300 Chapter 7 The Normal Distribution Properties of the z-Score 1. The z-score follows a standard normal distribution. 2. Values below the mean have negative z-scores, and values above the mean have positive z-scores. 3. The z-score tells how many standard deviations the original value is above or below the mean. Because the z-score follows a standard normal distribution, we can use the methods of Section 7.1 to find areas under any normal curve, by standardizing to convert the original values to z-scores. Rounding Off z-Scores The z-scores in Table A.2 are expressed to two decimal places. For this reason, when converting normal values to z-scores, we will round off the z-scores to two decimal places. EXAMPLE 7.12 Finding and interpreting a z-score Heights in a certain population of women follow a normal distribution with mean �� = 64 inches and standard deviation �� = 3 inches. a. A randomly selected woman has a height of x = 67 inches. Find and interpret the z-score of this value. b. Another randomly selected woman has a height of x = 63 inches. Find and interpret the z-score of this value. Solution a. The z-score for x = 67 is z = 67 − �� �� = 67 − 64 3 = 1.00 We interpret this by saying that a height of 67 inches is 1 standard deviation above the mean height of 64 inches. b. The z-score for x = 63 is z = 63 − �� �� = 63 − 64 3 = −0.33 We interpret this by saying that a height of 63 inches is 0.33 standard deviation below the mean height of 64 inches. Explain It Again Converting x-values to z-scores: After we convert an x-value to a z-score, we use the standard normal curve. This allows us to find areas under the normal curve by using Table A.2. Figure 7.14 illustrates the results of Example 7.12. Figure 7.14(a) is the normal curve that represents the heights of the population of women. It has a mean of 64. The heights of the two women are indicated at 63 and 67. Figure 7.14(b) is the standard normal curve. The mean is 0, and the heights are represented by their z-scores of −0.33 and 1.00. (a) 63 64 67 (b) −0.33 0 1.00 Figure 7.14 (a) This is the normal curve with mean 64 and standard deviation 3. It represents the population of heights of women. The heights of 63 and 67 are shown on the x-axis. (b) This is the standard normal curve. It also represents the population of heights of women, by using the z-scores instead of the actual heights. A height of 63 inches is represented by a z-score of −0.33, and a height of 67 inches is represented by a z-score of 1.00.
navidi_monk_elementary_statistics_2e_ch7-9
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