课程名称︰统计学与计量经济学暨实习上 课程性质︰系内必修 课程教师︰张胜凯 开课学院：社会科学院 开课系所︰经济学系 考试日期（年月日）︰2016/11/29 考试时限（分钟）：150 试题 : Problem 1. (5,5,5,5) Scores on a recent national statistics exam were normally distributed with a mean of 80 and a standard deviation of 6. a. What is the probability that a randomly selected exam will have a score of at least 71? b. What percentage of exams will have scores between 89 and 92? c. If the top 2.5% of test scores receive merit awards, what is the lowest score eligible for an award? d. If there were 334 exams with scores of at least 89, how many students took the exam? Problem 2. (10,10) a. the pdf of X is f(x;θ) = θx^(θ-1) for 0 <= x <= 1, and f(x;θ) = 0 otherwise. For random sampling with sample size n. Find the maximum likelihood estimator, θ_MLE, of θ. b. Suppose (X,Y) ~ SBVN(0.8). Find P(Y >= 3 | x = 3). Problem 3. (5,5,5,5) Let N follows Poisson(λ) and S_N = Σ(N;i = 1)X_i where X_1,...,X_N | N follows Benoulli(p) are i.i.d random variables. a. Find E(S_N | N = n). b. Find E(S_N). c. Find Var(S_N | N = n). d. Find Var(S_N). (Hint: if N follows Poisson(λ), E(N) = λ and Var(N) = λ) Problem 4. (10,10) a. A small stock brokerage firm wants to determine the average daily sales(in dollars) of stocks to their clients. A sample of the sales for 36 days revealed average daily sales of 200,000. Provide a 97% confidence interval estimate for the average daily sales. b. A quality control technician is checking the weights of a product. She takes a random sample of 8 units and weighs each unit. The observed weights are shown below. Assume the population has a normal distribution. The weight of products are 50, 48, 55, 52, 53, 46, 54 and 50. Provide a 95% confidence interval for the mean weight of the units. Problem 5. (6,7,7) A random sample follows N(μ, σ^2). Define μ_1 = 1/n(1/2*X_1 + 3/2*X_2 + ...+ 1/2*X_n-1 + 3/2*X_n) and μ_2 = 1/n*Σ(n;i = 1)X_i. a. Which one is an unbiased estimater of μ? b. Which one is relatively efficient? c. Which one is a consistent estimator of μ? --

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