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Unformatted text preview: Chapter 10: Estimation and Hypothesis Testing: Two Populations Bin Wang [email protected] Department of Mathematics and Statistics University of South Alabama Mann E6 1/37 Parameter(s) of interests The differences between the two population means: μ 1 μ 2 . What are we going to do? Or, what will you be required to do? Or what will be tested? 1 Find a (1 α )100% confidence interval for μ 1 μ 2 . 2 Test a hypothesis about μ 1 μ 2 . Mann E6 2/37 Notations Suppose we select two samples from two different populations that are referred to population 1 and population 2. Let μ 1 = the mean of population 1, μ 2 = the mean of population 2, σ 1 = the standard deviation of population 1, σ 2 = the standard deviation of population 2, n 1 = the size of sample drawn from population 1, n 2 = the size of sample drawn from population 2, ¯ x 1 = the mean of sample drawn from population 1, ¯ x 2 = the mean of sample drawn from population 2, s 1 = the standard deviation of sample 1, s 2 = the standard deviation of sample 2, Mann E6 3/37 Point estimator Can you find a point estimator for μ 1 μ 2 ? ¯ x 1 ¯ x 2 . 1 What is the mean of ¯ x 1 ¯ x 2 ? μ ¯ x 1 ¯ x 2 = μ ¯ x 1 μ ¯ x 1 = μ 1 μ 2 2 What is the standard deviation of ¯ x 1 ¯ x 2 : σ ¯ x 1 ¯ x 2 ? 3 Hypothesis Testing H : μ 1 = μ 2 or μ 1 μ 2 = versus H 1 : μ 1 μ 2 6 = 0 Could be H : μ 1 μ 2 = Δ versus H 1 : μ 1 μ 2 6 = Δ H : μ 1 μ 2 = versus H 1 : μ 1 μ 2 > H : μ 1 μ 2 = versus H 1 : μ 1 μ 2 < Mann E6 4/37 Notes Notes Notes Notes Independent versus Dependent Samples Definition (Independent versus Dependent Samples) Two samples drawn from two populations are independent if the selection of one sample from one population does not affect the selection of the second sample from the second population. Otherwise, the samples are dependent. Case I If the two samples are independent, ¯ x 1 and ¯ x 2 are independent. Hence Var (¯ x 1 ¯ x 2 ) = Var (¯ x 1 ) + Var (¯ x 2 ) Case D If the two samples are dependent, ¯ x 1 and ¯ x 2 are dependent, and Var (¯ x 1 ¯ x 2 ) = Var (¯ x 1 ) + Var (¯ x 2 ) 2 Cov (¯ x 1 , ¯ x 2 ) Mann E6 5/37 Independent sample case (Case I) Case I.1 If σ 1 and σ 2 are known, Var (¯ x 1 ¯ x 2 ) = Var (¯ x 1 ) + Var (¯ x 2 ) = σ 2 1 n 1 + σ 2 2 n 2 Case I.2 If σ 1 = σ 2 = σ but unknown, Var (¯ x 1 ¯ x 2 ) = σ 2 1 n 1 + 1 n 2 Estimate σ with s 2 pooled = ( n 1 1) s 2 1 +( n 2 1) s 2 2 ( n 1 1)+( n 2 1) . Case I.3 If σ 1 6 = σ 2 but unknown, Var (¯ x 1 ¯ x 2 ) = Var (¯ x 1 ) + Var (¯ x 2 ) = s 2 1 n 1 + s 2 2 n 2 Mann E6 6/37 Case I.1: independent samples with known population variances Test Statistic Z = (¯ x 1 ¯ x 2 ) ( μ 1 μ 2 ) q σ 2 1 / n 1 + σ 2 2 / n 2 ∼ N (0 , 1) , if either the two populations are normally distributed , or the two sample sizes are large enough (by CLT)....
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This note was uploaded on 02/05/2012 for the course ST 210 taught by Professor Wangs during the Fall '09 term at S. Alabama.
 Fall '09
 Wangs

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