Concept explainers
a.
Obtain the form of rejection region for a most powerful test for testing
a.
Explanation of Solution
In this context,
The likelihood
Neyman–Pearson Lemma:
It is assumed to test
In this context, the value of k is chosen so that the test will have the desired value for
The most powerful
Substitute the corresponding values in the above equation to get the most powerful test.
As n,
The inequality in the last step changes because
Here,
Therefore, the rejection region will have the form of
b.
Explain the way in which the given information used to find any constant associated with the rejection region derived in Part (a).
b.
Explanation of Solution
In this context, it is given that
From Part (a), it is clear that the rejection region will have the form of
Now, it is known that
In general, the k* is chosen to be small. Such that
c.
State whether the test derived in Part (a) is uniformly most powerful for testing
c.
Explanation of Solution
From Part (a), it has been found that the rejection region of the powerful test will be in the form of
As the rejection region does not depend on
Therefore, the test derived in Part (a) is uniformly most powerful for testing
d.
Obtain the form of rejection region for a most powerful test for testing
d.
Explanation of Solution
The test hypotheses are given below:
The likelihood function of the random variable Y is given below:
The most powerful
Substitute the corresponding values in the above equation to get the most powerful test.
As n,
The inequality in the last step changes because
Here,
Therefore, the rejection region will have the form of
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