We observe a random sample X12 X22 X1002 where Xi are No

We observe a random sample X1^2, X2^2, . . . , X100^2 where Xi are Normal(0, ^2). Find a two-sided confidence interval for ^2 based on this sample, with = 0.05.

Solution

First find degrees of freedom:
Degrees of Freedom = n - 1
Degrees of Freedom = 100 - 1
Degrees of Freedom = 99

Find low end confidence interval value:
_{low end} = /2
_{low end} = 0.05/2
_{low end} = 0.025

Chi square value for 0.025, 100 = 69.230

Low end = (n-1)s^2/chi square

= 99(sigma^2)/69.230

=1.4300 ^2

Chi square 128.422

HIgh end =99^2/ 128.422

= 0.7709^2

Hence confidence interval

= (0.7709^2, 1.4300 ^2)