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

Copyright © Cengage Learning. All rights reserved. 6.2 The Standard Normal Distribution

3 There are an unlimited number of normal probability distributions, but fortunately they are all related to one distribution, the standard normal distribution. The standard normal distribution is the normal distribution of the standard variable z (called “standard score” or “z-score”).

4 The Standard Normal Distribution Properties of the Standard Normal Distribution 1. The total area under the normal curve is equal to The distribution is mounded and symmetric; it extends indefinitely in both directions, approaching but never touching the horizontal axis. 3. The distribution has a mean of 0 and a standard deviation of The mean divides the area in half, 0.50 on each side. 5. Nearly all the area is between z = –3.00 and z = 3.00.

5 The Standard Normal Distribution Table 3 in Appendix B lists the probabilities associated with the cumulative area to the left of a specified value of z. Probabilities associated with other intervals may be found by using the table entries along with the operations of addition and subtraction, in accordance with the preceding properties. Let’s look at several examples demonstrating how to use Table 3 to find probabilities of the standard normal score, z. We have seen the standard normal distribution in earlier chapters, where it appeared as the empirical rule.

6 The Standard Normal Distribution When using the empirical rule, the values of z were typically integer values; see Figure 6.3. By using Table 3, the z-score will be measured to the nearest one-hundredth and allow increased accuracy. Figure 6.3 Standard Normal Distribution According to Empirical Rule

7 The Standard Normal Distribution We also know that one of the basic properties of a probability distribution is that the sum of all probabilities is exactly 1.0. Since the area under the normal curve represents the measure of probability, the total area under the bell-shaped curve is exactly 1. Notice in Figure 6.3 that the distribution is also symmetrical with respect to a vertical line drawn through z = 0.

8 The Standard Normal Distribution That is, the area under the curve to the left of the mean is one-half, 0.5, and the area to the right is also one-half, 0.5. Note z = 0.00 on Table 3 in Appendix B. Areas (probabilities, percentages) not given directly by the table can be found with the aid of these properties.

9 The Standard Normal Distribution Notes 1. Probabilities associated with positive z-values are greater than since they include the entire left half of the normal curve. 2. Always draw and label a sketch. It is most helpful. 3. Make it a habit to write the z-score with two decimal places, and the areas (probabilities, percentages) with four decimal places, as in Table 3. This will help with distinguishing between the two concepts.

10 The Standard Normal Distribution “The area under the entire normal distribution curve is equal to 1” is the key factor in determining probabilities associated with the values to the right of a z-value.

11 Example 4 – Finding Area Between Any Two z-Values Find the area under the normal curve between z = –1.36 and z = 2.14 : P (–1.36  z  2.14). Solution: The area between z = –1.36 and z = 2.14 is found using subtraction.

12 Example 4 – Solution The cumulative area to the left of the larger z, z = 2.14, includes both the area asked for and the area to the left of the smaller z, z = –1.36, from the area to the left of the larger z, z = 2.14: P (–1.36  z  2.14) = – = cont’d

13 The Standard Normal Distribution Note There are several situations that are similar to Example 4. As in Example 4, one z-score can be negative while the other is positive, or both can be negative, or both can be positive. In all three cases, one of the z-scores is larger (to the right on the figure), the other is smaller (to the left on the figure), and the area in between is found as shown in the example above.

14 The Standard Normal Distribution Table 3 can also be used to find the z-score (s) that bound (s) a specified area. By finding the area or probability within the table, the z-score can be read from along the left side and top margins.

15 Example 7 – Finding Two z-Scores that Bound an Area What z-scores bound the middle 95% of a normal distribution? Solution: The 95% is split into two equal parts by the mean, so is the area (percentage) between the z-score at the left boundary and z = 0, the mean (as well as the area between z = 0, the mean, and the right boundary).

16 Example 7 – Solution See Figure 6.5. Figure 6.5 cont’d

17 Example 7 – Solution The area that is not included in either tail can be found by recalling that the area for each half of the normal curve is equal to and that the curve is symmetric. Thus on the left side, – = is needed; and on the right side, = is needed. cont’d

18 Example 7 – Solution To find the left boundary z-score, use the area in Table 3 and find the “area” entry that is closest to ; this entry is exactly Reading the table, the z-score that corresponds to this area is found to be z = –1.96. A Portion of Table 3 (negative z-side) and A Portion of Table 3 (positive z-side) Table 6.5 cont’d

19 Example 7 – Solution Likewise, to find the right boundary z-score, use the area in Table 3 and find the “area” entry that is closest to ; this entry is exactly Reading this z-score gives z = Therefore, you can look up either one and utilize the symmetry of the normal distribution. z = –1.96 and z = 1.96 bound the middle 95% of a normal distribution. As a check, consider doing it one way, and then check the result using the other way. cont’d