326 Chapter 7 The Normal Distribution that X will be approximately normal whenever ̂p is approximately normal. This is in fact the case. The Normal Approximation to the Binomial Let X be a binomial random variable with n trials and success probability p. If np ≥ 10 and n(1 − p) ≥ √ 10, then X is approximately normal with mean ��X = np and standard deviation ��X = np(1 − p). The continuity correction The binomial distribution is discrete, whereas the normal distribution is continuous. The continuity correction is an adjustment, made when approximating a discrete distribution with a continuous one, that can improve the accuracy of the approximation. To see how it works, imagine that a fair coin is tossed 100 times. Let X represent the number of heads. Then X has the binomial distribution with n = 100 trials and success probability p = 0.5. Imagine that we want to compute the probability that X is between 45 and 55. This probability will differ depending on whether the endpoints, 45 and 55, are included or excluded. Figure 7.29 illustrates the case where the endpoints are included, that is, where we wish to compute P(45 ≤ X ≤ 55). The exact probability is given by the total area of the rectangles of the binomial probability histogram corresponding to the integers 45 to 55, inclusive. The approximating normal curve is superimposed. To get the best approximation, we should compute the area under the normal curve between 44.5 and 55.5. In contrast, Figure 7.30 illustrates the case where we wish to compute P(45 < X < 55). Here the endpoints are excluded. The exact probability is given by the total area of the rectangles of the binomial probability histogram corresponding to the integers 46 to 54. The best normal approximation is found by computing the area under the normal curve between 45.5 and 54.5. 40 45 50 55 60 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 44.5 55.5 Figure 7.29 To compute P(45 ≤ X ≤ 55), the areas of the rectangles corresponding to 45 and to 55 should be included. To approximate this probability with the normal curve, compute the area under the curve between 44.5 and 55.5. 40 45 50 55 60 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 45.5 54.5 Figure 7.30 To compute P(45 < X < 55), the areas of the rectangles corresponding to 45 and to 55 should be excluded. To approximate this probability with the normal curve, compute the area under the curve between 45.5 and 54.5. In general, to apply the continuity correction, determine precisely which rectangles of the discrete probability histogram you wish to include, then compute the area under the normal curve corresponding to those rectangles. EXAMPLE 7.25 Using the continuity correction to compute a probability Let X be the number of heads that appear when a fair coin is tossed 100 times. Use the normal curve to find P(45 ≤ X ≤ 55).
navidi_monk_elementary_statistics_2e_ch7-9
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