Problem 1. 8 points
Suppose that x1; x2; x3; x4; x5, is a sample drawn from the exponential distribution
f(x; ) :=
(
e x if x 0
0 otherwise;
where is an unknown parameter.
(a) [2 points] Find an expression for the joint likelihood function f(x1; x2; x3; x4; x5; ).
(b) [3 points] Is x a sucient statistic for ? Explain your answer.
(c) [3 points] Now suppose that x1 = 5; x2 = 3; x3 = 4; x4 = 2; x5 = 2. Find the maximum
likelihood estimate of .
Problem 2. 4 points
Let X1; : : : ;X120 be a collection of i.i.d. random variables and let Y1; : : : ; Y120 be the associated
order statistics.
(a) [4 points] Find a value of j such that the interval [Y30..j ; Y30+j ] is an approximate 95%
condence interval for the 1
4 quartile of the distribution.
Problem 3. 6 points
A political candidate hires a polling company to determine their level of support, i.e. the
probability p that a random person will vote for them. The polling company will contact n
people and collect the data x1; : : : ; xn where xi = 1 if person i supports the candidate and
xi = 0 if the person does not support the candidate. Thus, the people contacted by the polling
company can be treated as i.i.d. b(1; p) random variables. Let x be the sample mean of the
data.
(a) [3 points] How many people does the polling company need to contact in order to guar-
antee that (x :02; x + :02) is a 95% condence interval for p? (Use the overestimate
p(1 p) 1
4 to help estimate the variance).
(b) [3 points] The client believes that their true level of support is ^p = :7. How many fewer
people need to be contacted (compared to part (a)) if you instead estimate the variance
by replacing p(1 p) with ^p(1 ^p) = :21 instead?
Problem 4. 7 points
Suppose you have a normal distribution N(; 30) where is unknown. You want to test the
null hypothesis = 4 against the alternative hypothesis < 4. Suppose you create a test
where you collect n data points x1; : : : ; xn and reject the null hypothesis if the sample mean
x < c for some cuto value c.
(a) [3 points] Let K() be the power function of the test at some value 2 R. Give an
expression in terms of n; c; for K() (note your nal expression should either be in terms
of the error function erf OR it should be the probability of an N(0; 1) random variable
belonging to a certain region).
(b) [4 points] Find values for c and n that guarantee that the test has at most :005 type 1
error and at most :01 type 2 error when = 2.
Problem 5. 13 points
A company is testing the ecacy of a new vaccine against a certain virus. Let p0 represent
the probability that a random unvaccinated person will contract the virus over the course of
two months, and let p1 represent the probability that a random fully vaccinated person will
contract the virus over the course of two months. The company will test whether the vaccine
is eective by recruiting 20; 000 volunteers and splitting them into two equal groups of size
n = 10; 000. One group will receive the vaccine while the other group receives a placebo. After
tracking the participants for two months, the company collects the data n0; n1 where n0 is the
number of people in the placebo group who contracted the virus and n1 is the number of people
in the vaccine group who contracted the virus.
(a) [4 points] Suppose that n0 = 189. Find a 95% condence interval for p0 (Note since the
sample size is so large, if p0 shows up in the variance formula you can replace it with n0
n .)
(b) [4 points] Let X
be a random variable representing the sample mean of the placebo group
and let Y be a random variable representing the sample mean of the vaccine group. What
is the expected value and variance of W = Y 1
2
X
?
(c) [1 point] Express the probability that p1 < 1
2p0 in terms of W.
(d) [4 points] Suppose that n1 = 11. Using your answers from parts (b) and (c), how condent
can the company be that p1 < 1
2p0? (Again if p0 or p1 show up in a variance formula you
can replace them with n0
n and n1
n respectively).
Problem 6. 12 points
Suppose X1; : : : ;X60 are i.i.d. b(1; p) random variables where p is unknown. You are given the
null hypothesis H0 : p = 1=4.
(a) [3 points] Suppose you want to test H0 against the simple alternative hypothesis H1 : p =
1=2. Use the likelihood ratio test to nd a critical region C of (approximate) size :005 (i.e.
the probability of a type 1 error is :005).
(b) [4 points] Show that the likelihood ratio test gives you get the same critical region C from
part (a) if the alternative hypothesis is H1 : p = p1 where p1 is any number bigger than 1
4 .
(c) [3 points] What is the type 2 error associated with the region C when the alternative
hypothesis is p = 1
3?
(d) [2 points] Brie
y explain why any other critical region D with size :005 must make a
larger type 2 error than C when the alternative hypothesis is p = 1
3 .
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