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Practice exercises for Mathematical Statistics (Set 1)

1. 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 .

2. 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.

3. A complex electronic system is built with a certain number of backup components in its subsystems. One subsystem has four identical components, each with a probability of .2 of failing in less than 1000 hours. The subsystem will operate if any two of the four components are operating. Assume that the components operate independently. Find the probability that

a. exactly two of the four components last longer than 1000 hours.

b. the subsystem operates longer than 1000 hours.

4. The probability that a patient recovers from a stomach disease is .8. Suppose 20 people are known to have contracted this disease. What is the probability that

a. exactly 14 recover?

b. at least 10 recover?

c. at least 14 but not more than 18 recover?

d. at most 16 recover?

5. A multiple-choice examination has 15 questions, each with five possible answers, only one of which is correct. Suppose that one of the students who takes the examination answers each of the questions with an independent random guess. What is the probability that he answers at least ten questions correctly?

6. Two teams A and B play a series of games until one team wins four games.We assume that the games are played independently and that the probability that A wins any game is p. Compute the probability that the series lasts exactly five games.

7. Suppose that 30% of the applicants for a certain industrial job possess advanced training in computer programming. Applicants are interviewed sequentially and are selected at random from the pool. Find the probability that the first applicant with advanced training in programming is found on the fifth interview.

8. An oil prospector will drill a succession of holes in a given area to find a productive well. The probability that he is successful on a given trial is .2.

a. What is the probability that the third hole drilled is the first to yield a productive well?

b. If the prospector can afford to drill at most ten wells, what is the probability that he will fail to find a productive well?

9. The probability of a customer arrival at a grocery service counter in any one second is equal to .1. Assume that customers arrive in a random stream and hence that an arrival in any one second is independent of all others. Find the probability that the first arrival

a. will occur during the third one-second interval.

b. will not occur until at least the third one-second interval.

10. Ten percent of the engines manufactured on an assembly line are defective. If engines are randomly selected one at a time and tested, what is the probability that the first nondefective engine will be found on the second trial?