课程名称︰统计学与计量经济学暨实习上 课程性质︰必修 课程教师︰张胜凯 开课学院：社科院 开课系所︰经济系 考试日期（年月日）︰11/20/2015 考试时限（分钟）：180 min. 试题 : Problem 1. (25 points(5,10,10)) A new brand of chocolate bar is being market tested. Four hundred of the new chocolate bars were given to consumers to try. The consumers were asked whether they liked or disliked the chocolate bar. You are given their responses below. Among 400 consumers are tested, 300 of them liked the chocolate bar and 100 of them disliked it. a. What is the point estimate for the proportion of people who liked the chocolate bar? b. Construct a 95% confidence interval for the true proportion of people who liked the chocolate bar. c. With a 95% confidence level, how large of a sample needs to be taken to provide a margin of error of 3% or less? Problem 2. (25 points (8,8,9)) Let X ,X ,X be i.i.d. random variables from a population having mean μ and 1 2 3 2 X1+X2 X2+X3 finite variance σ . Let W =———,W =X , and W =———. 1 2 2 3 3 3 2 a. What are the expected values of W ,W and W in terms of μ and σ ? Are they 1 2 3 unbiased estimators for μ? 2 b. What are the variance of W ,W and W in terms of μ and σ ? 1 2 3 2 2 c. Suppose that you know μ = 2σ in population, which estimator of μ do you prefer, W ,W or W in terms of mean square errors (MSE) criterion? ( Hint: 1 2 3 2 MSE(W)=Var(W)+(Bias(W)) ). Problem 3. (25 points (12,13)) th th a. If the 20 and 60 percentiles of a normal random variable are -3.3 and th 10.8, respectively. What is the value of 95 percentiles of this mormal random variables? b. Below you are given ages that were obtained by taking a random sample of 9 undergraduate students. Assume the population has a normal distribution. 19,22,23,19,21,22,19,23,21 Find the point estimate of mean μ and construct a 98% confidence interval for the average age of undergraduate students. Problem 4. (25 points (15,10)) θ-1 a. The pdf of Y is f(y|θ)=θy for 0≦y≦1, and f(y|θ)= 0 otherwise. For random sampling with sample size n, Y ,Y ,...,Y ,find the maximum likelihood ︿ 1 2 n estimator (MLE),θ , of θ. MLE b. One observation is taken on a discrete random variable X with pmf f(χ|θ) as described below, where θ=｛1,2,3｝. Find the MLE of θ. ────────────── χ f(χ|1) f(χ|2) f(χ|3) ────────────── 0 1/3 0 1/4 1 1/6 1/4 1/4 2 1/3 0 0 3 1/6 3/4 1/2 ────────────── --

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