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Basic Business Statistics (9 th Edition)

Basic Business Statistics (9 th Edition). Chapter 6 The Normal Distribution and Other Continuous Distributions. Chapter Topics. The Normal Distribution The Standardized Normal Distribution Evaluating the Normality Assumption The Uniform Distribution The Exponential Distribution.

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Basic Business Statistics (9 th Edition)

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  1. Basic Business Statistics (9th Edition) Chapter 6 The Normal Distribution and Other Continuous Distributions © 2003 Prentice-Hall, Inc.

  2. Chapter Topics • The Normal Distribution • The Standardized Normal Distribution • Evaluating the Normality Assumption • The Uniform Distribution • The Exponential Distribution © 2003 Prentice-Hall, Inc.

  3. Continuous Probability Distributions • Continuous Random Variable • Values from interval of numbers • Absence of gaps • Continuous Probability Distribution • Distribution of continuous random variable • Most Important Continuous Probability Distribution • The normal distribution © 2003 Prentice-Hall, Inc.

  4. The Normal Distribution • “Bell Shaped” • Symmetrical • Mean, Median and Mode are Equal • Interquartile RangeEquals 1.33 s • Random VariableHas Infinite Range f(X) X  Mean Median Mode © 2003 Prentice-Hall, Inc.

  5. The Mathematical Model © 2003 Prentice-Hall, Inc.

  6. Many Normal Distributions There are an Infinite Number of Normal Distributions Varying the Parameters  and , We Obtain Different Normal Distributions © 2003 Prentice-Hall, Inc.

  7. The Standardized Normal Distribution When X is normally distributed with a mean and a standard deviation , follows a standardized (normalized) normal distribution with a mean 0 and a standard deviation 1. f(Z) f(X) X © 2003 Prentice-Hall, Inc.

  8. Finding Probabilities Probability is the area under the curve! f(X) X d c © 2003 Prentice-Hall, Inc.

  9. Which Table to Use? Infinitely Many Normal Distributions Means Infinitely Many Tables to Look Up! © 2003 Prentice-Hall, Inc.

  10. Solution: The Cumulative Standardized Normal Distribution Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .5478 .5000 0.0 .5040 .5080 .5398 .5438 .5478 0.1 0.2 .5793 .5832 .5871 Probabilities Z = 0.12 0.3 .6179 .6217 .6255 Only One Table is Needed © 2003 Prentice-Hall, Inc.

  11. Standardizing Example Standardized Normal Distribution Normal Distribution © 2003 Prentice-Hall, Inc.

  12. Example: Standardized Normal Distribution Normal Distribution © 2003 Prentice-Hall, Inc.

  13. Example: (continued) Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .5832 .5000 0.0 .5040 .5080 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 Z = 0.21 0.3 .6179 .6217 .6255 © 2003 Prentice-Hall, Inc.

  14. Example: (continued) Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .4168 .3821 -0.3 .3783 .3745 -0.2 .4207 .4168 .4129 -0.1 .4602 .4562 .4522 Z = -0.21 0.0 .5000 .4960 .4920 © 2003 Prentice-Hall, Inc.

  15. Normal Distribution in PHStat • PHStat | Probability & Prob. Distributions | Normal … • Example in Excel Spreadsheet © 2003 Prentice-Hall, Inc.

  16. Example: Standardized Normal Distribution Normal Distribution © 2003 Prentice-Hall, Inc.

  17. Example: (continued) Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .6179 .5000 0.0 .5040 .5080 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 Z = 0.30 0.3 .6179 .6217 .6255 © 2003 Prentice-Hall, Inc.

  18. Finding Z Values for Known Probabilities Cumulative Standardized Normal Distribution Table (Portion) What is Z Given Probability = 0.6217 ? .01 Z .00 0.2 0.0 .5040 .5000 .5080 .6217 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 .6179 .6255 .6217 0.3 © 2003 Prentice-Hall, Inc.

  19. Recovering X Values for Known Probabilities Standardized Normal Distribution Normal Distribution © 2003 Prentice-Hall, Inc.

  20. More Examples of Normal Distribution Using PHStat A set of final exam grades was found to be normally distributed with a mean of 73 and a standard deviation of 8. What is the probability of getting a grade no higher than 91 on this exam? 91 2.25 © 2003 Prentice-Hall, Inc.

  21. More Examples of Normal Distribution Using PHStat (continued) What percentage of students scored between 65 and 89? 65 89 -1 2 © 2003 Prentice-Hall, Inc.

  22. More Examples of Normal Distribution Using PHStat (continued) Only 5% of the students taking the test scored higher than what grade? ?=86.16 1.645 © 2003 Prentice-Hall, Inc.

  23. More Examples of Normal Distribution Using PHStat (continued) The middle 50% of the students scored between what two scores? .25 .25 67.6 78.4 -0.67 0.67 © 2003 Prentice-Hall, Inc.

  24. Assessing Normality • Not All Continuous Random Variables are Normally Distributed • It is Important to Evaluate How Well the Data Set Seems to Be Adequately Approximated by a Normal Distribution © 2003 Prentice-Hall, Inc.

  25. Assessing Normality (continued) • Construct Charts • For small- or moderate-sized data sets, do the stem-and-leaf display and box-and-whisker plot look symmetric? • For large data sets, does the histogram or polygon appear bell-shaped? • Compute Descriptive Summary Measures • Do the mean, median and mode have similar values? • Is the interquartile range approximately 1.33 s? • Is the range approximately 6 s? © 2003 Prentice-Hall, Inc.

  26. Assessing Normality (continued) • Observe the Distribution of the Data Set • Do approximately 2/3 of the observations lie between mean 1 standard deviation? • Do approximately 4/5 of the observations lie between mean 1.28 standard deviations? • Do approximately 19/20 of the observations lie between mean 2 standard deviations? • Evaluate Normal Probability Plot • Do the points lie on or close to a straight line with positive slope? © 2003 Prentice-Hall, Inc.

  27. Assessing Normality (continued) • Normal Probability Plot • Arrange Data into Ordered Array • Find Corresponding Standardized Normal Quantile Values • Plot the Pairs of Points with Observed Data Values on the Vertical Axis and the Standardized Normal Quantile Values on the Horizontal Axis • Evaluate the Plot for Evidence of Linearity © 2003 Prentice-Hall, Inc.

  28. Assessing Normality (continued) Normal Probability Plot for Normal Distribution 90 X 60 Z 30 -2 -1 0 1 2 Look for Straight Line! © 2003 Prentice-Hall, Inc.

  29. Normal Probability Plot Left-Skewed Right-Skewed 90 90 X X 60 60 Z Z 30 30 -2 -1 0 1 2 -2 -1 0 1 2 Rectangular U-Shaped 90 90 X X 60 60 Z Z 30 30 -2 -1 0 1 2 -2 -1 0 1 2 © 2003 Prentice-Hall, Inc.

  30. Obtaining Normal ProbabilityPlot in PHStat • PHStat | Probability & Prob. Distributions | Normal Probability Plot • Enter the range of the cells that contain the data in the Variable Cell Range window © 2003 Prentice-Hall, Inc.

  31. The Uniform Distribution • Properties: • The probability of occurrence of a value is equally likely to occur anywhere in the range between the smallest value a and the largest value b • Also called the rectangular distribution © 2003 Prentice-Hall, Inc.

  32. The Uniform Distribution (continued) • The Probability Density Function • Application: Selection of random numbers • E.g., A wooden wheel is spun on a horizontal surface and allowed to come to rest. What is the probability that a mark on the wheel will point to somewhere between the North and the East? © 2003 Prentice-Hall, Inc.

  33. Exponential Distributions E.g., Drivers arriving at a toll bridge; customers arriving at an ATM machine © 2003 Prentice-Hall, Inc.

  34. Exponential Distributions (continued) • Describes Time or Distance between Events • Used for queues • Density Function • Parameters f(X)  = 0.5  = 2.0 X © 2003 Prentice-Hall, Inc.

  35. Example E.g., Customers arrive at the checkout line of a supermarket at the rate of 30 per hour. What is the probability that the arrival time between consecutive customers will be greater than 5 minutes? © 2003 Prentice-Hall, Inc.

  36. Exponential Distributionin PHStat • PHStat | Probability & Prob. Distributions | Exponential • Example in Excel Spreadsheet © 2003 Prentice-Hall, Inc.

  37. Chapter Summary • Discussed the Normal Distribution • Described the Standard Normal Distribution • Evaluated the Normality Assumption • Defined the Uniform Distribution • Described the Exponential Distribution © 2003 Prentice-Hall, Inc.

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