Page 420 - Probability and Statistical Inference

8. Tests of Hypotheses 397

ones under given circumstances.
Once and for all, let us add that we freely interchange statements, for
example, “Reject H ” and “Accept H ” for i ≠ j ∈ {0, 1}.
i j
Next, let us explain what we mean by a test of H versus H . Having
1
0
observed the data X = (X , ..., X ), a test will guide us unambiguously to reject
n
1
H or accept H . This is accomplished by partitioning the sample space ℜ n
0
0
c
into two parts R and R corresponding to the respective final decision: “reject
H ” and “accept H ”
0 0
R is constructed so that we reject H whenever X ∈ R. The subset
0
R is called the rejection region or the critical region.
Example 8.2.1 Let us walk through a simple example. Consider a popula-
tion with the pdf N(θ, 1) where θ ∈ℜ is unknown. An experimenter postu-
lates two possible hypotheses H : θ = 5.5 and H : θ = 8. A random sample X
1
0
= (X , ..., X ) is collected and denote . Some examples of
1 9
“tests” are given below:

We may summarize these tests in a different fashion: Let us rewrite

Here, R is that part of the sample space ℜ where H is rejected by means of
9
i 0
the Test #i, i = 1, ..., 4. !
Whenever H , H are both simple hypotheses, we respectively write a and
0
1
ß for the Type I and II error probabilities: