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）︰2017.1.4
考試時限（分鐘）：50
試題 :
Introduction to Staatistics (Test 4)
1.(20%) State or define the following terms:
(1a) statistic. (1b) confidence interval. (1c) significance level.
(1d) power function. (1e) probability function.
2.(10%) Let X_1, X_2,..., X_n be a random sample from a normal distribution
with mean μ and variance σ^2. Let φ=Σ_{i=1}^{n}X_i/n and τ^2=Σ_{i=1}^{n}
(X_i-φ)^2/(n-1). What are the distributions of (n-1)(τ/σ)^2 and
√n(φ-μ)/τ.
3.(10%) Let X be a random variable from a normal distribution with mean μ and
known variance σ^2. Find the sample size required to achieve P(|φ-μ|≦E)
=1-α, where φ is the sample mean.
4.(10%) Let W_n be a test statistic such that large values of W_n tend to
reject the null hypothesis. Define a reasonable p-value based on W_n.
5. Let x_1,...,x_n be observed sample values from a normal distribution with
mean μ and variance σ^2, where μ and σ^2 are unknow constants.
(5a)(10%) Construct a (1-α), 0<α<1, confidence interval for μ.
(5b)(10%) Construct a (1-α), 0<α<1, confidence interval for σ^2.
(5c)(10%) Consider the null hypothesis H_0:μ=μ_0 versus the alternative
hypothesis H_A:μ>μ_0. What is the p-value of the corresponding hypothesis?
6. Let (X_11,...,X_1(n_1)) and (X_21,...,X_2(n_2)) be two independent random
samples from N(μ_1,σ^2) and N(μ_2,σ^2), respectively.
(6a)(10%) Construct a (1-α), 0<α<1, confidence interval for μ_1-μ_2.
(6b)(10%) State the testing procedure for the hypothesis H_0:μ_1=μ_2 versus
H_A:μ_1≠μ_2 with the significance level α.