Suppose Y1,Y2, .. ,Y8 are independent and identically distributed as Poisson random variables with mean lambda....

Suppose Y1,Y2, .. ,Y8 are independent and identically distributed as Poisson random variables with mean lambda.

a) Derive the most powerful test for testing Ho: lambda = 2, Ha: lambda = 3. Carefully show all work involved in the derivation.

i) Give the form of the test. (In other words, for what general values of Y1,Y2, .. ,Y8 will Ho be rejected?)

ii) Describe the rejection region as carefully as possible if alpha <= .05 (and is as close as possible to .05). (Here you will need to supply specific numbers. You will also need to use the fact that there are 8 observations.)

iii) Using the rejection region found in ii) above, find beta, the probability of a Type II error, for the specific alternative Ha: lambda = 3.

iv) Is there another test with the same alpha (probability of Type I error) as the test described in ii) that will have a smaller beta than was found in iii)? Why or why not?

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