Suppose that, on the average, 1/3 of the graduating

Each minute a machine produces a length of rope with mean of 4 feet and standard deviation of 5 inches. Assuming that the amounts produced in different minutes are independent and identically distributed, approximate the probability that the machine will produce at least 250 feet in one hour.

Suppose that 75 percent of the people in a certain metropolitan area live in the city and 25 percent of the people live in the suburbs. If 1200 people attending a certain concert represent a random sample from the metropolitan area, what is the probability that the number of people from the suburbs attending the concert will be fewer than 270?

Suppose that the distribution of the number of defects on any given bolt of cloth is the Poisson distribution with mean 5, and the number of defects on each bolt is counted for a random sample of 125 bolts. Determine the probability that the average number of defects per bolt in the sample will be less than 5.5.

Suppose that a random sample of size n is to be taken from a distribution for which the mean is and the standard deviation is 3. Use the central limit theorem to determine approximately the smallest value of n for which the following relation will be satisfied: Pr(|Xn | < 0.3) 0.95.

Suppose that the proportion of defective items in a large manufactured lot is 0.1. What is the smallest random sample of items that must be taken from the lot in order for the probability to be at least 0.99 that the proportion of defective items in the sample will be less than 0.13?

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

A physicist makes 25 independent measurements of the specific gravity of a certain body. He knows that the limitations of his equipment are such that the standard deviation of each measurement is units. a. By using the Chebyshev inequality, find a lower bound for the probability that the average of his measurements will differ from the actual specific gravity of the body by less than /4 units. b. By using the central limit theorem, find an approximate value for the probability in part (a).

A random sample of n items is to be taken from a distribution with mean and standard deviation . a. Use the Chebyshev inequality to determine the smallest number of items n that must be taken in order to satisfy the following relation: Pr |Xn | 4  0.99. b. Use the central limit theorem to determine the smallest number of items n that must be taken in order to satisfy the relation in part (a) approximately.

Suppose that, on the average, 1/3 of the graduating seniors at a certain college have two parents attend the graduation ceremony, another third of these seniors have one parent attend the ceremony, and the remaining third of these seniors have no parents attend. If there are 600 graduating seniors in a particular class, what is the probability that not more than 650 parents will attend the graduation ceremony?

Let Xn be a random variable having the binomial distribution with parameters n and pn. Assume that limn npn = . Prove that the m.g.f. of Xn converges to the m.g.f. of the Poisson distribution with mean .

Suppose that X1,...,Xn form a random sample from a normal distribution with unknown mean and variance 2. Assuming that = 0, determine the asymptotic distribution of X3 n.

Suppose that X1,...,Xn form a random sample from a normal distribution with mean 0 and unknown variance 2. a. Determine the asymptotic distribution of the statistic  1 n n i=1 X2 i 1 . b. Find a variance stabilizing transformation for the statistic 1 n n i=1 X2 i .

Let X1, X2,... be a sequence of i.i.d. random variables each having the uniform distribution on the interval [0, ] for some real number > 0. For each n, define Yn to be the maximum of X1,...,Xn. 6.4 The Correction for Continuity 371 a. Show that the c.d.f. of Yn is Fn(y) = 0 if x 0, (y/ )n if 0 . Hint: Read Example 3.9.6. b. Show that Zn = n(Yn ) converges in distribution to the distribution with c.d.f. F(z) =  exp(z/ ) if z < 0, 1 if z > 0. Hint: Apply Theorem 5.3.3 after finding the c.d.f. of Zn. c. Use Theorem 6.3.2 to find the approximate distribution of Y 2 n when n is large.