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Dr. Hackney STA Solutions pg 140

# Dr. Hackney STA Solutions pg 140 - Second Edition 8-19 8.45...

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Second Edition 8-19 8.45 The verification of size α is the same computation as in Exercise 8.37a. Example 8.3.3 shows that the power function β m ( θ ) for each of these tests is an increasing function. So for θ > θ 0 , β m ( θ ) > β m ( θ 0 ) = α . Hence, the tests are all unbiased. 8.47 a. This is very similar to the argument for Exercise 8.41. b. By an argument similar to part (a), this LRT rejects H + 0 if T + = ¯ X - ¯ Y - δ S 2 p ( 1 n + 1 m ) ≤ - t n + m - 2 . c. Because H 0 is the union of H + 0 and H - 0 , by the IUT method of Theorem 8.3.23 the test that rejects H 0 if the tests in parts (a) and (b) both reject is a level α test of H 0 . That is, the test rejects H 0 if T + ≤ - t n + m - 2 and T - t n + m - 2 . d. Use Theorem 8.3.24. Consider parameter points with μ X - μ Y = δ and σ 0. For any σ , P ( T + ≤ - t n + m - 2 ) = α . The power of the T - test is computed from the noncentral t distribution with noncentrality parameter | μ x - μ Y - ( - δ ) | / [ σ (1 /n + 1 /m )] = 2 δ/ [ σ (1 /n + 1 /m )] which converges to as σ 0. Thus, P ( T - t n + m - 2 ) 1 as σ 0. By Theorem 8.3.24, this IUT is a size
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