sta 2006 0708_chapter_5

sta 2006 0708_chapter_5 - STA 2006 Confidence Intervals for...

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Unformatted text preview: STA 2006 Confidence Intervals for Variances (Section 6.6) 2007-2008 Term II 1 One Sample Suppose X 1 ,...,X n are i.i.d. sampled from N ( μ,σ 2 ). Then, the sample variance is given by S 2 = ∑ n i =1 ( X i- ¯ X ) 2 n- 1 and it is an unbiased estimate of σ 2 . Its confidence interval can be derived by using the distribution ( n- 1) S 2 σ 2 ∼ χ 2 ( n- 1). Take a = χ 2 1- α/ 2 ( n- 1) and b = χ 2 α/ 2 ( n- 1). Then, Pr { a ≤ ( n- 1) S 2 σ 2 ≤ b } = 1- α ⇔ Pr { ( n- 1) S 2 b ≤ σ 2 ≤ ( n- 1) S 2 a } = 1- α. That is, the 100(1- α )% C.I. of σ 2 is [ ( n- 1) s 2 b , ( n- 1) s 2 a ]. Remark: Note that the cutoff value χ 2 p ( r ) is determined by: Z ∞ χ 2 p ( r ) (1 / 2) r/ 2 Γ( r/ 2) y r/ 2- 1 exp {- y 2 } dy = p where 0 < p < 1. Example 1: In a sample of credit card holders the mean monthly value of credit card purchases was $ 232 and the sample variance was 620. Assume that the population distribution is normal. Suppose the sample results were 1 obtained from a random sample of 12 credit card holders. Find a 95% confi-obtained from a random sample of 12 credit card holders....
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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_5 - STA 2006 Confidence Intervals for...

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