[Solved] SOLVED:EXAM 3

Problems 1. (28 points total, 4 points each) True/False Questions. Circle either True or False. (a) True or False: The statistic XX S/n , where X is the sample mean, X is the population mean, S is the sample standard deviation, and n is the sample size, is Gaussian-distributed. (b) True or False: When data X1,,Xn are Gaussian distributed with variance 2, and a sample size n has sample variance S2, the statistic (n1)S2/2 has the Chi-squared distribution with n 1 degrees of freedom. (c) True or False: If X and Y are independent, then Var [X Y ] = Var [X] Var [Y ]. (d) True or False: The sample mean is always equal to the population mean. (e) True or False: The bias of the sample mean is zero. (f) True or False: The T distribution is symmetric about zero. (g) True or False: A false alarm is deciding H0 when H1 is true.

2. (24 total points) Assume that random variable X has mean and standard deviation , but that X is not Gaussian. We can only collect n = 9 samples. Since n < 30, one may not use the CLT to approximate X as Gaussian. Instead, one may use Chebychevs theorem to nd a condence interval on the population mean of X. If we apply Chebychevs theorem to the random variable X (the sample mean), then P  X k X < X < X + k X 1 1 k2 (1) Use this expression to develop a 95% condence interval on , as follows. (a) (8 points) Write expressions for the mean X and standard deviation X of the sample mean X as functions of and . Answer: (b) (8 points) What k is needed so that the probability that X is in the interval is 0.95? Answer: (c) (8 points) Rearrange the inequality inside of the probability operator in Equation (1) above so that it is a 95% condence interval on , with limits that are a function of only X and . Answer: 3. (30 total points) A communications system manufacturer tests the bit error rate (BER) of receivers they manufacture (under certain test conditions). A sample size of 19 receivers is tested and the sample mean BER is 1.4 102. The BER is approximately Gaussian distributed. (a) (12 points) Test the null hypothesis that the population mean is = 1102, vs. the alternative hypthothesis that is greater than that. They will accept a false alarm rate of 1%, and assume that is known to be 0.7 102. Should H0 be rejected? (b) (12 points) Now assume that is unknown and s = 0.7102. Find a two-sided 95% condence interval for . (c) (6 points) In part (b), would a 99% condence interval would be narrower or wider than a 95% condence interval? Answer narrower or wider:4. (18 total points) Let X be a Gaussian random variable. You collect a sample of size n = 21 and nd the sample mean x = 12.2 and sample standard deviation s = 4.1. (a) (12 points) Find a 90% upper one-sided condence interval on the population standard deviation . (b) (6 points) If n was increased to 31, assuming s remained the same, would the upper limit of the 90% condence interval increase or decrease? Answer increase or decrease: