solution

Suppose that a level a test based on a test statistic T rejects if T> to. Suppose that g is a monotone increasing function and let S = g(T). Is the test that rejects if S> g(to) a level a test?

Suppose the null hypothesis is true, that the distribution of the test statistic, T say, is continuous with cdfF and that the test rejects for large values of T. Let V denote the p-value of the test.

a Show that V = 1 — F(T).

b Conclude that the null distribution of V is uniform. (Hint: See Proposition C of Section 2.3.)

c If the null hypothesis is true, what is the probability that the p-value is greater than .1?

d Show that the test that rejects if V  has significance level .

 
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