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L25posted

# L25posted - Tests Comparing 1 with 2 Independent random...

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Tests Comparing μ 1 with μ 2 Independent random samples { X 1 , . . . , X n 1 } and { Y 1 , . . . , Y n 2 } from N ( μ 1 , σ 2 1 ) , N ( μ 2 , σ 2 2 ) popula- tions respectively. Null hypothesis is: H 0 : μ 1 = μ 2 If σ 2 1 , σ 2 2 known then test based on normal stan- dardized score Z = If σ 2 1 , σ 2 2 not known but sample sizes large then use approximate normal standardized score Z = 1

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Example 4.6-1 . Test H 0 : μ 1 = μ 2 versus μ 1 > μ 2 As for testing a single mean where θ = μ, here θ = μ 1 - μ 2 and the estimator is Θ = ¯ X - ¯ Y . The null value of θ = 0 because μ 1 = μ 2 . So reject H 0 if ¯ X - ¯ Y is so large that the standardized score Z > z α , equivalently ¯ X - ¯ Y > = 2 . 84 z α 2
OC curve Continue previous example. How good is this test when θ 6 = 0? OC ( θ ) = P (accept H 0 | θ ) = P ( ¯ X - ¯ Y < 2 . 84 z α θ ) = P ( ¯ X - ¯ Y - θ 2 . 84 < 2 . 84 z α - θ 2 . 84 ) = P ( Z < z α - θ 2 . 84 ) = Φ( z α - θ 2 . 84 ) 3

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Suppose α = 0 . 25 so z α = 1 . 96 . OC ( θ ) = Φ(1 . 96 - θ 2 . 84 ) . Consider a difference θ = μ 1 - μ 2 = 8 , meaning that we don’t want to accept H 0 . How likely is this test to accept H 0 , that is yield a
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L25posted - Tests Comparing 1 with 2 Independent random...

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