Description
1. The number of successes in n independent trails is to be used to test the null hypothesis
that the parameter θ of a binomial population equals 1/2 against the alternative that it
does not equal 1/2.
(i) Find an expression for the likelihood ratio statistic;
(ii) Use the result of part (i) to show that the critical region of the likelihood ratio test
can be writeen as
x ln x + (n − x) ln(n − x) ≥ K
where x is the observed number of successes;
(iii) Study the graph of f(x) = x ln x + (n − x) ln(n − x), in particular its minimum and
its symmetry, to show that the critical region of this likelihood ratio test can also be
written as
|x − n/2| ≥ K.
2. Given a random sample of size n from a normal population with unknown mean and variance, find an expression for the likelihood ratio statistic for testing H0 : σ = σ0 against
H1 : σ 6= σ0.
3. Consider a random sample of size 4 from the uniform distribution U(0, θ). Let Y1 < Y2 <
Y3 < Y4 be the order statistics. Let the observed value of Y4 be y4. We reject H0 : θ = 1
and accept H1 : θ 6= 1 if either y4 ≤ 1/2 or y4 > 1. Find the power function Q(θ), θ > 0, of
the test.
4. Given a random sample of size n from a normal population with µ = 0, use the NeymanPearson theorem to construct the most powerful critical region of size α to test the null
hypothesis σ = σ0 against the alternative σ = σ1, where σ1 > σ0.
5. Let X1, · · · , X20 be a random sample of size 20 from a Poisson distribution with mean θ.
Show that the critical region defined by P20
i=1 Xi ≥ 5 is a uniformly most powerful critical
region for testing H0 : θ = 0.1 against H1 : θ > 0.1. What is α, the significance level of
the test? Find the power function.
