Fundamentals of Probability and Statistics for Engineers

9.3.2.3 Confidence Interval for^2 in N(m^2 )

An unbiased point estimator for population variance^2 is S^2. For the con-
struction of confidence intervals for^2 , let us use the random variable

which has been shown in Section 9.1.2 to have a chi-squared distribution with
(n 1) degrees of freedom. Letting be the value such that
with n degrees of fr eedom, we can write (see Figure 9.9)

which gives, upon substituting Equation (9.142) for D,

Let us note that the [100(1 )]% confidence interval for^2 as defined by
Equation (9.144) is not the minimum-width interval on the basis of a given
sample. As we see in Figure 9.9, a shift to the left, leaving area to the left
and area to the right under the fD(d) curve, where is an appropriate
amount, will result in a smaller confidence interval. This is because the width
needed at the left to give an increase of in the area is less than the correspond-
ing width eliminated at the right. The minimum interval width for a given

d

fD(d)

1–

2 2

n^2 ,1– (/2) n^2 , /2

Figure 9. 9 [100(1 )]% confidence limits for D with n degrees of freedom

302 Fundamentals of Probability and Statistics for Engineers

s ,s





Dˆ

…n 1 †S^2

^2

; … 9 : 142 †

^2 n, /2
PD>^2 n, /2)ˆ /2

P…^2 n 1 ; 1 …= 2 †<D<^2 n 1 ;= 2 †ˆ 1 ; … 9 : 143 †



P

…n 1 †S^2

^2 n 1 ;= 2

<^2 <

…n 1 †S^2

^2 n 1 ; 1 …= 2 †

"

ˆ 1 : … 9 : 144 †

 

/2"

/2‡" "

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