# stat400lec28 - 2-Case 3 σ x 2 ≠σ Y 2 are unknown W= m S...

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Statistics 400 Section 7.3 Confidence Intervals for Difference of Two Means If X1, X2, …, Xn are observations of a random sample of size n from a normal distribution N( μ x , σ x 2 ), then we have X is N( μ x , σ x 2 /n) If Y1, Y2, …, Ym are observations of an independent random sample of size m from a normal distribution N( μ Y , σ Y 2 ), then we have Y is N( μ Y , σ Y 2 /m) Objective: construct confidence interval for μ x - μ Y Case 1: σ x 2 and σ Y 2 are known Thus we have W= X - Y ~N( μ x - μ Y , σ x 2 /n+, σ Y 2 /m) and Ping Ma Lecture 28 Fall 2005 - 1 -

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Case 2: σ x 2 = σ Y 2 = σ 2 are unknown Z= m n Y X Y X / / ) ( 2 2 σ μ + - - - ~ N(0,1) U= 2 2 2 2 ) 1 ( ) 1 ( Y X S m S n - + - ~ χ 2 (n+m-2) ) 2 /( - + = m n U Z T ~ t (n+m-2) Ping Ma Lecture 28 Fall 2005

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Unformatted text preview: - 2 -Case 3: σ x 2 ≠σ Y 2 are unknown W= m S n S Y X Y X Y X / / ) ( 2 2 +---μ Welch’s approximation W ~ t(r) where 2 2 2 2 2 2 2 1 1 1 1 -+ - + = m s m n s n m s n s r Y X Y X Ping Ma Lecture 28 Fall 2005- 3 -Case 4: confidence interval for paired data Let (X1, Y2), (X2,Y2), …, (Xn, Yn) be n pairs of dependent measurements. Let Di=Xi-Yi, μ D = μ x-μ Y n S D D D / μ-Ping Ma Lecture 28 Fall 2005- 4 -...
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## This note was uploaded on 07/24/2008 for the course STAT 400 taught by Professor Tba during the Fall '05 term at University of Illinois at Urbana–Champaign.

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stat400lec28 - 2-Case 3 σ x 2 ≠σ Y 2 are unknown W= m S...

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