Question

Find the confidence level associated to the confidence interval extending from 4.94 to 26.2 for the population variance of a Normally distributed random variable X if a sample of 13 observations has a standard deviation of 3.1.

Answer 1

Sample size = n = 13

Sample standard deviation = s = 3.1

Confidence interval for population variance = [4.94, 26.2]

Now, the confidence interval for the variance of the normal variate is given by -

Putting the values in the formula of confidence interval, we get,

= [4.94, 26.2]

= [4.94, 26.2]

Hence, ,therefore, = 24.87

Now, the critical value of chi square with 12 degrees of freedom is 24.87 for the probability between 0.95 and 0.975 (as obtained from the chi square table)

Hence, lies between 0.95 and 0.975

So, lies between 0.025 and 0.0125 ____ eqn 1

And, , therefore, = 4.69

Now, the critical value of chi square with 12 degrees of freedom is 4.69 for probability between (0.01, 0.025) as obtained from the chi square table).

Hence, lies between 0.01 and 0.025

So, lies between 0.02 and 0.05 ______ eqn 2

From equation 1 and 2 , we get, lying between 0.02 and 0.025

So, confidence level is between 0.98 and 0.975.