108 CHAPTER 2 Descriptive Statistics Finding z-Scores The mean speed of vehicles along a stretch of highway is 56 miles per hour with a standard deviation of 4 miles per hour. You measure the speeds of three cars traveling along this stretch of highway as 62 miles per hour, 47 miles per hour, and 56 miles per hour. Find the z-score that corresponds to each speed. Assume the distribution of the speeds is approximately bell-shaped. SOLUTION The z-score that corresponds to each speed is calculated below. x = 62 mph x = 47 mph x = 56 mph z = 62 - 56 4 = 1.5 z = 47 - 56 4 = -2.25 z = 56 - 56 4 = 0 Interpretation From the z-scores, you can conclude that a speed of 62 miles per hour is 1.5 standard deviations above the mean; a speed of 47 miles per hour is 2.25 standard deviations below the mean; and a speed of 56 miles per hour is equal to the mean. The car traveling 47 miles per hour is said to be traveling unusually slowly, because its speed corresponds to a z-score of -2.25. TRY IT YOURSELF 7 The monthly utility bills in a city have a mean of $70 and a standard deviation of $8. Find the z-scores that correspond to utility bills of $60, $71, and $92. Assume the distribution of the utility bills is approximately bell-shaped. Answer: Page A37 Comparing z-Scores from Different Data Sets The table shows the mean heights and standard deviations for a population of men and a population of women. Compare the z-scores for a 6-foot-tall man and a 6-foot-tall woman. Assume the distributions of the heights are approximately bell-shaped. Men’s heights Women’s heights m = 69.9 in. m = 64.3 in. s = 3.0 in. s = 2.6 in. SOLUTION Note that 6 feet = 72 inches. Find the z-score for each height. z-score for 6-foot-tall man z-score for 6-foot-tall woman z = x - m s = 72 - 69.9 3.0 = 0.7 z = x - m s = 72 - 64.3 2.6 ≈ 3.0 Interpretation The z-score for the 6-foot-tall man is within 1 standard deviation of the mean (69.9 inches). This is among the typical heights for a man. The z-score for the 6-foot-tall woman is about 3 standard deviations from the mean (64.3 inches). This is an unusual height for a woman. TRY IT YOURSELF 8 Use the information in Example 8 to compare the z-scores for a 5-foot-tall man and a 5-foot-tall woman. Answer: Page A37 EXAMPLE 7 EXAMPLE 8
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