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.