1. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-1 Chapter 1 Section 13-5 The Normal Distribution

2. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-2 The Normal Distribution Discrete and Continuous Random Variables Definition and Properties of a Normal Curve A Table of Standard Normal Curve Areas Interpreting Normal Curve Areas

3. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-3 Discrete and Continuous Random Variables A random variable that can take on only certain fixed values is called a ___________________. A variable whose values are not restricted in this way is a _______________.

4. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-4 Definition and Properties of a Normal Curve A _____________ is a symmetric, bell-shaped curve. Any random variable whose graph has this characteristic shape is said to have a ________________. On a normal curve, if the quantity shown on the horizontal axis is the number of standard deviations from the mean, rather than values of the random variable itself, then we call the curve the _____________________________.

5. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-5 Normal Curves 0 S B A C

6. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-6 Properties of Normal Curves The graph of a normal curve is bell-shaped and symmetric about a vertical line through its center. The mean, median, and mode of a normal curve are all equal and occur at the center of the distribution. Empirical Rule About 68% of all data values of a normal curve lie within 1 standard deviation of the mean (in both directions), and about 95% within 2 standard deviations, and about 99.7% within 3 standard deviations.

7. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-7 Empirical Rule

8. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-8 Example: Applying the Empirical Rule Suppose that 280 sociology students take an exam and that the distribution of their scores can be treated as normal. Find the number of scores falling within 2 standard deviations of the mean. Solution

9. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-9 A Table of Standard Normal Curve Areas To answer questions that involve regions other than 1, 2, or 3 standard deviations of the mean we can refer to the table on page 829 or other tools such as a computer or calculator. The table gives the fraction of all scores in a normal distribution that lie between the mean and z standard deviations from the mean. Because of symmetry, the table can be used for values above the mean or below the mean.

10. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-10 Example: Applying the Normal Curve Table Use the table to find the percent of all scores that lie between the mean and the following values. a) 1.5 standard deviation above the mean b) 2.62 standard deviations below the mean

11. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-11 Example: Applying the Normal Curve Table Solution

12. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-12 Example: Applying the Normal Curve Table Find the total area indicated in the region in color below. Solution z = –1.7z = 2.55

13. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-13 Example: Applying the Normal Curve Table Find the total area indicated in the region in color below. Solution z = 2.63z =.61

14. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-14 Example: Applying the Normal Curve Table Find the total area indicated in the region in color below. Solution z = 2.14

15. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-15 Interpreting Normal Curve Areas In a standard normal curve, the following three quantities are equivalent.

16. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-16 Example: Probability with Volume The volumes of soda in bottles from a small company are distributed normally with mean 12 ounces and standard deviation.15 ounce. If 1 bottle is randomly selected, what is the probability that it will have more than 12.33 ounces? Solution

17. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-17 Example: Finding z-Values for Given Areas Assuming a normal distribution, find the z-value meeting the condition that 39% of the area is to the right of z. Solution

18. © 2008 Pearson Addison-Wesley. All rights reserved 13-5-18 Example: Finding z-Values for Given Areas Assuming a normal distribution, find the z-value meeting the condition that 76% of the area is to the left of z. Solution