sta 2006 0708_chapter_4

sta 2006 0708_chapter_4 - STA 2006 Confidence Intervals for...

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Unformatted text preview: STA 2006 Confidence Intervals for Difference of Two Means (Section 6.5) 2007-2008 Term II 1 Known variances Suppose X 1 , .. . ,X n are i.i.d. sampled from N ( μ X , σ 2 X ) and Y 1 , .. . ,Y m are i.i.d. sampled from N ( μ Y , σ 2 Y ). Also, X i and Y j are independent to each other for i = 1 , .. . ,n and j = 1 , .. . ,m . If the variances are known, then μ X- μ Y is in the following interval with 100(1- α )% probability: [ ¯ X- ¯ Y- z α/ 2 r σ 2 X n + σ 2 Y m , ¯ X- ¯ Y + z α/ 2 r σ 2 X n + σ 2 Y m ]. (1) The realization of the above interval is called the 100(1- α )% confidence in- terval of μ X- μ Y . It can be easily computed by using [¯ x- ¯ y- z α/ 2 r σ 2 X n + σ 2 Y m , ¯ x- ¯ y + z α/ 2 r σ 2 X n + σ 2 Y m ]. Proof of (1): Note that ¯ X ∼ N ( μ X , σ 2 X /n ) and ¯ Y ∼ N ( μ Y , σ 2 Y /m ) and they are indep. to each other. That implies, ¯ X- ¯ Y ∼ N ( μ X- μ Y , σ 2 X n + σ 2 Y m ) ⇒ ¯ X- ¯ Y- ( μ X- μ Y ) q σ 2 X n + σ 2 Y m ∼ N (0 , 1) and Pr {- z α/ 2 ≤ ¯ X- ¯ Y- ( μ X- μ Y ) q σ 2 X n + σ 2 Y m ≤ z α/ 2 } = 1- α . That is, Pr { ¯ X- ¯ Y- z α/ 2 r σ 2 X n + σ 2 Y m ≤ μ X- μ Y ≤ ¯ X- ¯ Y + z α/ 2 r σ 2 X n + σ 2 Y m } = 1- α 1 2 Unknown but equal variances Same assumption as in the previous section. Now both σ 2 X and σ 2 Y are un- known but equal, i.e. σ 2 X = σ 2 Y . Then μ X- μ Y is in the following interval with 100(1- α )% probability: [ ¯ X- ¯ Y- t n + m- 2 ,α/ 2 S p r 1 n + 1 m , ¯ X- ¯ Y + t n + m- 2 ,α/ 2 S p r...
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This note was uploaded on 06/05/2011 for the course STATISTICS 2006 taught by Professor Ho during the Spring '11 term at CUHK.

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sta 2006 0708_chapter_4 - STA 2006 Confidence Intervals for...

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