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1、Chap 6-1Chap 6-1Chapter 6The Normal DistributionBasic Business Statistics12th EditionChap 6-2Chap 6-2Learning ObjectivesIn this chapter,you learn:nTo compute probabilities from the normal distributionnHow to use the normal distribution to solve business problemsnTo use the normal probability plot to
2、 determine whether a set of data is approximately normally distributedChap 6-3Chap 6-3Continuous Probability DistributionsnA continuous random variable is a variable that can assume any value on a continuum(can assume an uncountable number of values)nthickness of an itemntime required to complete a
3、taskntemperature of a solutionnheight,in inchesnThese can potentially take on any value depending only on the ability to precisely and accurately measureChap 6-4Chap 6-4The Normal Distributionn Bell Shapedn Symmetrical n Mean,Median and Mode are EqualLocation is determined by the mean,Spread is dete
4、rmined by the standard deviation,The random variable has an infinite theoretical range:+to Mean=Median=ModeXf(X)Chap 6-5Chap 6-5The Normal DistributionDensity FunctionnThe formula for the normal probability density function is=the population mean=the population standard deviationX=any value of the c
5、ontinuous variableChap 6-6Chap 6-6By varying the parameters and,we obtain different normal distributionsMany Normal DistributionsChap 6-7Chap 6-7The Normal Distribution ShapeXf(X)Changing shifts the distribution left or right.Changing increases or decreases the spread.Chap 6-8Chap 6-8The Standardize
6、d NormalnAny normal distribution(with any mean and standard deviation combination)can be transformed into the standardized normal distribution(Z)nNeed to transform X units into Z unitsnThe standardized normal distribution(Z)has a mean of 0 and a standard deviation of 1Chap 6-9Chap 6-9Translation to
7、the Standardized Normal DistributionnTranslate 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=1Chap 6-10Chap 6-10The Standardized Normal DistributionnAlso known as th
8、e“Z”distributionnMean is 0nStandard Deviation is 1Zf(Z)01Values above the mean have positive Z-values,values below the mean have negative Z-valuesChap 6-11Chap 6-11ExamplenIf X is distributed normally with mean of$100 and standard deviation of$50,the Z value for X=$200 isnThis says that X=$200 is tw
9、o standard deviations(2 increments of$50 units)above the mean of$100.Chap 6-12Chap 6-12Comparing X and Z unitsZ$1000$200$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,=$5
10、0)(=0,=1)Chap 6-13Chap 6-13Finding Normal Probabilities abXf(X)P aXb()Probability is measured by the area under the curveP aXb()=(Note that the probability of any individual value is zero)Chap 6-14Chap 6-14f(X)XProbability as Area Under the CurveThe,and the curve is symmetric,so half is above the me
11、an,half is belowChap 6-15Chap 6-15The Standardized Normal Tablen 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)Z0Example:Chap 6-16Chap 6-16The Standardized Normal Table(Table E2)The va
12、lue within the table gives the probability from Z=up to the desired Z-value.9772P(Z 2.00)=The row shows the value of Z to the first decimal point The column gives the value of Z to the second decimal point.(continued)Z 0.00 0.01 0.02 Chap 6-17Chap 6-17General Procedure for Finding Normal Probabiliti
13、esn Draw the normal curve for the problem in terms of Xn Translate X-values to Z-valuesn Use the Standardized Normal Table(Table E2)To find Prob(a X b)=P(a X b)when X is distributed normally:Chap 6-18Chap 6-18Finding Normal ProbabilitiesnLet X represent the time it takes(in seconds)to download an im
14、age file from the internet.nSuppose X is normal with a mean of 18.0 seconds and a standard deviation of 5.0 seconds.Find P(X 18.6)18.6X18.0Chap 6-19Chap 6-19nLet X represent the time it takes,in seconds to download an image file from the internet.nSuppose X is normal with a mean of 18.0 seconds and
15、a standard deviation of 5.0 seconds.Find P(X 18.6)Z 0X 18=18 =5=0=1(continued)Finding Normal ProbabilitiesP(X 18.6)P(Z 0.12)Chap 6-20Chap 6-20ZZ.00.01.5000.5040.5080.5398.5438.5793.5832.5871.6179.6217.6255Solution:Finding P(Z 0.12).02.5478Standardized Normal Probability Table(Portion)=P(Z 0.12)P(X 1
16、8.6)XChap 6-22Chap 6-22nNow Find P(X 18.6)(continued)Z 0Z 0P(X)=P(Z)=1.0-P(Z .12)=1-.5478=.4522Finding NormalUpper Tail Probabilities(Table E2)Chap 6-23Chap 6-23Finding a Normal Probability Between Two ValuesnSuppose X is normal with mean 18.0 and standard deviation 5.0.Find P(18 X 18.6)P(18 X 18.6)
17、=P(0 Z 0.12)Z 0X 18Calculate Z-values:Chap 6-24Chap 6-24ZSolution:Finding P(0 Z 0.12)=P(0 Z 0.12)P(18 X 18.6)=P(Z 0.12)P(Z 0)=0.5478-0.5000=Z.00.01.5000.5040.5080.5398.5438.5793.5832.5871.6179.6217.6255.02.5478Standardized Normal Probability Table(Portion)Chap 6-25Chap 6-25nSuppose X is normal with
18、mean 18.0 and standard deviation 5.0.nNow Find P(17.4 X 18)XProbabilities in the Lower Tail Chap 6-26Chap 6-26Probabilities in the Lower Tail Now Find P(17.4 X 18)X P(17.4 X 18)=P(-0.12 Z 0)=P(Z 0)P(Z -0.12)=0.5000-0.4522=(continued)Z 0The Normal distribution is symmetric,so this probability is the
19、same as P(0 Z 0.12)Chap 6-27Chap 6-27nSteps to find the X value 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 ValueChap 6-28Chap 6-28Finding the X value for a Known ProbabilityExample:nLet X represent t
20、he time it takes(in seconds)to download an image file from the internet.nFind X such that 20%of download times are less than X.X?Z?0(continued)Chap 6-29Chap 6-29Find the Z-value for 20%in the Lower Tailn20%area in the lower tail is consistent with a Z-value of Z.03.1762.1736.2033.2327.2296.04.2005St
21、andardized Normal Probability Table(Portion).05.1711.1977.2266X?Z 01.Find the Z-value for the known probabilityChap 6-30Chap 6-302.Convert to X units using the formula:Finding the X value Chap 6-31Chap 6-31Evaluating NormalitynNot all continuous distributions are normalnIt is important to evaluate h
22、ow well the data set is approximated by a normal distribution.nNormally distributed data should approximate the theoretical normal distribution:nThe normal distribution is bell shaped(symmetrical)where the mean is equal to the median.nThe empirical rule applies to the normal distribution.nThe interq
23、uartile range of a normal distribution is 1.33 standard deviations.Chap 6-32Chap 6-32Evaluating NormalityComparing data characteristics to theoretical propertiesnConstruct charts or graphsnFor small-or moderate-sized data sets,construct a stem-and-leaf display or a boxplot to check for symmetrynFor
24、large data sets,does the histogram or polygon appear bell-shaped?nCompute descriptive summary measuresnDo the mean,median and mode have similar values?nIs the interquartile range approximately 1.33?nIs the range approximately 6?(continued)Chap 6-33Chap 6-33Evaluating NormalityComparing data characte
25、ristics to theoretical propertiesn Observe the distribution of the data setnDo approximately 2/3 of the observations lie within mean 1 standard deviation?nDo approximately 80%of the observations lie within mean 1.28 standard deviations?nDo approximately 95%of the observations lie within mean 2 stand
26、ard deviations?n Evaluate normal probability plotnIs the normal probability plot approximately linear(i.e.a straight line)with positive slope?(continued)Chap 6-34Chap 6-34Constructing A Quantile-Quantile Normal Probability PlotnNormal probability plotnArrange data into ordered arraynFind correspondi
27、ng standardized normal quantile values(Z)nPlot the pairs of points with observed data values(X)on the vertical axis and the standardized normal quantile values(Z)on the horizontal axisnEvaluate the plot for evidence of linearityChap 6-35Chap 6-35A quantile-quantile normal probability plot for data f
28、rom a normal distribution will be approximately linear:306090-2-1012ZXThe Quantile-Quantile Normal Probability Plot InterpretationChap 6-36Chap 6-36Quantile-Quantile Normal Probability Plot InterpretationLeft-SkewedRight-SkewedRectangular306090-2-1 012ZX(continued)306090-2-1 012ZX306090-2-1 012ZXNonlinear plots indicate a deviation from normality