Section 7.5 The Normal Approximation to the Binomial Distribution 327 Solution This situation is illustrated in Figure 7.29. Area = 0.7286 44.5 55.5 z = −1.10 z = 1.10 Figure 7.31 Step 1: Check the assumptions: The number of trials is n = 100. Since the coin is fair, the success probability is p = 0.5. Therefore, np = (100)(0.5) = 50 ≥ 10 and n(1 − p) = (100)(1 − 0.5) = 50 ≥ 10. We can use the normal approximation. Step 2: We compute the mean and standard deviation of X: ��X = np = (100)(0.5) = 50 ��X = √ np(1 − p) = √ (100)(0.5)(1 − 0.5) = 5 Step 3: Because the probability is P(45 ≤ X ≤ 55), we want to include both 45 and 55. Therefore, we set the left endpoint to 44.5 and the right endpoint to 55.5. Step 4: We sketch a normal curve, label the mean of 50, and the endpoints 44.5 and 55.5. Step 5: We use Table A.2 to find the area. The z-scores for 44.5 and 55.5 are z = 44.5 − 50 5 = −1.1 z = 55.5 − 50 5 = 1.1 From Table A.2, we find that the probability is 0.7286. See Figure 7.31. In Example 7.25, we used the normal approximation to compute a probability of the form P(a ≤ X ≤ b). We can also use the normal approximation to compute probabilities of the form P(X ≤ a), P(X ≥ a), and P(X = a). EXAMPLE 7.26 Illustrate areas to be found for the continuity correction A fair coin is tossed 100 times. Let X be the number of heads that appear. Illustrate the area under the normal curve that represents each of the following probabilities. a. P(X ≤ 55) b. P(X ≥ 55) c. P(X = 55) Solution a. We find the area to the left of 55.5, as illustrated in Figure 7.32. b. We find the area to the right of 54.5, as illustrated in Figure 7.33. c. We find the area between 54.5 and 55.5, as illustrated in Figure 7.34. 30 35 40 45 50 55 60 65 70 0.08 0.06 0.04 0.02 0 55.5 Figure 7.32 To approximate P(X ≤ 55), find the area to the left of 55.5. 30 35 40 45 50 55 60 65 70 0.08 0.06 0.04 0.02 0 54.5 Figure 7.33 To approximate P(X ≥ 55), find the area to the right of 54.5. 30 35 40 45 50 55 60 65 70 0.08 0.06 0.04 0.02 0 54.5 55.5 Figure 7.34 To approximate P(X = 55), find the area between 54.5 and 55.5.
navidi_monk_elementary_statistics_2e_ch7-9
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