294 CHAPTERS 3–5 CUMULATIVE REVIEW CHAPTER 5 Normal Probability Distributions 1. A survey of adults in the United States found that 61% ate at a restaurant at least once in the past week. You randomly select 30 adults and ask them whether they ate at a restaurant at least once in the past week. (Source: Gallup) (a) Verify that a normal distribution can be used to approximate the binomial distribution. (b) Find the probability that at most 14 adults ate at a restaurant at least once in the past week. (c) Is it unusual for exactly 14 out of 30 adults to have eaten in a restaurant at least once in the past week? Explain your reasoning. In Exercises 2 and 3, find the (a) mean, (b) variance, (c) standard deviation, and (d) expected value of the probability distribution. Interpret the results. 2. The table shows the distribution of household sizes in the United States for a recent year. (Adapted from U.S. Census Bureau) x 1 2 3 4 5 6 7 8 9 P1x2 0.282 0.348 0.151 0.127 0.058 0.023 0.007 0.003 0.002 3. The table shows the distribution of personal fouls per game for DeMar DeRozan in a recent NBA season. (Source: National Basketball Association) x 0 1 2 3 4 5 6 P1x2 0.044 0.162 0.221 0.368 0.103 0.088 0.015 4. Use the probability distribution in Exercise 3 to find the probability of randomly selecting a game in which DeMar DeRozan had (a) fewer than four personal fouls, (b) at least three personal fouls, and (c) between two and four personal fouls, inclusive. 5. From a pool of 16 candidates, 9 men and 7 women, the offices of president, vice president, secretary, and treasurer will be filled. (a) In how many different ways can the offices be filled? (b) What is the probability that all four of the offices are filled by women? In Exercises 6–11, find the indicated area under the standard normal curve. If convenient, use technology to find the area. 6. To the left of z = 0.72 7. To the left of z = -3.08 8. To the right of z = -0.84 9. Between z = 0 and z = 2.95 10. Between z = -1.22 and z = -0.26 11. To the left of z = 0.12 or to the right of z = 1.72 12. Forty-nine percent of U.S. adults think that human activity such as burning fossil fuels contributes a great deal to climate change. You randomly select 25 U.S. adults. Find the probability that the number who think that human activity contributes a great deal to climate change is (a) exactly 12, (b) between 8 and 11, inclusive, and (c) less than two. (d) Are any of these events unusual? Explain your reasoning. (Source: Pew Research Center)
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