SECTION 5.3 Normal Distributions: Finding Values 255 Finding a Specific Data Value for a Given Probability You can also use the normal distribution to find a specific data value (x@value) for a given probability, as shown in Examples 4 and 5. Finding a Specific Data Value Scores for the California Peace Officer Standards and Training test are normally distributed, with a mean of 50 and a standard deviation of 10. An agency will only hire applicants with scores in the top 10%. What is the lowest score an applicant can earn and still be eligible to be hired by the agency? (Source: State of California) SOLUTION Exam scores in the top 10% correspond to the shaded region shown. 0 1.28 z 50 ? x 10% Test score Scores for the California Peace Officer Standards and Training Test A test score in the top 10% is any score above the 90th percentile. To find the score that represents the 90th percentile, you must first find the z@score that corresponds to a cumulative area of 0.9. In the Standard Normal Table, the area closest to 0.9 is 0.8997. So, the z@score that corresponds to an area of 0.9 is z = 1.28. To find the x@value, note that m = 50 and s = 10, and use the formula x = m + zs, as shown. x = m + zs = 50 + 1.281102 = 62.8 You can check this answer using technology. For instance, you can use a TI-84 Plus to find the x-value, as shown at the left. Interpretation The lowest score an applicant can earn and still be eligible to be hired by the agency is about 63. TRY IT YOURSELF 4 A researcher tests the braking distances of several cars. The braking distance from 60 miles per hour to a complete stop on dry pavement is measured in feet. The braking distances of a sample of cars are normally distributed, with a mean of 132 feet and a standard deviation of 5.18 feet. What is the longest braking distance one of these cars could have and still be in the bottom 1%? (Adapted from Consumer Reports) Answer: Page A39 EXAMPLE 4 Picturing the World Many investors choose mutual funds as a way to invest in the stock market.The mean annual rate of return for large growth mutual funds during a recent five-year period was about 21.2% with a standard deviation of 3.5%. (Adapted from Morningstar) Annual Rate of Return for Large Growth Mutual Funds x Rate of return = 0.212 μ 0.14 0.21 0.28 Between what two values does the middle 90% of the data lie? TI-84 PLUS invNorm(.9,50,10) 62.81551567
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