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1A.

In each situation below, is it reasonable to use a binomial distribution for the random variable X? Give reasons for your answer in each case. (a An auto manufacturer chooses one car from each hour's production for a detailed quality inspection. One variable recorded is the count X of finish defects (dimple, ripples, etc.) in the car's paint. 3 Each observation falls into a "success" or "failure." :I There is a fixed number n of observations. :I The n observations are all independent. 3 The probability of success p is the same for each observation. :| None of the binomial conditions are met. (b) The pool of potential jurors for a murder case contains 100 persons chosen at random from the adult residents of a large city. Each person in the pool is asked whether he or she opposes the death penalty. X is the number who say "Yes." :I Each observation falls into a "success" or "failure." 3 "here is a fixed number n of observations. 3 "he n observations are all independent. :I "he probability of success ,0 is the same for each observation. 3 None of the binomial conditions are met. (c) Joe buys a ticket in his state's "Pick 3" lottery game every week; X is the number of times in a year that he wins a prize. 3 Each observation falls into a "success" or "failure." :I There is a fixed number n of observations. 3 The n observations are all independent. :I The probability of success p is the same for each observation. :| None of the binomial conditions are met. Suppose that James guesses on each question of a 43-item true-false quiz. Find the probability that James passes if each of the following is true. (a) A score of 24 or more correct is needed to pass. E (b) A score of 29 or more correct is needed to pass. S (c) A score of 31 or more correct is needed to pass. E According to a study by the Bureau of Justice Statistics, approximately 2% of the nation's 72 million children had a parent behind bars - nearly 1.4 million minors. Let X be the number of children who had an incarcerated parent. Suppose that 130 children are randomly selected. (a) Does X satisfy the requirements for a binomial setting? Explain. O Each observation falls into a "success" or "failure." O There is a fixed number n of observations. O The n observations are all independent. O The probability of success p is the same for each observation. O None of the binomial conditions are met. If X = B(n, p), what are n and p? n = p = (b) Describe P(X = 0) in words. O the probability that none of the children with incarcerated parents were missed by the study O the probability that there are no children without incarcerated parents O the probability of no incarcerated parents O the probability of none of the children having an incarcerated parent Then find P(X = 0) and P(X = 1). P(X = 0) = P(X = 1) = (c) What is the probability that 2 or more of the 130 children have a parent behind bars?Among employed women, 25% have never been married. Select 13 employed women at random. (a) The number in your sample who have never been married has a binomial distribution. What are n and p? n = p = (b) What is the probability that exactly 3 of the 13 women in your sample have never been married? :I (c) What is the probability that 2 or fewer women have never been married? :I Suppose you purchase a bundle of 14 bare-root broccoli plants. The sales clerk tells you that on average you can expect 3% of the plants to die before producing any broccoli. Assume that the bundle is a random sample of plants. Use the binomial formula to nd the probability that you will lose at most 1 of the broccoli plants. S A university claims that 70% of its basketball players get degrees. An investigation examines the fate of all 15 players who entered the program over a period of several years. The number of athletes who graduate is 5(15, 0.70). Use the binomial probability formula to find the probability that all 15 graduate. S What is the probability that not all of the 15 graduate? E A factory employs several thousand workers, of whom 20% are Hispanic. If the 19 members of the union executive committee were chosen from the workers at random, the number of Hispanics on the committee would have the binomial distribution with n = 19 and p = 0.2. (a) What is the mean number of Hispanics on randomly chosen committees of 19 workers? |:| (b) What is the standard deviation 0 of the count X of Hispanic members? l:| (c) Suppose that 10% of the factory workers were Hispanic. Then p = 0.1. What is a in this case? |:| What is a ifp = 0.01? l:| What does your work show about the behavior of the standard deviation of a binomial distribution as the probability of a success gets closer to 0? 0 As ,0 decreases, 0 increases. 0 As ,0 increases, 0 increases. 0 As ,0 increases, 0 decreases. 0 As ,0 decreases, 0 decreases