Statistics 103: Midterm 2 Test Form A (Dated: May 2, 2016) Instructions: Before you do anything, make sure your name, ID and your test form type is on your scantron. You have until 11:50am to finish this test. You may use one sheet of notes and a calculator. 1. Let X be a normal random variable with E(X) = 10 and var(X) = 36. Find P (4 < X < 16). (a) 0.34 (b) 0.0000317 (c) 0.132 (d) 0.123 (e) 0.68 Answer: 1e 2. The lifetime of a small electric motor can be modeled as a random variable with the following probability distribution table: lifetime probability 1 year 0.6 2 years 0.2 3 years 0.1 0.1 4 years Suppose we randomly select 25 motors and run them to see how long they last. Suppose the average lifetime, X, of these 25 randomly selected motors is 3.5 years. Find the z-score of this observation. (a) 8.9553 (b) 1.7910 (c) 4.5573 (d) 4.9752 (e) 0.0 Answer: 2a 3. Let Z be a standard normal random variable. Find the number a such that P (Z a) = 0.025. (a) 0.33 (b) 1.88 (c) 0.33 (d) 1.96 (e) 1.64 Answer: 3d 4. Suppose X is a continuous random variable with probability density function given by ( 3 2 x , when 1 x < 1 pX (x) = 2 0 otherwise. Find E(X). (a) 0 (c) 1 (e) 3 Answer: 4a (b) 3/2 (d) 3/8 5. Assume that X1 , X2 , . . . , X100 are independent normal random variables with mean and variance 2 . Find the variance of X1 X2 + X3 X4 + X100 . (a) 502 (c) / 100 (e) 2 Answer: 5b 2 (b) 100 (d) 100 6. Suppose X1 , ..., X100 are random samples (with replacement) from some population. Suppose E(X1 ) = 2.2 and sd(X1 ) = 10. Approximate P (X > 3). (a) 0.2119 (b) 0.7881 (c) 0.1587 (d) 0.0013 (e) 0.2327 Answer: 6a 7. Let X1 , . . . , X35 denote random samples with replacement from some population of numbers. Suppose further that E(X1 ) = 10 and var(X1 ) = 40. What is var(X)? (a) 40 (b) 40/ 35 (c) 40/ 35 (d) 40/35 (e) 40/35 Answer: 7e 8. Let X1 , . . . , X100 denote random samples with replacement from a population of numbers. Suppose E(X1 ) = 0 and sd(X1 ) = 1. Approximate the probability that X1 + X2 + + X100 > 20. (a) 0.4207 (b) 0.0456 (c) 0.9772 (d) 0.8414 (e) 0.0228 Answer: 8e 9. Suppose X and Y represent the the future profit (in dollars) of two investments. In addition, suppose X and Y have the same expected value and variance, but cov(X, Y ) is negative. Is it better to double your investment in X (so your profit will be 2X) or diversify between the two invesments (so your profit will be X + Y )? (a) X + Y is better. (b) 2X is better. (c) Neither one is better than the other. Answer: 9a 10. Suppose X N (2, 5). Find P (|X| > 3). (a) 0.421 (b) 0.340 (c) 1.0 (d) 0.579 (e) 0.013 Answer: 10b 2 11. Suppose X and Y are both normal random variables which satisfy E(X) = 0, var(X) = 2 E(Y ) = 1, var(Y ) = 4 cov(X, Y ) = 0.5 Let W = X + 2Y . Find P (W > 0) (Hint: you can assume that W is a normal random variable). (a) 0.0786 (c) 0.6915 (e) 0.5987 Answer: 11c (b) 0.9213 (d) 0.8413 12. Let X be a normal random variable such that E(X) = 22 and sd(X) = 4.5. What is the value of X that corresponds to a Z-score of 1.7. (a) 20.3 (b) 14.35 (c) 29.65 (d) 32.9 (e) 7.65 Answer: 12b