Section 7.5 The Normal Approximation to the Binomial Distribution 329 Sometimes we want to use the normal approximation to compute a probability of the form P(X < a) or P(X > a). One way to do this is by changing the inequality to a form involving ≤ or ≥. Example 7.28 provides an illustration. EXAMPLE 7.28 Using the continuity correction to compute a probability The Statistical Abstract of the United States reported that 66% of students who graduated from high school in 2012 enrolled in college. One hundred high school graduates are sampled. Approximate the probability that more than 60 enroll in college. Solution Let X be the number of students in the sample who enrolled in college. We need to find P(X > 60). We change this to an inequality involving ≥ by noting that P(X > 60) = P(X ≥ 61). We therefore find P(X ≥ 61). 60.5 66 z = –1.16 Figure 7.36 Step 1: Check the assumptions. The number of trials is n = 100 and the success probability is p = 0.66. Therefore np = (100)(0.66) = 66 ≥ 10 and n(1 − p) = (100)(1 − 0.66) = 34 ≥ 10. We can use the normal approximation. Step 2: We compute the mean and standard deviation of X: ��X = np = (100)(0.66) = 66 √ ��X = np(1 − p) = √ (100)(0.66)(1 − 0.66) = 4.73709 Step 3: Since the probability is P(X ≥ 61), we compute the area to the right of 60.5. Step 4: We sketch a normal curve, and label the mean of 66 and the point 60.5. Step 5: We use Table A.2 to find the area. The z-score for 60.5 is z = 60.5 − 66 4.73709 = −1.16 From Table A.2 we find that the probability is 0.8770. See Figure 7.36. Check Your Understanding 1. X is a binomial random variable with n = 50 and p = 0.15. Should the normal approximation be used to find P(X > 10)? Why or why not? No 2. X is a binomial random variable with n = 72 and p = 0.90. Should the normal approximation be used to find P(X ≤ 60)? Why or why not? No 3. Let X have a binomial distribution with n = 64 and p = 0.41. If appropriate, use the normal approximation to find P(X ≤ 20). If not, explain why not. 0.0721 Tech: 0.0723 4. Let X have a binomial distribution with n = 379 and p = 0.09. If appropriate, use the normal approximation to find P(X > 40). If not, explain why not. 0.1251 Tech: 0.1257 Answers are on page 331. SECTION 7.5 Exercises Exercises 1– 4 are the Check Your Understanding exercises located within the section. Understanding the Concepts In Exercises 5 and 6, fill in each blank with the appropriate word or phrase. 5. If X is a binomial random variable and if np ≥ 10 and n(1 − p) ≥ 10, then X is approximately normal with ��X = and ��X = . np, √ np(1 − p) 6. The adjustment made when approximating a discrete random distribution with a continuous one is called the correction. continuity In Exercises 7 and 8, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement. 7. If technology is to be used, exact binomial probabilities can be calculated and the normal approximation is not necessary. True
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