## Confidence Interval for Variance

Let `$X_1, X_2, \cdots , X_n$`

be a random sample of size $n$ from $N(\mu, \sigma^2)$.

## Formula

$100(1-\alpha)$% confidence interval estimate of population variance $\sigma^2$ is

`$\bigg(\dfrac{(n-1)s^2}{\chi^2_{(\alpha/2,n-1)}}, \dfrac{(n-1)s^2}{\chi^2_{(1-\alpha/2,n-1)}}\bigg)$`

where,

`$\overline{X}=\frac{1}{n} \sum X_i$`

is the sample mean,`$s^2= \frac{1}{n-1}\sum (X_i-\overline{X})^2$`

is the sample variance,`$1-\alpha$`

is the confidence coefficient,`$\chi^2_{(1-\alpha/2,n-1)}$`

and`$\chi^2_{(1-\alpha/2,n-1)}$`

are the table values of`$\chi^2$`

.