Fdist - n 2-1 S 2 2 σ 2 2 then Z 1 and Z 2 are independent...

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Statistics 351 (Fall 2007) Fisher’s F -distribution and Statistical Hypothesis Testing For Problem #4 on page 27 you are asked to manipulate Fisher’s F -distribution. The F -distribution arises in the following context. (See Section 10.9 in the Stat 251 textbook by Wackerly, et al.) Let Y 1 , . . . , Y n 1 be i.i.d. N ( μ 1 , σ 2 1 ), and let X 1 , . . . , X n 2 be i.i.d. N ( μ 2 , σ 2 2 ) with both μ i and σ 2 i unknown. Suppose that we wish to test H 0 : σ 2 1 = σ 2 2 vs. H A : σ 2 1 > σ 2 2 . The likelihood ratio test gives the rejection region as RR = S 2 1 S 2 2 > k where k is a suitably chosen constant, S 2 1 = 1 n 1 - 1 n 1 j =1 ( Y j - Y ) 2 with Y = 1 n 1 n 1 j =1 Y j , and S 2 2 = 1 n 2 - 1 n 2 j =1 ( X j - X ) 2 with X = 1 n 2 n 2 j =1 X j . Fact. If Z 1 = ( n 1 - 1) S 2
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Unformatted text preview: n 2-1) S 2 2 σ 2 2 , then Z 1 and Z 2 are independent random variables with Z i ∈ χ 2 ( n i ). If F = Z 1 / ( n 1-1) Z 2 / ( n 2-2) = ( n 1-1) S 2 1 σ 2 1 ( n 1-1) ( n 2-1) S 2 2 σ 2 2 ( n 2-1) = S 2 1 /σ 2 1 S 2 2 /σ 2 2 = S 2 1 σ 2 2 S 2 2 σ 2 1 , then F has the F-distribution with ( n 1-1) numerator degrees-of-freedom and ( n 2-1) denominator degrees-of-freedom. This is written as F ∈ F ( n 1-1 ,n 2-1). Problem #10 on page 28 asks you to verify this fact....
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