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L19posted_rev3

# L19posted_rev3 - Condence Intervals for 1 2 Compare...

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Confidence Intervals for μ 1 - μ 2 Compare responses in two groups Can arise in two ways 1. 2. 1

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Population Sample Variable Mean St. dev. Size 1 n 1 X μ 1 σ 1 2 n 2 Y μ 2 σ 2 Parameter of interest θ = Natural estimator c Θ = Then E[ ¯ X - ¯ Y ] = 3

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Assume independent samples. Then Var [ ¯ X - ¯ Y ] = Finally assume both populations are nor- mal . Then ¯ X - ¯ Y Then standardized Z = Note. Make sure you understand why means, variances, and Z statements hold. These are properties of linear combi- nations or random variables and nor- mals. 4
Therefore C.I. construction reduces to that for a single population in which Z = replaces the previous Z = ¯ X - μ σ/ n As before, start with P ( - z α/ 2 Z z α/ 2 ) = 1 - α and use simple algebra to rewrite this as P ( ¯ X - ¯ Y - z α/ 2 r σ 2 1 n 1 + σ 2 2 n 2 μ 1 - μ 2 ¯ X - ¯ Y + z α/ 2 r σ 2 1 n 1 + σ 2 2 n 2 = 1 - α to end with the confidence interval ¯ X - ¯ Y ± z α/ 2 v u u t σ 2 1 n 1 + σ 2 2 n 2 5

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What if population cannot be as- sumed normal? For large n 1 , n 2 Cen- tral Limit Theorem. Replace σ 1 with S 1 and σ 2 with S 2 where S 1 , S 2 are the sample standard deviations for X, Y samples respectively. to get approxi- mate 100(1 - α )% confidence interval ¯ X - ¯ Y ± z

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L19posted_rev3 - Condence Intervals for 1 2 Compare...

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