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Let ?1, ?2, … , ?? denote a random sample from a normal population distribution with...

Let ?1, ?2, … , ?? denote a random sample from a normal population distribution with a known value of ?.

(a) For testing the hypotheses ?0: ? = ?0 versus ??: ? > ?0 where ?0 is a fixed number, show that the test with test statistic ?̅ and rejection region ?̅≥ ?0 + 2.33(?⁄√?) has a significance level 0.01.

(b) Suppose the procedure of part (a) is used to test ?0: ? ≤ ?0 versus ??: ? > ?0 . If ?0 = 100, ? = 25 and ? = 5,

(1) What is the probability of committing a type I error when ? = 99?

(2) What is the probability of committing a type I error when ? = 98?

(3) In general, what can be said about the probability of a type I error when the actual value of ? is less than ?0 ?

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