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A researcher routinely tests using a nominal P(type I error)=0.05, rejecting H0 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 Tprovides more evidence against the null.
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.
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 H0 when T=2 and U≤5/8 . Show the actual P(type I error)=0.05. Is this a sensible test?

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