276 CHAPTER 5 Normal Probability Distributions Approximating a Binomial Distribution Two binomial experiments are listed. Determine whether you can use a normal distribution to approximate the distribution of x, the number of people who reply yes. If you can, find the mean and standard deviation. If you cannot, explain why. 1. 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. (Source: Inside Higher Ed) 2. In a survey of high schools in a certain state, it was reported that 18% of seniors were off track to graduate because of at least one course failure. You randomly select 20 seniors from that state and ask them whether they are off track to graduate because of at least one course failure. (Source: Inside Higher Ed) SOLUTION 1. In this binomial experiment, n = 45, p = 0.40, and q = 0.60. So, np = 4510.402 = 18 and nq = 4510.602 = 27. Because np and nq are greater than 5, you can use a normal distribution with m = np = 18 and s = 1npq = 24510.40210.602 ≈ 3.29 to approximate the distribution of x. In the figure at the left, notice that the binomial distribution is approximately bell-shaped, which supports the conclusion that you can use a normal distribution to approximate the distribution of x. 2. In this binomial experiment, n = 20, p = 0.18, and q = 0.82. So, np = 2010.182 = 3.6 and nq = 2010.822 = 16.4. Because np 6 5, you cannot use a normal distribution to approximate the distribution of x. In the figure at the left, notice that the binomial distribution is skewed right, which supports the conclusion that you cannot use a normal distribution to approximate the distribution of x. TRY IT YOURSELF 1 A binomial experiment is listed. Determine whether you can use a normal distribution to approximate the distribution of x, the number of people who reply yes. If you can, find the mean and standard deviation. If you cannot, explain why. In a survey of teen drivers in the United States, 35% admit to texting while driving. You randomly select 100 teen drivers in the United States and ask them whether they admit to texting while driving. (Source: American Automobile Association) Answer: Page A39 EXAMPLE 1 23 25 27 9 111315171921 0.06 0.04 0.02 0.08 0.10 0.12 x P(x) 0 2 4 6 8 10121416 x 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 P(x)
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