課程名稱︰統計導論 課程性質︰數學系選修 課程教師︰江金倉 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2017.1.9 考試時限（分鐘）：110 試題 : Introduction to Statistics (Final Exam) 1.(20%) Explain of define the following terms: (1a) central limit theorem. (1b) random variable. (1c) resistance. (1d) standard error. (1e) sampling distribution. 2.(10%) Suppose that P(D)～0 in a case-control study. Show that the relative risk of E versus E' can be approximated by the odds ratio of E versus E'. 3.(10%) What is the total number of possible arrangements of r un-ordered objects drawing with replacement from n subjects? 4.(10%) Give an example to illustrate that both of the random variables X and Y are uncorrelated but dependent. 5.(10%) Let X_1,...,X_n be a random sample from a normal distribution with mean μ and variance σ_0^2, where σ_0^2 is an unknown constant. Consider the null hypothesis H_0:μ≧μ_0 versus the alternative hypothesis H_A:μ<μ_0. Compute the power at μ_1 with μ_1<μ_0. 6. Let X_1,...,X_n be a random sample from Bernoulli(π), 0<π<1. (6a)(10%) Supose that the sample size is large enough. Construct an approximated (1-α), 0<α<1, confidence interval for π. (6b)(10%) Find the smallest sample size to achieve P(|φ-π|≦e)～1-α, where φ is the sample mean. 7. Let X_11,...,X_1(n_1),...,X_k1,...,X_k(n_k) be k, k>2, independent random samples from N(μ_1,σ^2),...,N(μ_k,σ^2), respectively. (7a)(5%) Write an unbiased estimator of σ^2. (7b)(5%) Compute the corresponding residuals of X_ij's. (7c)(5%)(5%) Diagnose the assumptions of constant variance and normal distribution. --

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