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sta 2006 0708_chapter_5

# sta 2006 0708_chapter_5 - STA 2006 Condence Intervals for...

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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: 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

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obtained from a random sample of 12 credit card holders. Find a 95% confi- dence interval for the variance of the monthly value of credit card purchases of all card holders.
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sta 2006 0708_chapter_5 - STA 2006 Condence Intervals for...

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