© 2002 Prentice-Hall, Inc.Chap 6-1 Basic Business Statistics (8 th Edition) Chapter 6 The Normal Distribution and Other Continuous Distributions

© 2002 Prentice-Hall, Inc. Chap 6-2 Chapter Topics The normal distribution The standardized normal distribution Evaluating the normality assumption The exponential distribution

© 2002 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

© 2002 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) 

© 2002 Prentice-Hall, Inc. Chap 6-5 The Mathematical Model

© 2002 Prentice-Hall, Inc. Chap 6-6 Many Normal Distributions By varying the parameters  and , we obtain different normal distributions There are an infinite number of normal distributions

© 2002 Prentice-Hall, Inc. Chap 6-7 Finding Probabilities Probability is the area under the curve! c d X f(X)f(X)

© 2002 Prentice-Hall, Inc. Chap 6-8 Which Table to Use? An infinite number of normal distributions means an infinite number of tables to look up!

© 2002 Prentice-Hall, Inc. Chap 6-9 Solution: The Cumulative Standardized Normal Distribution Z Cumulative Standardized Normal Distribution Table (Portion) Probabilities Shaded Area Exaggerated Only One Table is Needed Z = 0.12

© 2002 Prentice-Hall, Inc. Chap 6-10 Standardizing Example Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

© 2002 Prentice-Hall, Inc. Chap 6-11 Example: Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

© 2002 Prentice-Hall, Inc. Chap 6-12 Z Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = 0.21 Example: (continued)

© 2002 Prentice-Hall, Inc. Chap 6-13 Z Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = Example: (continued)

© 2002 Prentice-Hall, Inc. Chap 6-14 Normal Distribution in PHStat PHStat | probability & prob. distributions | normal … Example in excel spreadsheet

© 2002 Prentice-Hall, Inc. Chap 6-15 Example: Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

© 2002 Prentice-Hall, Inc. Chap 6-16 Example: (continued) Z Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = 0.30

© 2002 Prentice-Hall, Inc. Chap Finding Z Values for Known Probabilities Z Cumulative Standardized Normal Distribution Table (Portion) What is Z Given Probability = ? Shaded Area Exaggerated.6217

© 2002 Prentice-Hall, Inc. Chap 6-18 Recovering X Values for Known Probabilities Normal Distribution Standardized Normal Distribution

© 2002 Prentice-Hall, Inc. Chap 6-19 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

© 2002 Prentice-Hall, Inc. Chap 6-20 Assessing Normality Construct charts For small- or moderate-sized data sets, do 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)

© 2002 Prentice-Hall, Inc. Chap 6-21 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)

© 2002 Prentice-Hall, Inc. Chap 6-22 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)

© 2002 Prentice-Hall, Inc. Chap 6-23 Assessing Normality Normal Probability Plot for Normal Distribution Look for Straight Line! Z X (continued)

© 2002 Prentice-Hall, Inc. Chap 6-24 Normal Probability Plot Left-SkewedRight-Skewed RectangularU-Shaped Z X Z X Z X Z X

© 2002 Prentice-Hall, Inc. Chap 6-25 Exponential Distributions e.g.: Drivers arriving at a toll bridge; customers arriving at an ATM machine

© 2002 Prentice-Hall, Inc. Chap 6-26 Exponential Distributions Describes time or distance between events Used for queues Density function Parameters (continued) f(X) X = 0.5 = 2.0

© 2002 Prentice-Hall, Inc. Chap 6-27 Example e.g.: Customers arrive at the check out line of a supermarket at the rate of 30 per hour. What is the probability that the arrival time between consecutive customers to be greater than 5 minutes?

© 2002 Prentice-Hall, Inc. Chap 6-28 Exponential Distribution in PHStat PHStat | probability & prob. distributions | exponential Example in excel spreadsheet

© 2002 Prentice-Hall, Inc. Chap 6-29 Chapter Summary Discussed the normal distribution Described the standard normal distribution Evaluated the normality assumption Defined the exponential distribution