Suppose a researcher routinely conducts significance tests by rejecting H0 if the P-value satisfies P ≤ 0.05. Suppose a test using a test statistic T and righttail probability for the P-value has null distribution P(T = 0) = 0.30, P(T =

3) = 0.62, and P(T = 9) = 0.08.

a. Show that with the usual P-value, the actual probability of type I error is 0 rather than 0.05.

b. Show that with the mid P-value, the actual probability of type I error equals 0.08.

c. Repeat

(a) and

(b) using P(T = 0) = 0.30, P(T = 3) = 0.66, and P(T =

9) = 0.04. Note that the test with mid P-value can be “conservative” [having actual P(type I error) below the desired value] or “liberal” [having actual P(type I error) above the desired value]. The test with the ordinary P-value cannot be liberal.