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

© 2003 Prentice-Hall, Inc. Chap 6-2 Chapter Topics The Normal Distribution The Standardized Normal Distribution Evaluating the Normality Assumption The Uniform Distribution The Exponential Distribution

© 2003 Prentice-Hall, Inc. Chap 6-3 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. Chap 6-4 The Normal Distribution “Bell Shaped” Symmetrical Mean, Median and Mode are Equal Interquartile Range Equals 1.33  Random Variable Has Infinite Range Mean Median Mode X f(X) 

© 2003 Prentice-Hall, Inc. Chap 6-5 The Mathematical Model

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

© 2003 Prentice-Hall, Inc. Chap 6-7 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. X f(X)f(X) f(Z)f(Z)

© 2003 Prentice-Hall, Inc. Chap 6-8 Finding Probabilities Probability is the area under the curve! c d X f(X)f(X)

© 2003 Prentice-Hall, Inc. Chap 6-9 Which Table to Use? Infinitely Many Normal Distributions Means Infinitely Many Tables to Look Up!

© 2003 Prentice-Hall, Inc. Chap 6-10 Solution: The Cumulative Standardized Normal Distribution Z Cumulative Standardized Normal Distribution Table (Portion) Probabilities Only One Table is Needed Z = 0.12

© 2003 Prentice-Hall, Inc. Chap 6-11 Standardizing Example Normal Distribution Standardized Normal Distribution

© 2003 Prentice-Hall, Inc. Chap 6-12 Example: Normal Distribution Standardized Normal Distribution

© 2003 Prentice-Hall, Inc. Chap 6-13 Z Cumulative Standardized Normal Distribution Table (Portion) Z = 0.21 Example: (continued)

© 2003 Prentice-Hall, Inc. Chap 6-14 Z Cumulative Standardized Normal Distribution Table (Portion) Z = Example: (continued)

© 2003 Prentice-Hall, Inc. Chap 6-15 Normal Distribution in PHStat PHStat | Probability & Prob. Distributions | Normal … Example in Excel Spreadsheet

© 2003 Prentice-Hall, Inc. Chap 6-16 Example: Normal Distribution Standardized Normal Distribution

© 2003 Prentice-Hall, Inc. Chap 6-17 Example: (continued) Z Cumulative Standardized Normal Distribution Table (Portion) Z = 0.30

© 2003 Prentice-Hall, Inc. Chap Finding Z Values for Known Probabilities Z Cumulative Standardized Normal Distribution Table (Portion) What is Z Given Probability = ?.6217

© 2003 Prentice-Hall, Inc. Chap 6-19 Recovering X Values for Known Probabilities Normal Distribution Standardized Normal Distribution

© 2003 Prentice-Hall, Inc. Chap 6-20 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?

© 2003 Prentice-Hall, Inc. Chap 6-21 What percentage of students scored between 65 and 89? More Examples of Normal Distribution Using PHStat (continued)

© 2003 Prentice-Hall, Inc. Chap 6-22 Only 5% of the students taking the test scored higher than what grade? ? =86.16 (continued) More Examples of Normal Distribution Using PHStat

© 2003 Prentice-Hall, Inc. Chap 6-23 The middle 50% of the students scored between what two scores? More Examples of Normal Distribution Using PHStat (continued)

© 2003 Prentice-Hall, Inc. Chap 6-24 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. Chap 6-25 Assessing Normality 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  ? Is the range approximately 6  ? (continued)

© 2003 Prentice-Hall, Inc. Chap 6-26 Assessing Normality 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? (continued)

© 2003 Prentice-Hall, Inc. Chap 6-27 Assessing Normality 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 (continued)

© 2003 Prentice-Hall, Inc. Chap 6-28 Assessing Normality Normal Probability Plot for Normal Distribution Look for Straight Line! Z X (continued)

© 2003 Prentice-Hall, Inc. Chap 6-29 Normal Probability Plot Left-SkewedRight-Skewed RectangularU-Shaped Z X Z X Z X Z X

© 2003 Prentice-Hall, Inc. Chap 6-30 Obtaining Normal Probability Plot 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. Chap 6-31 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. Chap 6-32 The Uniform Distribution 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? (continued)

© 2003 Prentice-Hall, Inc. Chap 6-33 Exponential Distributions E.g., Drivers arriving at a toll bridge; customers arriving at an ATM machine

© 2003 Prentice-Hall, Inc. Chap 6-34 Exponential Distributions Describes Time or Distance between Events Used for queues Density Function Parameters (continued) f(X) X = 0.5 = 2.0

© 2003 Prentice-Hall, Inc. Chap 6-35 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. Chap 6-36 Exponential Distribution in PHStat PHStat | Probability & Prob. Distributions | Exponential Example in Excel Spreadsheet

© 2003 Prentice-Hall, Inc. Chap 6-37 Chapter Summary Discussed the Normal Distribution Described the Standard Normal Distribution Evaluated the Normality Assumption Defined the Uniform Distribution Described the Exponential Distribution