Page 430 - Probability and Statistical Inference

8. Tests of Hypotheses 407

problems, we may discover that we reject H when an appropriate test statis-
0
tic T falls under some number k. This was the situation in the Example 8.3.2
where we had and k = -z . Here, the alternative hypoth-
α
esis was on the lower side (of µ ) and the rejection region R (for H ) fell on
0
0
the lower side too.
In general, the cut-off point k has to be determined from the distribution
g(t), that is the pmf or pdf of the test statistic T under H . The pmf or pdf of
0
T under H specifies what is called a null distribution. We have summarized
0
the upper- and lower-sided critical regions in the Figure 8.3.3.
Example 8.3.3 Suppose that X , ..., X are iid with the common pdf b -1
1
n
+
+
exp(-x/b) for x ∈ ℜ with unknown b ∈ ℜ . With preassigned α ∈ (0, 1), we
wish to obtain the MP level a test for H : b = b versus H : b = b (> b ) where
0
1
1
0
0
b , b are two positive numbers. Both H , H are simple hypothesis and the
0 1 0 1
Neyman-Pearson Lemma applies. The likelihood function is given by
The MP test will have the following form:
that is, we will reject the null hypothesis H if and only if
0

Now since b > b , the condition in (8.3.10) can be rephrased as:
1 0

Figure 8.3.4. Chi-Square Upper 100a% Point with
Degrees of Freedom 2n

But, the MP test described by (8.3.11) is not yet in the implementable
form. Under H , observe that are iid standard exponential random
0