Stat GR5204: Practice Final Name (Print):
Fall 2022 Student UNI:
Time Limit: 2 hours 45 minutes Signature:
This exam contains 7 problems. Answer all of them. Point values are in parentheses. You
must show your work to get credit for your solutions — correct answers without work
will not be awarded points.
1 10 pts
2 20 pts
3 15 pts
4 15 pts
5 10 pts
6 15 pts
7 15 pts
TOTAL 100 pts
Stat GR5204 Fall 2022 Practice Final - Page 2 of 12 December 05, 2022
1. (10 points) (5 + 5) Suppose that we have a random sample X1, . . . , Xn from a N(0, σ
2)
population.
(i) Find the form of the most powerful test of H0 : σ
2 = σ20 versus H1 : σ
2 = σ21, where
0 < σ0 < σ1.
(ii) For a given value of α, the size of the Type I error, find the explicit form of the
critical value of the above test.
Stat GR5204 Fall 2022 Practice Final - Page 3 of 12 December 05, 2022
2. (20 points) (7 + 3 + 5 + 5) Suppose that we observe data (X,Z) where Z has a Bernoulli
distribution with parameter 1/2 and X|Z has distribution
X|Z ∼
{
N(θ, 100) if Z = 1
N(θ, 1) if Z = 0.
where θ ∈ R is the unknown mean of the two normal distributions.
(a) Construct level 1 − α confidence interval for θ using both (X,Z) (the confidence
interval should depend on Z) [Hint: Consider Z = 0 and Z = 1 separately].
(b) Find the marginal distribution of X.
Stat GR5204 Fall 2022 Practice Final - Page 4 of 12 December 05, 2022
(c) Use only X to test the hypothesis (at level α)
H0 : θ = 0 versus H1 : θ > 0.
Show explicitly how the critical value of the test should be determined. [Note that
the test should NOT depend on Z now.]
(d) Express the power function of this test (as a function of θ) in terms of Φ(·) (the
standard normal c.d.f.).
Stat GR5204 Fall 2022 Practice Final - Page 5 of 12 December 05, 2022
3. (15 points) (5 + 7 + 3) Observations (Xi, Yi), i = 1, . . . , n, are made from a bivariate
normal population with parameters (µX , µY , σ
2
X , σ
2
Y , ρ), and the model
Y = α + βX + ,
is going to be fit to the observed data.
(i) Argue that the hypothesis H0 : β = 0 is true if and only if the hypothesis H0 : ρ = 0
is true.
(ii) Show algebraically that
βˆ
σ˜/
√
Sx
=
√
n− 2 r√
1− r2 ,
where
r =
∑n
i=1(Xi − X¯n)Yi√∑n
i=1(Xi − X¯n)2
∑n
i=1(Yi − Y¯n)2
is the sample correlation coefficient (the MLE of ρ), σ˜2 = 1
n−2
∑n
i=1(Yi− αˆ− βˆXi)2
and S2X =
∑n
i=1(Xi − X¯n)2.
Stat GR5204 Fall 2022 Practice Final - Page 6 of 12 December 05, 2022
(iii) Show how to test H0 : ρ = 0, given only r
2 and n, using t distribution with n − 2
degrees of freedom.
Stat GR5204 Fall 2022 Practice Final - Page 7 of 12 December 05, 2022
4. (15 points) (5 + 5 + 5) Suppose that X1, . . . , Xn are i.i.d Exp(1/µ), where E(X1) =
µ > 0.
(i) Find the mean and variance of X¯n =
∑n
i=1Xi/n. Hence, find the asymptotic
distribution of X¯n (properly standardized).
(ii) Let T = log X¯n. Find the corresponding asymptotic distribution of T (properly
standardized).
(iii) How can the asymptotic distribution of T be used to construct an approximate
(1− α) confidence interval (CI) for µ? Explain your answer and give desired CI.
Stat GR5204 Fall 2022 Practice Final - Page 8 of 12 December 05, 2022
5. (10 points) Let X1, . . . , Xn be a random sample from an exponential distribution with
the density function
f(x|θ) = θ exp(−θx), for x > 0,
and 0 otherwise. Derive the likelihood ratio test of
H0 : θ = θ0 versus θ 6= θ0,
and show that the rejection region is of the form {X¯n exp(−θ0X¯n) ≤ c}, where X¯n is the
sample mean and c is a constant. Show all your steps.
Stat GR5204 Fall 2022 Practice Final - Page 9 of 12 December 05, 2022
6. (15 points) (3 + 4 + 8) Suppose that
Yij ∼ N(θi, σ2), i = 1, . . . , k; j = 1, . . . , ni,
where θi, for i = 1, . . . , k, and σ
2 are unknown.
(a) Find the distribution of Y¯i· = 1ni
∑ni
i=1 Yij.
(b) Suppose that the goal is to estimate τ =
∑k
i=1 aiθi, where ai, for i = 1, . . . , k, are
known constants. Note that τˆ =
∑k
i=1 aiY¯i· is a natural estimator of τ . Find the
distribution of τˆ .
Stat GR5204 Fall 2022 Practice Final - Page 10 of 12 December 05, 2022
(c) Let us consider testing for H0 : τ = 0 versus H1 : τ 6= 0. Describe a test procedure
— the test statistic, its exact distribution (under H0) and the critical value.
Stat GR5204 Fall 2022 Practice Final - Page 11 of 12 December 05, 2022
7. (15 points) (3 + 3 + 4 + 5) The multinomial distribution is a generalization of the
binomial distribution. It models the probability of counts for each side of a k-sided die
rolled n times.
Let k be a fixed known number. We have k possible mutually exclusive outcomes, with
corresponding probabilities p1, . . . , pk, and n independent trials (n is known). Since the
k outcomes are mutually exclusive and one must occur we have pi ≥ 0, for i = 1, . . . , k,
and
∑k
i=1 pi = 1. Then if the random variable Xi indicates the number of times outcome
number i is observed over the n trials, the vector X = (X1, . . . , Xk) follows a multinomial
distribution with parameters n and p, where p = (p1, . . . , pk). While the trials are
independent, their outcomes X are dependent because they must be summed to n.
(a) Find the probability mass function of X.
(b) What is the distribution of X1?
Stat GR5204 Fall 2022 Practice Final - Page 12 of 12 December 05, 2022
(c) Find and expression for Cov(Xi, Xj) (in terms of n, pi and pj).
(d) Find the maximum likelihood estimator of p = (p1, . . . , pk).