# sol1 - PSTAT 120C Solutions to Assignment 1 1 We have 9...

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PSTAT 120C: Solutions to Assignment # 1 April 14, 2009 1. We have 9 independent observations from a normal distribution with standard deviation 10, and we want ttest H 0 : μ = 150 vs. H A : μ 6 = 150 The GLRT with level α = 0 . 05 has the critical set C 1 = { ¯ x < 143 . 5 or ¯ x > 156 . 5 } . In all of these calculations ¯ x ∼ N ( μ, 100 9 ) . (a) The power of this test against the alternative μ = 151 is P { ¯ x < 143 . 5 } + P { ¯ x > 156 . 5 } = P ± ¯ x - 151 10 / 3 < 143 . 5 - 151 10 / 3 ² + P ± ¯ x - 151 10 / 3 > 156 . 5 - 151 10 / 3 ² = P { Z < - 2 . 25 } + P { Z > 1 . 65 } = . 0122 + 0 . 0495 = 0 . 0617 (b) The power of this test against the alternative μ = 123 is P { ¯ x < 143 . 5 } + P { ¯ x > 156 . 5 } = P ± ¯ x - 123 10 / 3 < 143 . 5 - 123 10 / 3 ² + P ± ¯ x - 123 10 / 3 > 156 . 5 - 123 10 / 3 ² = P { Z < 6 . 15 } + P { Z > 10 . 05 } 1 (c) Alternatively, consider the test with C 2 = { ¯ x < 144 . 5 or ¯ x > 160 . 9 } . i. The level of this second test is the probability when the mean is 150:

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sol1 - PSTAT 120C Solutions to Assignment 1 1 We have 9...

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