280 CHAPTER 5 Normal Probability Distributions Approximating a Binomial Probability A study of former National Football League (NFL) players found that 12.3% have undergone knee joint replacement surgery. You randomly select 100 former NFL players and ask them whether they have undergone knee joint replacement surgery. What is the probability that exactly 15 will say yes? (Source: BMJ Open Sport & Exercise Medicine) SOLUTION Because np = 10010.1232 = 12.3 and nq = 10010.8772 = 87.7, the binomial variable x is approximately normally distributed, with m = np = 12.3 and s = 1npq = 210010.123210.8772 ≈ 3.28. Using the continuity correction, you can rewrite the discrete probability P1x = 152 as the continuous probability P114.5 6 x 6 15.52. The figure shows a normal curve with m = 12.3, s ≈ 3.28, and the shaded area under the curve between 14.5 and 15.5. x μ= 12.3 Number responding yes 14.5 15.5 5 0 10 15 20 25 The z@score that corresponds to 14.5 is z1 = 14.5 - 12.3 2 10010.123210.8772 ≈0.67 and the z@score that corresponds to 15.5 is z2 = 15.5 - 12.3 2 10010.123210.8772 ≈0.97. So, the probability that exactly 15 former NFL players will say they have undergone knee joint replacement surgery is P114.5 6 x 6 15.52 = P10.67 6 z 6 0.972 = P1z 6 0.972 - P1z 6 0.672 = 0.8340 - 0.7486 = 0.0854. Interpretation The probability that exactly 15 former NFL players will say they have undergone knee joint replacement surgery is approximately 0.0854, or about 8.5%. TRY IT YOURSELF 5 The study in Example 5 found that 8.1% of former NFL players have undergone hip joint replacement surgery. You randomly select 100 former NFL players and ask them whether they have undergone hip joint replacement surgery. What is the probability that exactly four will say yes? (Source: BMJ Open Sport & Exercise Medicine) Answer: Page A39 Tech Tip The approximation in Example 5 is comparable to the probability found using the binomial probability feature of a technology tool. For instance, compare the result in Example 5 with the one found on a TI-84 Plus shown below. binompdf(100,.123,15) 0.0807687299 EXAMPLE 5
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