課程名稱︰統計導論
課程性質︰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）︰2010/1/11
考試時限（分鐘）：120min
是否需發放獎勵金：Yes
（如未明確表示，則不予發放）
試題 :
1.(10%) Let x_1,...,x_n be realizations of a random sample from N(μ,σ^2),
where μ and σ^2 are unknown. Consider the hypotheses H_o:μ≧μ_0 versus
H_A:μ≦μ_o. Compute the power at μ_1, where μ_1≦μ_o, for the test rule
╴
base on the sample mean X_n with the significance level α.
2.(10%) Let X_1,...,X_n be a random sample from a population with median M.
Suppose that the sample size n is small, state the testing procedure for the
hypotheses H_o: M = M_o versus H_A: M ≠ M_o with thw significance level α.
3.(7%) Let X_1,...,X_n be a random sample from a symmetric probability
density function f(x). Write the test statistic of the Wilcoxon sign-rank
rest for H_o : f(x) is symmetric about μ_o versus H_A : f(x) is symmetric
about another value.
4.(10%) Let X_1～N(μ_1, σ^2_1), X_2～N(μ_2, σ^2_2), and (X_11,X_21),...,
(X_1n,X_2n) be matched pair random sample of size n. Construct a (1-α),
0 < α < 1, confidece interval for μ_1-μ_2.
5.(8%) Let X_11,...,X_1n_1 and X_21,...,X_2n_2 be independent random samples
from population 1 and population 2, respectively. Let F_1(x) and F_2(x) be
the corresponding distributions. Write the test statistic f the Mann-Whitney
-Willcoxon test for H_o : F_1(x) = F_2(x) versus H_A : F_1(x) ≠ F_2(x) for
some x.
6. Let (X_11,...,X_1n_1),...,(X_k1,...,X_kn_k), k > 2, be indipendent random
samples from populations 1,...,k, respectively. Let F_1(x),...,F_k(x) be
the corresponding distributions.
(6a) (8%) Write the test statistic of th Kruskall_Wallis teat for H_o :
F_1(x)＝...＝F_k(x) versus H_A : F_i(x)≠F_j(x) for some x and i≠j.
(6b) (7%) What is the approxmated distribution of the test statistic as
k
n_T = Σ n_i is large enough?
i=1
(6c) (10%) Let x_11,...,x_1n_1,...,x_k1,...,x_kn_k be realivations of k,
k > 2, independent random samples from N(μ_1,σ^2_1),...,N(μ_k,σ^2_k),
respectively. Compute the corresponding p-value of the proposed test rule
for the hypotheses H_o: μ_1 =...=μ_k versus H_A :μ_i≠μ_j for some i≠j
with the significance level α.
7.(20%) State or define the following terms:
(7a) Bayes rule. (7b) probabiility function. (7c) random variable.
(7d) case-control design. (7e) odds ratio.
8.(10%) Give n example to illustrate that both of the random variables X and Y
are uncorrelated byt dependent.
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