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Unformatted text preview: Two sample tests Inferences Based on Two Samples Artin Armagan STA 113 Chapter 9 of Devore November 5, 2009 Artin Armagan Inferences Based on Two Samples Two sample tests Table of contents 1 Two sample tests Normal known variance Large sample tests Normal unknown variance Analysis of Paired Data Difference Between Population Proportions Artin Armagan Inferences Based on Two Samples Two sample tests Normal known variance Large sample tests Normal unknown variance Analysis of Paired Data Difference Between Population Proportions Two sample tests A very common situation is the following setup 1 X 1 ,..., Xm iid ∼ p 1 ( θ 1 ) . 2 Y 1 ,..., Yn iid ∼ p 2 ( θ 2 ) . 3 X and Y are independent. Questions: μ 1 μ 2 = Δ μ 1 μ 2 > Δ μ 1 μ 2 < Δ μ 1 μ 2 6 = Δ . Artin Armagan Inferences Based on Two Samples Two sample tests Normal known variance Large sample tests Normal unknown variance Analysis of Paired Data Difference Between Population Proportions Normal population known σ X 1 ,..., Xm iid ∼ No( μ 1 ,σ 2 1 ) Y 1 ,..., Yn iid ∼ No( μ 2 ,σ 2 2 ) The null and alternative hypotheses are H : μ 1 μ 2 = Δ , Ha : μ 1 μ 2 > Δ . We want to compute rejection regions for this test. We first have to specify the the level of type I error or the critical α of the test, say α = . 05. We specify the test statistic as ¯ X ¯ Y . We now need to compute the value ‘ for which . 05 = Pr( ¯ X ¯ Y > ‘  X i ∼ No( μ = μ 1 ,σ 2 1 ) and Y i ∼ No( μ = μ 2 ,σ 2 2 )) . To do this we need the “distribution of the test statistic under the null hypothesis.” Artin Armagan Inferences Based on Two Samples Two sample tests Normal known variance Large sample tests Normal unknown variance Analysis of Paired Data Difference Between Population Proportions Normal population known σ If the null hypothesis is true and the data comes from a normal with known σ then we know that the following test statistic is distributed as a standard normal z = ¯ X ¯ Y Δ r σ 2 1 m + σ 2 2 n ∼ No(0 , 1) . As we did in the case of the one sample test we use z α values. Artin Armagan Inferences Based on Two Samples Two sample tests Normal known variance Large sample tests Normal unknown variance Analysis of Paired Data Difference Between Population Proportions Normal population known σ X 1 ,..., Xm iid ∼ No( μ 1 ,σ 2 1 ) Y 1 ,..., Yn iid ∼ No( μ 2 ,σ 2 2 ) Null hypothesis H : μ 1 μ 2 = Δ ....
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This note was uploaded on 01/17/2010 for the course STA 113 taught by Professor Staff during the Fall '08 term at Duke.
 Fall '08
 Staff
 Variance

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