Homework 8 _SOLUTION

1. Assume X and Y are independent random variables with means E [X] = 1, E [Y ] = 5, and second moments EX2 = 5, EY 2 = 41. Find the mean and variance of each of the following linear combinations:(a) 2X + 20 (b) X + Y (c) X − Y (d) 10X + 3Y (e) Find the mean only of: 10X + 3Y 2 2. Walpole 4.57 3. Walpole 4.76 4. From Tolga Tasdizen: For each of the scenarios described below, answer whether the central limit theorem can be used reliably to compute the probability that is asked. If the answer is yes, use Table A.3 to determine the numerical value of the probability that is asked. (a) A manufacturing process for resistors has unknown population distribution f(x) but we know that the mean µ = 100 Ohms and the standard deviation σ = 5 Ohms. If a random sample of 10 resistors are picked, what is the probability that the sample mean will be larger than 105 Ohms? (b) A manufacturing process for resistors is known to have an approximately normal distribution f(x) with mean µ = 200 Ohms and standard deviation σ = 12 Ohms. If a random sample of 9 are resistors is picked, what is the probability that the sample mean will be between 190 and 210 Ohms? (c) Let X be the number of computers sold at the U. bookstore on any given day. X has the following population distribution: f(x) =            3/8, x = 0 3/8, x = 1 1/8, x = 2 1/8, x = 3 0, o.w. If a random sample of 100 days are recorded, what is the probability that the sample mean will be greater than 1.2? 5. Let X be the lifetime of a product. (a) First, assume that the standard deviation of X is 2 years. What sample size n is needed so that the standard deviation of the sample mean is (i) 6 months or (ii) 1 month?   (b) Next, assume that the standard deviation of X is not known (and not necessarily 2 years). However, past work by the manufacturer shows that when n = 20, the standard deviation of the sample mean is 4.0 months. Given this information, how large of an n is needed to acheive a standard deviation of the sample mean of 1 month? (c) In general, for a given standard deviation of X, what happens to the standard deviation of the sample mean when n is (i) doubled; (ii) multiplied by 4; and (iii) multiplied by 100? 6. Walpole 8.22