In any case, we reject
H
0
if
S
2
is sufficiently less than or sufficiently
greater than
X
̄
.
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∙
Without specifying the population distribution, we can only use
asymptotic analysis to derive a general test statistic in this example – so
we hold off for now.
∙
Another way to generate a composite null (and alternative) is
H
0
:
2
≤
H
1
:
2
so the null is that either the Poisson variance holds or the variance is
underdispersed (relative to the Poisson distribution). The alternative is
overdispersion.
41

Summary of Classical Hypothesis Testing
1. Choose the null and alternative.
2. Choose the size of the test (maximum probability of Type I error)
you are willing to tolerate.
3. Choose a test statistic (often based on whether its distribution can be
calculated, at least under the null hypothesis).
4. Choose a rejection rule based on the null and size of the test.
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