STAT 234 Final, Fall 2005

STAT 234 Final Exam Feb 2, 2006 Time = 1 hour 1. Fish are caught in a lake until m fish of a certain type have been obtained. Let X denote the total number of fish caught at this point. Assume that the lake is so large so that the extraction of these fish does not change the proportion of this type of fish in the lake. (a) Find the probabilty density function of the random variable X. (b) Show that n−1 X−1 is an unbiased estimator for p. Note that this estimator differs from the intuitive estimator n X of p. which is biased for this problem. 2. Let X have the pdf f(x; θ) = θ−1 exp(−x/θ) if x > 0 and θ > 0. (a) Compute E(1/√x) as a function of the parameter θ. (b) Find the maximum likelihood estimator of E(1/√x) based on a random sample X = (x1, · · · , xn)′ 3. Let X = (x1, · · · , xm)′ be a random sample from the pdf f(x; µ) = 1 4 √ 2π exp[− 1 32(x − µ)2] (a) If n = 25 and ΣX = 500, find a 90% confidence interval for µ. (b) How large a sample would be needed if the resulting confidence interval were to be half as large? 4. Let X = (x1, · · · , xn)′ be a random sample from a normal population with an unknown mean µ and standard deviation σ = 6.079. We would like to test the hypothesis H0 : µ = 10 vs Ha : µ < 10. The significance level α has been set at 5% and the critical region was: reject H0 if ¯X < 9. Find the value of n , the sample size needed to produce the above rejection region. 1