課程名稱︰統計學∕數理統計
課程性質︰選修 (非數學系需有統計基礎)
課程教師︰鄭少為 教授
開課系所︰數學系
考試時間︰2005/06/21 14:20-17:20
試題：
(考試時可以攜帶計算機或電子字典及一張A4大小之小抄 (雙面)
考試為close book, close note, close homework solutions)
1.(10pts, 2pts for each) For the following statements, please answer true
or false. If false, please explain why.
(a) Suppose that a 99% confidence interval for the mean μ of a normal
distribution is found to be (-2.0, 3.0). The probability that
(-2.0, 3.0) contains μ equals 0.99.
(b) Suppose that a 99% confidence interval for the mean μ of a normal
distribution is found to be (-2.0, 3.0). A corresponding test of
H_0: μ＝-3 versus H_A: μ≠-3 would be accepted at the 0.01
significance level.
(c) The p-value of a test is a random variable because it is a function
of the test statistic.
(d) For non-parametric methods, the dimension of parameters in the
statistical model is infinite.
(e) The probability that the null hypothesis is true equals significance
level α.
2. Let X_1,...,X_n be i.i.d. random variables from the Gamma distribution,
Γ(α,λ), with α known, and λ屬於Ω＝(0,∞) unknown. The pdf of
Γ(α,λ) is
λ^α
───‧x^(α-1)‧e^(-λx), where x≧0.
Γ(α)
1 n
(a)(15pts) Show that the UMVUE of θ＝1/λ is ── Σ X_i.
nα i=1
n
[Hint. (i) Let Y_1,...,Y_n be i.i.d. Γ(α,λ), then Σ Y_i～Γ(nα,λ)
i=1
(ii) the mean of Γ(α,λ) is α/λ]
(b)(5pts) Find a function of the UMVUE that is a pivotal quantity.
[Hint. Let Y～Γ(α,λ), then aY～Γ(α,λ/a) for a constant a]
(c)(10pts) Use the pivotal quantity from part (b) to find a 100(1-β)%
confidence interval for θ.
3. Let X_1,...,X_n i.i.d. from a Poisson distribution P(λ), where pdf is
e^(-λ)‧λ^x
───────, where x＝0,1,2,..., and λ＞0.
x!
(a)(15pts) Find the likelihood ratio for testing H_0: λ＝λ_0 v.s. H_A:
λ＝λ_1, where λ_1＞λ_0. Use the fact that the sum of independent
Poisson random variables follows a Poisson distribution (ie.
n
Σ X_i～P(nλ)) to explain how to determine a rejection region for
i=1
a test at level α.
(b)(5 pts) Show that the test of (a) is uniformly most powerful for
testing H_0: λ＝λ_0 versus H_A: λ＞λ_0.
4.(20 pts) If gene frequencies are in equilibrium, the genotypes AA, Aa,
and aa occur with probabilities p_1(θ)＝(1-θ)^2, p_2(θ)＝2θ(1-θ),
p_3(θ)＝θ^2, respectively. Plato et al. (1964) published the following
data on haptoglobin type in a sample of n＝190 people:
┌──────────────┐
│ Haptoglobin Type │
├────┬────┬────┤
│ AA │ Aa │ aa │
├────┼────┼────┤
│X_1＝ 10│X_2＝ 68│X_3＝112│
└────┴────┴────┘
Derive and carry out a generalized likelihood ratio test, based on Chi-
square approximation, of the hypothesis H_0: θ＝1/2 versus H_A: θ≠1/2
for significance level 0.05.
n! x_1 x_k
[Hint. (i) The pdf of multinomial(n,p_1,...,p_k) is ────── p_1...p_k
︿ x_1!...x_k!
(ii) The MLE of θ is θ＝(2X_3＋X_2)/2n≒0.77
╭ p_1(0.5) ╮ ╭ p_2(0.5) ╮
(iii) log│─────│＝1.55, log│─────│＝0.34,
╰ p_1(0.77)╯ ╰ p_2(0.77)╯
╭ p_3(0.5) ╮
log│─────│＝-0.86
╰ p_3(0.77)╯
2 2 2
(iv) χ_1(0.05)＝3.841, χ_2(0.05)＝5.991, χ_3(0.05)＝7.814 i.e.,
2 2 2
P(χ_1＞3.841)＝P(χ_2＞5.991)＝P(χ_3＞7.814)＝0.05]
5.(10pts) Suppose that n measurements are to be taken under a treatment
condition and another n measurements are to be taken independently under
a contal condiction. It is thought that the standard deviation of a
single observation is about 10 under both conditions. How large should n
be so that the test of H_0: μ_X＝μ_Y versus the one-sided alternative
H_A: μ_X＞μ_Y has a power of 0.9 if μ_X－μ_Y＝2 and α＝0.1? Use the
normal distribution rather than the t distribution since n will turn out
to be rather large. _ _ ___
[Hint. (i) The rejection region for the test is X－Y＞z(α)σ√2/n
_ _ ___
(ii) power＝P(X－Y＞z(α)σ√2/n | μ_X－μ_Y＝△)
(iii) z(0.1)＝1.282, i.e., P(N(0,1)＞1.282)＝0.1]
6. An oil company wishes to compare the effect of 4 different gasoline
additives on average milesge. Twenty four cars of the same make and model
in approximately the same condition were selected. Six cars were randomly
assigned to each of the 4 additive groups, called additive A_1, A_2, A_3,
A_4.
(a)(2pts) Let Y_ij be jth observation of the ith group, j＝1,...,6, and
i＝1,...,4. Assume that Y_ij follow the model
Y_ij＝μ＋α_i＋ε_ij,
4
where Σ α_i＝0 and ε_ij's are i.i.d. N(0,σ^2). Explain the meanings
i=1
of parameters μ and α_i's in the model.
(b)(6pts, 1pts for each cell) To obtain the F-statistic for testing H_0:
α_1＝...＝α_4＝0, complete the following ANOVA table
┌─────┬────┬─────┬────┬────┐
│Source of │Sum of │Degree of │Mean │ F │
│ Variation│ Squares│ Freedom │ Squares│ │
├─────┼────┼─────┼────┼────┤
│ Between │ 210.34 │ │ │ │
├─────┼────┼─────┼────┼────┘
│ Within │ │ │ │
├─────┼────┼─────┴────┘
│ Total │ 620.48 │
└─────┴────┘
(c)(1pt) What is the distribution of the F statistic under the assumption
that the null hypothesis is true?
(d)(1pt) The p-value of the F-test is 0.037. For significance level 0.05,
state your conclusion on the mean comparsion of the 4 different
gasoline additives.
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