1.12 A researcher routinely tests using a nominal P (type I error) = 0.05, rejecting Ho if the P-value < 0.05. An exact test using test statistic T has null distribution P(T =

0) = 0.30, P(T = 1) = 0.62, and P(T = 2) = 0.08, where a higher T provides more evidence against the null.

Table 1.3 Data on Deaths by Mule Kicks, for Exercise 1.10 Number of Deaths Number of Corps-Years 0

109 1

65 2

22 3

3 4

1

≥5 0

a. With the usual P-value, show that the actual P(type I error) = 0.

b. With the mid P-value, show that the actual P(type I error) = 0.08.
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c. Find P(type I error) in parts

(a) and

(b) when P(T = 0) = 0.30, P(T = 1) = 0.66, P(T = 2) = 0.04. Note that the test with mid P-value can be conservative or liberal. The exact test with ordinary P-value cannot be liberal.

d. In part (a), a randomized-decision test generates a uniform random variable U from [0, 1] and rejects Ho if both T = 2 and U ≤3. Show the actual P(type I error) = 0.05. Is this a sensible test?
8