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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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 Spring '11
 Stepanov
 Degrees Of Freedom

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