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 â€” 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 Â has significance level .