256 CHAPTER 5 Normal Probability Distributions Finding a Specific Data Value In a randomly selected sample of U.S. adults ages 20 and over, the mean total cholesterol level is 190 milligrams per deciliter with a standard deviation of 40.9 milligrams per deciliter. Assume the total cholesterol levels are normally distributed. Find the highest total cholesterol level an adult aged 20 or over can have and still be in the bottom 1%. (Adapted from National Center for Health Statistics) SOLUTION Total cholesterol levels in the lowest 1% correspond to the shaded region shown. −2.33 0 z ? 190 x Total Cholesterol Levels in U.S. Adults Ages 20 and Over 1% Total cholesterol level (in mg/dL) A total cholesterol level in the lowest 1% is any level below the 1st percentile. To find the level that represents the 1st percentile, you must first find the z@score that corresponds to a cumulative area of 0.01. In the Standard Normal Table, the area closest to 0.01 is 0.0099. So, the z@score that corresponds to an area of 0.01 is z = -2.33. To find the x@value, note that m = 190 and s = 40.9, and use the formula x = m + zs, as shown. x = m + zs = 190 + 1-2.332140.92 ≈ 94.7 You can check this answer using technology. For instance, you can use Excel to find the x@value, as shown below. EXCEL 94.85237195 =NORM.INV(0.01,190,40.9) A 1 Interpretation The value that separates the lowest 1% of total cholesterol levels for U.S. adults ages 20 and over from the highest 99% is about 95 milligrams per deciliter. TRY IT YOURSELF 5 The lengths of time employees have worked at a corporation are normally distributed, with a mean of 11.2 years and a standard deviation of 2.1 years. In a company cutback, the lowest 10% in seniority are laid off. What is the maximum length of time an employee could have worked and still be laid off? Answer: Page A39 EXAMPLE 5
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