91Ó°ÊÓ

Suppose that \({X_1},...,{X_n}\)form a random sample of size n from a distribution for which the mean is 6.5 and the variance is 4. Determine how large the value of n must be in order for the following relation to be satisfied:

\({\bf{P}}\left( {{\bf{6}} \le {{{\bf{\bar X}}}_{\bf{n}}} \le {\bf{7}}} \right) \ge {\bf{0}}{\bf{.8}}\)

Suppose that three girls, A, B, and C throw snowballs at a target. Suppose also that girl A throws 10 times, and the probability that she will hit the target on any given throw is 0.3; girl B throws 15 times, and the probability that she will hit the target on any given throw is 0.2, and girl C throws 20 times, and the probability that she will hit the target on any given throw is 0.1. Determine the probability that the target will be hit at least 12 times

Suppose that 16 digits are chosen at random with replacement from the set {0,...,9}. What is the probability that their average will lie between 4 and 6?

Suppose people attending a party pour drinks from a bottle containing 63 ounces of a particular liquid. Suppose also that the expected size of each drink is 2 ounces, the standard deviation of each drink is 1/2 ounce, and all drinks are poured independently. Determine the probability that the bottle will not be empty after 36 drinks have been poured.

Suppose that 30 percent of the items in a large manufactured lot are of poor quality. Suppose also that a random sample of n items is to be taken from the lot, and let \({Q_n}\) denote the proportion of the items in the sample that are of poor quality. Find a value of n such that Pr(0.2 ≤ \({Q_n}\)≤ 0.4) ≥ 0.75 by using

(a) the Chebyshev inequality and

(b) the tables of the binomial distribution at the end of this book.