利用Excel进行统计分析Chapter The Normal Distribution and Other Continuous Distributions

1、利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsIn this chapter, you will learn:To compute probabilities from the normal distribution.To use the normal probability plot to determine whether a set of data is approximately normally distributed.To compute probabilities f

2、rom the uniform distribution.To compute probabilities from the exponential distribution2利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsA continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values)thic

3、kness of an itemtime required to complete a tasktemperature of a solutionheightThese can potentially take on any value, depending only on the ability to measure precisely and accurately.3利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous Distributions Bell Shaped Symmetrical Mean, Me

4、dian and Mode are equal Location is characterized by the mean, Spread is characterized by the standard deviation, The random variable has an infinite theoretical range: - to + Mean = Median = Modef(X)4利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsThe formula for the

5、 normal probability density function is = the population mean = the population standard deviationX = any value of the continuous variable5利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsBy varying the parameters and , we obtain different normal distributions6利用Excel进展

6、统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsXf(X)Changing shifts the distribution left or right.Changing increases or decreases the spread.7利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsAny normal distribution (with any mean and standard d

7、eviation combination) can be transformed into the standardized normal distribution (Z).Need to transform X units into Z units.The standardized normal distribution has a mean of 0 and a standard deviation of 1.8利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsTranslate

8、from X to the standardized normal (the “Z distribution) by subtracting the mean of X and dividing by its standard deviation:9利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsThe formula for the standardized normal probability density function isZ = any value of the sta

9、ndardized normal distribution10利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsZf(Z)01Also known as the “Z distributionMean is 0Standard Deviation is 1Values above the mean have positive Z-values, values below the mean have negative Z-values11利用Excel进展统计分析Chapter 6The

10、 Normal Distribution and Other Continuous DistributionsIf X is distributed normally with mean of 100 and standard deviation of 50, the Z value for X = 200 isThis says that X = 200 is two standard deviations (2 increments of 50 units) above the mean of 100.12利用Excel进展统计分析Chapter 6The Normal Distribut

11、ion and Other Continuous DistributionsZ1000200X( = 100, = 50)( = 0, = 1)Note that the distribution is the same, only the scale has changed. We can express the problem in original units (X) or in standardized units (Z)13利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous Distributions1

12、4利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsThe total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below.f(X)15利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsExample: P(Z 2.00) = .977

13、2 The Standardized Normal table in the textbook (Appendix table E.2) gives the probability less than a desired value for Z (i.e., from negative infinity to Z)Z0.977216利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous Distributions The value within the table gives the probability fro

14、m Z = up to the desired Z value.9772P(Z 2.00) = .9772 The row shows the value of Z to the first decimal point The column gives the value of Z to the second decimal point. Z 0.00 0.01 0.02 17利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous Distributions Draw the normal curve for the

15、 problem in terms of X. Translate X-values to Z-values. Use the Standardized Normal Table.To find P(a X b) when X is distributed normally:18利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous Distributions19利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous Distributio

16、ns20利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsZ.00.01.020.0.5000.5040.50800.1.5398.5438.54780.2.5793.5832.58710.3.6179.6217.6255Standardized Normal Probability Table (Portion)Z 0 = 0 = 1.5478= P(Z 0.12)P(X 8.6)21利用Excel进展统计分析Chapter 6The Normal Distribution and

17、Other Continuous Distributions22利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous Distributions23利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsZ.00.01.020.0.5000.5040.50800.1.5398.5438.54780.2.5793.5832.58710.3.6179.6217.6255Standardized Normal Prob

18、ability Table (Portion)Z.0478= P(0 Z 0.12)P(8 X 8.6)= P(Z 0.12) P(Z 0)= .5478 - .5000 = .0478.500024利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous Distributions25利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous Distributions26利用Excel进展统计分析Chapter 6The Normal Dis

19、tribution and Other Continuous DistributionsSecond, convert the Z value to X units using the following formula.So 20% of the download times from the distribution with mean 8.0 and standard deviation 5.0 are less than 3.80 seconds.27利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous D

20、istributionsIt 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 the median.The empirical rule applie

21、s to the normal distribution.The interquartile range of a normal distribution is 1.33 standard deviations.28利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsConstruct charts or graphsFor small- or moderate-sized data sets, do stem-and-leaf display and box-and-whisker p

22、lot look symmetric?For 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 approximately 6 ?29利用Excel进展统计分析Chapter 6The Normal Distribution a

23、nd Other Continuous DistributionsObserve 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 mean 1.28 standard deviations?Do approximately 95% of the observations lie within mean 2 standard

24、 deviations?Evaluate normal probability plotIs the normal probability plot approximately linear with positive slope?30利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsNormal probability plot (steps):Arrange data into ordered arrayFind corresponding standardized normal

25、quantile (Z) valuesPlot the pairs of points with observed data values (X) on the vertical axis and the standardized normal quantile (Z) values on the horizontal axisEvaluate the plot for evidence of linearity31利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous Distributions32利用Excel进

26、展统计分析Chapter 6The Normal Distribution and Other Continuous Distributions33利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsThe uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variableBecause of its

27、 shape it is also called a rectangular distribution34利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsThe Continuous Uniform Distribution:wheref(X) = value of the density function at any X valuea = minimum value of Xb = maximum value of Xf(X) =35利用Excel进展统计分析Chapter 6T

28、he Normal Distribution and Other Continuous DistributionsThe mean of a uniform distribution is:The standard deviation is:36利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsExample: Uniform probability distribution over the range 2 X 6:f(X) = = .25 for 2 X 66 - 2126.25X

29、f(X)37利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsUsed to model the length of time between two occurrences of an event (the time between arrivals)Examples: Time between trucks arriving at an unloading dockTime between transactions at an ATM MachineTime between phone calls to the main operator38利用Excel进展统计分析Chapter 6The Normal Distribution and Other Continuous DistributionsDefined by a single parameter, its mean (lambda)The probability that an a