sta 2006 0708_chapter_4

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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 06/05/2011 for the course STATISTICS 2006 taught by Professor Ho during the Spring '11 term at CUHK.

### Page1 / 5

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online