Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 10 Confidence intervals for variance Moo K. Chung [email protected] August 13, 2004 1. Suppose the fat content of a hotdog follows nor- mal distribution. 10 measurements are given. > x<-c(25.2,21.3,22.8,17.0,29.8,21.0, 25.5,16.0,20.9,19.5) We are interested in constructing interval estimate of the unknown population variance. To solve this problem, we need to know the following fact. 2. Let X 1 , · · · , X n be a random sample from N ( μ, σ 2 ) . ( n - 1) S 2 σ 2 = 1 σ 2 n X j =1 ( X j - ¯ X ) 2 χ 2 n - 1 Quiz. What is the expectation of χ 2 n - 1 ? > y<- 0:50 > par(mfrow=c(2,2)) > plot(y,dchisq(y,1),type=’l’) > plot(y,dchisq(y,5),type=’l’) > plot(y,dchisq(y,10),type=’l’) > plot(y,dchisq(y,20),type=’l’) 3. Critical values for χ 2 n distribution are defined as numbers that gives P ( χ 2 1 - α/ 2 ,n < χ 2 n < χ 2 α/ 2 ,n ) = 1 - α To find χ 2 0 . 975 , 9 and χ 2 0 . 025 , 9 that is need to con-
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