Applied Statistics and Probability for Engineers

8-4 CONFIDENCE INTERVAL ON THE VARIANCE AND STANDARD DEVIATION OF A NORMAL POPULATION 263

10 degrees of freedom. We may write this as a probability statement as follows:

Conversely, a lower5% point of chi-square with 10 degrees of freedom would be ^2 0.95,103.94

(from Appendix Table III). Both of these percentage points are shown in Figure 8-9(b).

The construction of the 100(1 )% CI for ^2 is straightforward. Because

is chi-square with n1 degrees of freedom, we may write

so that

This last equation can be rearranged as

This leads to the following definition of the confidence interval for ^2.

P a

1 n 12 S^2

^2  2,n 1

^2 

1 n 12 S 2

^21  2,n 1

b 1 

P a^21  2,n 1 

1 n 12 S 2

^2

^2  2,n 1 b 1 

P 1 ^21  2,n 1 X^2 ^2  2,n 12  1 

X 2 

1 n 12 S 2

^2

P 1 X 2 

^2 0.05,10 2 P 1 X 2

 18.31 2 0.05

(a)

α,k

α

0 ^2

f(x) f(x)

x

(b)

0 ^2

0.05 0.05

0.95, 10

= 3.94

^2 0.05, 10

= 18.31

Figure 8-9 Percentage point of the ^2 distribution. (a) The percentage point ^2 ,k. (b) The upper

percentage point ^2 0.05,1018.31 and the lower percentage point ^2 0.95,103.94.

If s^2 is the sample variance from a random sample of nobservations from a normal dis-

tribution with unknown variance ^2 , then a 100(1)% confidence interval on ^2 is

(8-21)

where and are the upper and lower 1002 percentage points of

the chi-square distribution with n1 degrees of freedom, respectively. A confidence

interval forhas lower and upper limits that are the square roots of the correspon-

ding limits in Equation 8-21.

^2  2,n 1 ^21  2,n 1

1 n 12 s^2

^2  2,n 1

^2 

1 n 12 s^2

^21  2,n 1

Definition

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