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www.downloadslide.com 3.4 The Binomial Probability Distribution 109 is the success. As a consequence, q may be used erroneously in place of p. Carefully define a success and make certain that p equals the probability of a success for each application. Thus far in this section we have assumed that the number of trials, n, and the probability of success, p, were known, and we used the formula p(y) = ( n y) p y q n−y to compute probabilities associated **with** binomial random variables. In Example 3.8 we obtained a value for P(Y ≥ 9) and used this probability to reach a conclusion about the effectiveness of the medication. The next example exhibits another statistical, rather than probabilistic, use of the binomial distribution. EXAMPLE 3.10 Solution Suppose that we survey 20 individuals working for a large company and ask each whether they favor implementation of a new policy regarding retirement funding. If, in our sample, 6 favored the new policy, find an estimate for p, the true but unknown proportion of employees that favor the new policy. If Y denotes the number among the 20 who favor the new policy, it is reasonable to conclude that Y has a binomial distribution **with** n = 20 for some value of p. Whatever the true value for p, we conclude that the probability of observing 6 out of 20 in favor of the policy is ( ) 20 P(Y = 6) = p 6 (1 − p) 14 . 6 We will use as our estimate for p the value that maximizes the probability of observing the value that we actually observed (6 in favor in 20 trials). How do we find the value of p that maximizes P(Y = 6)? Because ( ) 20 6 is a constant (relative to p) and ln(w) is an increasing function of w, the value of p that maximizes P(Y = 6) = ( ) 20 6 p 6 (1 − p) 14 is the same as the value of p that maximizes ln[p 6 (1 − p) 14 ] = [6 ln(p) + 14 ln(1 − p)]. If we take the derivative of [6 ln(p) + 14 ln(1 − p)] **with** respect to p, we obtain d[6 ln(p) + 14 ln(1 − p)] dp = ( 6 p ) − ( 14 1 − p The value of p that maximizes (or minimizes) [6 ln(p) + 14 ln(1 − p)] [and, more important, P(Y = 6)] is the solution to the equation 6 p − 14 1 − p = 0. Solving, we obtain p = 6/20. Because the second derivative of [6 ln(p) + 14 ln(1 − p)] is negative when p = 6/20, it follows that [6 ln(p) + 14 ln(1 − p)] [and P(Y = 6)] ismaximized when p = 6/20. Our estimate for p, based on 6 “successes” in 20 trials is therefore 6/20. The ultimate answer that we obtained should look very reasonable to you. Because p is the probability of a “success” on any given trial, a reasonable estimate is, indeed, ) .

www.downloadslide.com 110 Chapter 3 Discrete Random Variables and Their Probability Distributions the proportion of “successes” in our sample, in this case 6/20. In the next section, we will apply this same technique to obtain an estimate that is not initially so intuitive. As we will see in Chapter 9, the estimate that we just obtained is the maximum likelihood estimate for p and the procedure used above is an example of the application of the method of maximum likelihood. Exercises 3.35 Consider the population of voters described in Example 3.6. Suppose that there are N = 5000 voters in the population, 40% of whom favor Jones. Identify the event favors Jones as a success S. It is evident that the probability of S on trial 1 is .40. Consider the event B that S occurs on the second trial. Then B can occur two ways: The first two trials are both successes or the first trial is a failure and the second is a success. Show that P(B) = .4. What is P(B| the first trial is S)? Does this conditional probability differ markedly from P(B)? 3.36 The manufacturer of a low-calorie dairy drink wishes to compare the taste appeal of a new formula (formula B) **with** that of the standard formula (formula A). Each of four judges is given three glasses in random order, two containing formula A and the other containing formula B. Each judge is asked to state which glass he or she most enjoyed. Suppose that the two formulas are equally attractive. Let Y be the number of judges stating a preference for the new formula. a Find the probability function for Y . b What is the probability that at least three of the four judges state a preference for the new formula? c Find the expected value of Y . d Find the variance of Y . 3.37 In 2003, the average combined SAT score (math and verbal) for college-bound students in the United States was 1026. Suppose that approximately 45% of all high school graduates took this test and that 100 high school graduates are randomly selected from among all high school grads in the United States. Which of the following random variables has a distribution that can be approximated by a binomial distribution? Whenever possible, give the values for n and p. a b c d e The number of students who took the SAT The scores of the 100 students in the sample The number of students in the sample who scored above average on the SAT The amount of time required by each student to complete the SAT The number of female high school grads in the sample 3.38 a A meteorologist in Denver recorded Y = the number of days of rain during a 30-day period. Does Y have a binomial distribution? If so, are the values of both n and p given? b A market research firm has hired operators who conduct telephone surveys. A computer is used to randomly dial a telephone number, and the operator asks the answering person whether she has time to answer some questions. Let Y = the number of calls made until the first person replies that she is willing to answer the questions. Is this a binomial experiment? Explain.

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714 Chapter 14 Analysis of Categori

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742 Chapter 15 Nonparametric Statis

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