278 CHAPTER 5 Normal Probability Distributions x Number responding yes μ= 18 8 1216202428 19.5 Approximating Binomial Probabilities Using a Normal Distribution to Approximate Binomial Probabilities In Words In Symbols 1. Verify that a binomial distribution Specify n, p, and q. applies. 2. Determine whether you can use a normal Is np Ú 5? distribution to approximate x, the Is nq Ú 5? binomial variable. 3. Find the mean m and standard deviation s m = np for the distribution. s = 1npq 4. Apply the appropriate continuity Add 0.5 to (or subtract correction. Shade the corresponding 0.5 from) the binomial area under the normal curve. probability. 5. Find the corresponding z@score(s). z = x - m s 6. Find the probability. Use the Standard Normal Table. GUIDELINES Approximating a Binomial Probability In a survey of high schools in a certain state, it was reported that 40% of students failed at least one class taken through distance learning. You randomly select 45 students from that state and ask them whether they failed at least one class taken through distance learning. What is the probability that fewer than 20 of them respond yes? (Source: Inside Higher Ed) SOLUTION From Example 1, you know that you can use a normal distribution with m = 18 and s ≈ 3.29 to approximate the binomial distribution. To use a normal distribution, note that the probability is “fewer than 20.” So, apply the continuity correction by subtracting 0.5 from 20 and write the probability as P1x 6 20 - 0.52 = P1x 6 19.52. The figure at the left shows a normal curve with m = 18, s ≈ 3.29, and the shaded area to the left of 19.5. The z-score that corresponds to x = 19.5 is z = x - m s z ≈ 19.5 - 18 3.29 ≈ 0.46. Using the Standard Normal Table, P1z 6 0.462 = 0.6772. Interpretation The probability that fewer than twenty students respond yes is approximately 0.6772, or about 67.72%. EXAMPLE 3 Picturing the World In a survey of U.S. adults in a romantic relationship, 60.4% responded that they have hidden purchases from their spouses or partners, as shown in the pie chart. (Adapted from MyBankTracker) Have You Ever Hidden Purchases from Your Spouse or Partner? Yes 60.4% No 39.6% Assume that this survey is a true indication of the proportion of the population who say they have hidden purchases from their spouses or partners.You sample 50 adults with spouses or partners at random. What is the probability that between 20 and 25, inclusive, would say they have hidden purchases from their spouses or partners?
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