Let X be a Gaussian random variable with unknown mean and un

Let X be a Gaussian random variable with unknown mean and unknown variance. A set of 20 independent measurements of X yields a sample variance of 6. Find a 95% confidence interval of the variance of X.

Solution

Confidence Interval
CI = (n-1) S^2 / ^2 right < ^2 < (n-1) S^2 / ^2 left
Where,
S^2 = Variance
^2 right = (1 - Confidence Level)/2
^2 left = 1 - ^2 right
n = Sample Size

Since aplha =0.05
^2 right = (1 - Confidence Level)/2 = (1 - 0.95)/2 = 0.05/2 = 0.025
^2 left = 1 - ^2 right = 1 - 0.025 = 0.975
the two critical values ^2 left, ^2 right at 19 df are 32.8523 , 8.907
Variacne( S^2 )=6
Sample Size(n)=20
Confidence Interval = [ 19 * 6/32.8523 < ^2 < 19 * 6/8.907 ]
= [ 114/32.8523 < ^2 < 114/8.9065 ]
[ 3.4701 , 12.7996 ]