09_25_09ans - STAT 409 Fall 2009 Examples for If U and V...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

This note was uploaded on 10/12/2011 for the course STATISTICS stat 410 taught by Professor Stepanov during the Spring '11 term at University of Illinois, Urbana Champaign.

Ask a homework question - tutors are online