Math 4620 Spring 2009 Homework 4 Jason Stover
These problems were copied from J. Rice’s book, Mathematical Statistics and Data Anal-
ysis, second edition, chapter 9.
1. A coin is thrown independently 10 times to test the hypothesis that the probability
of heads is 1/2 versus the alternative that the probability is not 1/2. The test
rejects if either 0 or 10 heads are observed.
(a) What is the significance level of this test?
(b) If in fact the probability of heads is 0.1, what is the power of the test?
2. Let Xhave one of the following distributions:
X H0HA
x10.2 0.1
x20.3 0.4
x30.3 0.1
x40.2 0.4
(a) Compare the likelihood ratio, Λ, for each possible value Xand sort the xi
according to Λ.
(b) What is the likelihood ratio test of H0versus HAat level α= 0.2? What is
the test at level α= 0.5?
3. Let X1, ..., Xnbe a sample from a Poisson distribution. Find the likelihood ratio
for testing H0:λ=λ0versus HA:λ=λ0, where λ1> λ0. Use the fact that
the sum of independent Poisson random variables follows a Poisson distribution to
explain how to determine a rejection region for a test at level α.
4. Show that the test of the previous problem is uniformly most powerful for testing
H0:λ=λ0versus HA:λ > λ0.
5. Suppose that X1, ..., Xnform a random sample from a density function, f(x|θ),
for which Tis a sufficient statistic for θ. Show that the likelihood ratio test of
H0:θ=θ0versus HA:θ=θ1is a function of T. Explain how, if the distribution
of Tis known under H0, the rejection region of the test may be chosen so that the
test has the level α.