09_25_09ans - STAT 409 Fall 2009 Examples for 09/25/2009 If...

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Unformatted text preview: STAT 409 Fall 2009 Examples for 09/25/2009 If U and V are independent χ 2 ( r 1 ) and χ2( r 2 ) random variables, respectively, then F= U r1 V r2 has an F distribution with r 1 and r 2 degrees of freedom. 2 X 1 , … , X n ~ N µ X ,σ X , 2 SY 2 σY = 2 SX 2 σX ⇒ 2 Y 1 , … , Y m ~ N µ Y ,σ Y , 2 ( m −1 ) S Y 2 σY 2 ( n −1 ) S X 2 σX ( m −1 ) ( n −1 ) the variance of the lifetime distribution of components he manufactures is equal to that of his major competitor; that is, he believes that the ratio of the two population variances is equal to 1. A random sample of size 31 from the first manufacturer has a variance of 32.6, and a random sample of size 25 from the competitor has a variance of 21.7. Assume the lifetimes of the electronic components produced by these two manufacturers are normally distributed. Find a 90% confidence interval for the ratio of the two population variances. Is it reasonable for the manufacturer to express his belief as an advertising claim? n = 31, ~ F ( m – 1, n – 1 ). 2 2 100 ( 1 – α ) % confidence interval for σ X σ Y 1. A manufacturer of electronic components believes that 2 s2 1 • x,F ( m −1, n −1 ) • sx α2 2 Fα 2 ( n −1, m −1 ) s 2 sy y 2 sx = 32.6, m = 25, 2 s2 1 • x ,F ( m −1, n −1 ) • sx 2 Fα 2 ( n −1, m −1 ) s 2 α 2 sy y F 0.05 ( 30, 24 ) = 1.94, 2 s y = 21.7. 1 ⋅ 32.6 , 1.89 ⋅ 32.6 1.94 21.7 21.7 F 0.05 ( 24, 30 ) = 1.89. ( 0.774 , 2.839 ) The claim that the variance ratio is equal to 1 is consistent with the data, since a ratio of 1 falls within this 90% confidence interval. ...
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