[試題] 105-2 鄭明燕 統計學導論 期末考
課程名稱︰統計學導論
課程性質︰數學系選修
課程教師︰鄭明燕
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/6/22
考試時限（分鐘）：110
試題 :
1.(15 pts.) Let X ,..., X be a sample from an Exponential(λ) distribution.
1 n
(a) Find the likelihood ratio test for testing H : λ = λ versus
0 0
H : λ = λ , where λ > λ . Explain how to determine the critical
A 1 1 0
point at significance level α.
(b) Show that the test in part (a) is uniformly most powerful for testing
H : λ = λ versus H : λ > λ .
0 0 A 0
2.(15 pts.) Let X ~ Binomial(n , p ), i = 1,..., m, be independent random
i i i
variables.
(a) Derive a likelihood ratio test statistic for the hypothesis
H : p = p = ... = p against the alternative that the p are not
0 1 2 m i
all equal.
(b) What is the large-sample distribution of the test statistic in part(a)?
3.(15 pts.) Let X ,..., X be a sample from a cdf F, and let F denote
1 n n
the ecdf.
(a) Determine the distribution of F (x) and find E[F (x)], Var[F (x)].
n n n
(b) Find Cov[F (u),F (v)].
n n
4.(15 pts.) Suppose a nonnegative random variable X has hazard funciton h(t).
(a) Find the hazard function of aX where a is a positive constant.
(b) Find the hazard function h(t) when X is uniform(0,1).
1
(c) Find the density function of X when h(t) = -----, t > 0.
1+t
5.(15 pts.) Let X ,..., X be a sample from an Exponential(λ ) distribution
1 n 1
and Y ,..., Y be an indepedent sample from an Exponential(λ ) distribution.
1 n
(a) Determine the expected rank sum of the X's.
(b) Determine the variance of the rank sum of the X's.
2
6.(15 pts.) Let X ,..., X be i.i.d. N(μ , σ ), and let Y ,..., Y be i.i.d.
1 n X X 1 m
2
N(μ , σ ), where the two samples are independent.
Y Y
2 2 2 2
(a) Find the distribution of s /s , where s and s are sample variance.
X Y X Y
2 2
(b) Construct a level 100(1-α)% confidence interval for the ratio σ /σ .
X Y
(c) Test the hypothesis H : σ = σ versus H : σ ≠ σ
0 X Y 1 X Y
at significance level α.
7.(30 pts.) Consider the one-way analysis of variance model Y = μ+α +ε ,
ij i ij
i = 1,..., I, j = 1,..., J, where Y is the jth observation for the ith
ij
treatment level, μ is the overall mean, α is the effect of the ith
i
I 2
treatment with Σ α = 0, and ε 's are independent N(0, σ ) errors.
i=1 i ij
(a) Find the mle's of the parameters μ and α , i = 1,..., J.
i
(b) Find the expectation of the sum of squares within and between groups
respectively.
(c) Write down the One-Way Analysis of Variance table.
(d) Test the null hypothesis H : α = α = ... = α = 0 at significant
0 1 2 I
level α.
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