Problem 4.
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.
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.
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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