Page 453 - Probability and Statistical Inference

430 8. Tests of Hypotheses

Find Type I and Type II error probabilities for each test and compare the
tests.
8.2.3 Suppose that X , X , X , X are iid random variables from the N(θ, 4)
3
2
1
4
population where θ(∈ ℜ) is the unknown parameter. We wish to test H : θ =
0
3 versus H : θ = 1. Consider the following tests:
1

Find Type I and Type II error probabilities for each test and compare the
tests.
8.2.4 Suppose that X , X , X , X are iid random variables from a popula-
3
2
1
4
+
tion with the exponential distribution having unknown mean θ(∈ ℜ ). We
wish to test H : θ = 6 versus H : θ = 2. Consider the following possible
0
1
tests:

Find Type I and Type II error probabilities for each test and compare the
tests.
8.2.5 Suppose that X , X are iid with the common pdf
1 2

where θ(> 0) is the unknown parameter. In order to test the null hypothesis
H : θ = 1 against the alternative hypothesis H : θ = 2, we propose the critical
0 1
region
(i) Show that the level α = ¼ + ¾log(¾);
(ii) Show that power at θ = 2 is 7/16 + 9/8log(¾).
{Hints: Observe that which is written

as Similarly, power