1. Copyright © Cengage Learning. All rights reserved. 6 Normal Probability Distributions

2. Copyright © Cengage Learning. All rights reserved. 6.3 Applications of Normal Distributions

3. 3 We have learned how to use Table 3 in Appendix B to convert information about the standard normal variable z into probability, or the opposite, to convert probability information about the standard normal distribution into z-scores. Now we are ready to apply this methodology to all normal distributions. The key is the standard score, z. The information associated with a normal distribution will be in terms of x values or probabilities. We will use the z-score and Table 3 as the tools to “go between” the given information and the desired answer.

4. 4 Applications of Normal Distributions

5. 5 Example 8 – Converting to a Standard Normal Curve to Find Probabilities Consider the intelligence quotient (IQ) scores for people. IQ scores are normally distributed, with a mean of 100 and a standard deviation of 16. If a person is picked at random, what is the probability that his or her IQ is between 100 and 115? That is, what is P(100 < x < 115)? Solution: P(100 < x < 115) is represented by the shaded area in the figure. The variable x must be standardized using formula (6.3).

6. 6 Example 8 – Solution The z-values are shown on the figure to the left. When x = 100: When x = 115: cont’d

7. 7 Example 8 – Solution Therefore, P(100 < x < 115) = P(0.00 < z < 0.94) = 0.8264 – 0.5000 = 0.3264 Thus, the probability is 0.3264 that a person picked at random has an IQ between 100 and 115. cont’d

8. 8 Applications of Normal Distributions The normal table, Table 3, can be used to answer many kinds of questions that involve a normal distribution. Many times a problem will call for the location of a “cutoff point,” that is, a particular value of x such that there is exactly a certain percentage in a specified area. Example 12 concerns a normal distribution in which you are asked to find the standard deviation  when given the related information.

9. 9 Example 12 – Using the Normal Curve and z to determine Population Parameters The incomes of junior executives in a large corporation are approximately normally distributed. A pending cutback will not discharge those junior executives with earnings within $4900 of the mean. If this represents the middle 80% of the incomes, what is the standard deviation for the salaries of this group of junior executives?

10. 10 Example 12 – Solution Table 3 indicates that the middle 80%, or 0.8000, of a normal distribution is bounded by –1.28 and 1.28. Consider point B shown in the figure.

11. 11 Example 12 – Solution 4900 is the difference between the x value at B and the value of the mean, the numerator of formula (6.3): x –  = 4900. Using formula (6.3) we can find the value of  : That is, the current standard deviation for the salaries of junior executives is $3828. cont’d

12. 12 Applications of Normal Distributions Additional Insight Referring again to the IQ scores, what is the probability that a person picked at random has an IQ of 125, P (x = 125)? (IQ scores are normally distributed, with a mean of 100 and a standard deviation of 16.) This situation has two interpretations: (1) theoretical and (2) practical. Let’s look at the theoretical interpretation first. We know that the probability associated with an interval for a continuous random variable is represented by the area under the curve.

13. 13 Applications of Normal Distributions That is, P (a  x  b) is equal to the area between a and b under the curve. P (x = 125) (that is, x is exactly 125) is then P (125  x  125), or the area of the vertical line segment at x = 125. This area is zero. However, this is not the practical meaning of x = 125, which generally means 125 to the nearest integer value. Thus, P (x = 125) would most likely be interpreted as P(124.5 < x < 125.5)

14. 14 Applications of Normal Distributions The interval from 124.5 to 125.5 under the curve has a measurable area and is then nonzero. In situations of this nature, you must be sure of the meaning being used.