

30、 data sets, construct a stem-and-leaf display or a boxplot to check for symmetryFor large data sets, does the histogram or polygon appear bell-shaped?Compute descriptive summary measuresDo the mean, median and mode have similar values?Is the interquartile range approximately 1.33?Is the range approx

31、imately 6?(continued)Evaluating NormalityComparing Evaluating NormalityComparing data characteristics to theoretical properties Observe the distribution of the data setDo approximately 2/3 of the observations lie within mean 1 standard deviation?Do approximately 80% of the observations lie within me

32、an 1.28 standard deviations?Do approximately 95% of the observations lie within mean 2 standard deviations? Evaluate normal probability plotIs the normal probability plot approximately linear (i.e. a straight line) with positive slope?(continued)Evaluating NormalityComparing ConstructingA Normal Pro

33、bability PlotNormal probability plotArrange data into ordered arrayFind corresponding standardized normal quantile values (Z)Plot the pairs of points with observed data values (X) on the vertical axis and the standardized normal quantile values (Z) on the horizontal axisEvaluate the plot for evidenc

34、e of linearityConstructingA Normal ProbabilA normal probability plot for data from a normal distribution will be approximately linear:306090-2-1012ZXThe Normal Probability PlotInterpretationA normal probability plot for Normal Probability PlotInterpretationLeft-SkewedRight-SkewedRectangular306090-2-

35、1012ZX(continued)306090-2-1012ZX306090-2-1012ZXNonlinear plots indicate a deviation from normalityNormal Probability PlotInterpEvaluating NormalityAn Example: Mutual Fund ReturnsThe boxplot is skewed to the right. (The normal distribution is symmetric.)Evaluating NormalityAn ExamplEvaluating Normali

36、tyAn Example: Mutual Fund ReturnsDescriptive Statistics(continued)The mean (21.84) is approximately the same as the median (21.65). (In a normal distribution the mean and median are equal.)The interquartile range of 6.98 is approximately 1.09 standard deviations. (In a normal distribution the interq

37、uartile range is 1.33 standard deviations.)The range of 59.52 is equal to 9.26 standard deviations. (In a normal distribution the range is 6 standard deviations.)77.04% of the observations are within 1 standard deviation of the mean. (In a normal distribution this percentage is 68.26%.)86.79% of the

38、 observations are within 1.28 standard deviations of the mean. (In a normal distribution this percentage is 80%.)96.86% of the observations are within 2 standard deviations of the mean. (In a normal distribution this percentage is 95.44%.)The skewness statistic is 1.698 and the kurtosis statistic is 8.467. (In a normal distribution, each of these stati