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八十九學年度第二學期 統計學(二) Mid-term Exam (C 卷) 17/05/04 一、Multiple Choice and True/False (9%, 3% each): 1. If the mean and variance for the population are 900 and 9 respectively, what are the mean and variance of the sample mean if the sample size is 16? a. 900, 144 b. 900, 0.5625 c. 900, 1296 d. 900, 9 2. A random sample of size 100 is drawn from a binomial distribution with mean 32 and variance 16. What is the distribution of ( X 32) /( 4 / 10) ? a. b. c. d. Normal with mean 32 and variance 1.6 Normal with mean 0 and variance 1 Binomial with n = 64 and p = 0.5 None of the above 3. X is distributed with = 10 and 2 = 4. Y is distributed with = 50 and 2 = 100. Which one has higher coefficient of variation? a. X b. Y c. the same d. couldn’t tell 二、Short Answer: (21%) 1. (6%) 若某路口每月(以 30 天計)發生車禍之次數為波以松分布,平均次數為 6 次,則該 路口發生車禍之間隔時間為________分布,平均間隔時間為_________天。 2. (3%) List the formula for Chebyshev’s Inequality: 3. (3%) If sample mean = 100, sample standard deviation = 10, and sample size = 16, what is the stand error of the sample mean?_____________________________________________ 4. (6%) 某電子公司生產一產品,該產品之電阻值呈常態分布( = 100, = 10) a. 試問該產品一批(10,000 個),約有多少個超過 120?__________________ b. 隨機抽樣 25 個,其平均電阻值小於 96之機率為多少?_______________ 5. (3%) A random sample of 36 observations has been drawn from a normal distribution with mean 50 and standard deviation 12. The probability that the sample mean is in the interval [47, 53] is __________. 1 班級: 學號: 姓名: 三、Open Question: (70%) 1. (15%) Assume that one has 2 independent random samples of sizes n1, n2 from the same n X n2 X 2 X X2 population. Show that wt 1 1 is a better estimator for than 1 . n1 n2 2 2. (15%) 請說明: a. 中央極限定理; b. 大數法則; c. 常態分布之標準化。 3. (10%) Let X be a random variable with the following probability distribution: e x f x , x 1,2,.... x! Find the maximum likelihood estimator of , based on a random sample of size n. 2 八十九學年度第二學期 統計學(二) Mid-term Exam (C 卷) 17/05/04 4. (15%) The following data are the temperatures of effluent at discharge from a sewage treatment facility on consecutive days: 43 47 51 48 52 50 46 49 45 52 46 51 44 49 46 51 49 45 44 50 48 50 49 50 a. Calculate the sample mean and sample variance. b. Construct a stem-and-leaf plot of the data. c. Construct a box plot of the data and comment on the information in this display. X n 5. (15%) Show that i 1 i X 2 n is a biased estimator of 2. this estimator. 3 In addition, what is the bias of