Suppose you are interested in the probability that a person with allergies responds to a particular medicine. You want to test the null hypothesis that this probability is 0.6. Find a level 0.1 likelihood ratio test, assuming you test n = 8 people with allergies, and the number Y who respond to the medicine is a Binomial random variable. Do this in two different ways:

(a) First, compute the exact rejection region. This has the form {? < k}, where ? is the likelihood ratio statistic. List all values of k that give a level 0.1 test. Also give the rejection region in terms of the number Y = y of observed successes.

(b) Now assume that the number of observations is large enough that you can use the ?2- approximation. Compute the rejection region in terms of both the test statistic, and Y . How close is this to the exact value you calculated above?

(c) Suppose you observe y = 1. What is the p-value, in each of the two tests? Com- ment on how the approximate value compares to the true value.

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