222 CHAPTER 4 Discrete Probability Distributions Extending Concepts 27. Poisson Approximation of a Binomial Distribution An automobile manufacturer finds that 1 in every 2500 automobiles produced has a specific manufacturing defect. (a) Use a binomial distribution to find the probability of finding 4 cars with the defect in a random sample of 6000 cars. (b) The Poisson distribution with a mean m = np can be used to approximate the binomial distribution for large values of n and small values of p. Repeat part (a) using the Poisson distribution and compare the results. 28. Hypergeometric Distribution Binomial experiments require that any sampling be done with replacement because each trial must be independent of the others. The hypergeometric distribution also has two outcomes: success and failure. The sampling, however, is done without replacement. For a population of N items having k successes and N - k failures, the probability of selecting a sample of size n that has x successes and n - x failures is given by P1x2 = 1 kCx21N-kCn-x2 NCn . In a shipment of 15 microchips, 2 are defective and 13 are not defective. A sample of three microchips is chosen at random. Use the above formula to find the probability that (a) all three microchips are not defective, (b) one microchip is defective and two are not defective, and (c) two microchips are defective and one is not defective. Geometric Distribution: Mean and Variance In Exercises 29 and 30, use the fact that the mean of a geometric distribution is m = 1 p and the variance is s 2 = q p2. 29. Daily Lottery A daily number lottery chooses three balls numbered 0 to 9. The probability of winning the lottery is 1/1000. Let x be the number of times you play the lottery before winning the first time. (a) Find the mean, variance, and standard deviation. (b) How many times would you expect to have to play the lottery before winning? (c) The price to play is $1 and winners are paid $500. Would you expect to make or lose money playing this lottery? Explain. 30. Paycheck Errors A company assumes that 0.5% of the paychecks for a year were calculated incorrectly. The company has 200 employees and examines the payroll records from one month. (a) Find the mean, variance, and standard deviation. (b) How many employee payroll records would you expect to examine before finding one with an error? Poisson Distribution: Variance In Exercises 31 and 32, use the fact that the variance of the Poisson distribution is s 2 = m. 31. Golf In a recent year, the mean number of strokes per hole for golfer Bubba Watson was about 3.9. (a) Find the variance and standard deviation. Interpret the results. (b) Identify any numbers of strokes on a hole that you would consider unusual. (Source: PGATour.com) 32. Bankruptcies The mean number of bankruptcies filed per hour by businesses in the United States in 2020 was about 2.5. (a) Find the variance and the standard deviation. Interpret the results. (b) Identify any numbers of bankruptcies during an hour that you would consider unusual. (Source: Administrative Office of the U.S. Courts)
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