According to the U.S. Golf Association (USGA), "The weight of the [golf] ball shall not be greater than 1.620 ounces avoirdupois (45.93 grams). The diameter of the ball shall not be less than 1.680 inchesThe velocity of the ball shall not be greater than 250 feet per second" (USGA, 2002). The USGA periodically checks the specifications of golf balls sold in the United States by randomly sampling balls from pro shops around the country. Two dozen of each kind are sampled, and if more than three do not meet size and/or velocity requirements, that kind of ball is removed from the USGA's approved-ball list.
a. What assumptions must be made and what information must be known in order to use the binomial probability distribution to calculate the probability that the USGA will remove a particular kind of golf ball from its approved-ball list?
b. Suppose 10% of all balls produced by a particular manufacturer are less than 1.680 inches in diameter, and assume that the number of such balls, x, in a sample of two dozen balls can be adequately characterized by a binomial probability distribution. Find the mean and standard deviation of the binomial distribution.
c. Refer to part b. If x has a binomial distribution, then so does the number, y, of balls in the sample that meet the USGA's minimum diameter. [x + y = 24.] Describe the distribution of y. In particular, what are p, q, and n? Also, find E(y) and the standard deviation of y.