## Confidence Interval for ratio of variances

Let $X_1, X_2, \cdots , X_{n_1}$ be a random sample of size $n_1$ from $N(\mu_1, \sigma_1^2)$ and $Y_1, Y_2, \cdots , Y_{n_2}$ be a random sample of size $n_2$ from $N(\mu_2, \sigma_2^2)$. Moreover, $X$ and $Y$ are independently distributed.

## Formula

$100(1-\alpha)$% confidence interval estimate for the ratio of variances is

### $\bigg(\dfrac{s_1^2}{s_2^2}\dfrac{1}{F_{(\alpha/2,n_1-1,n_2-1)}}, \dfrac{s_1^2}{s_2^2}\dfrac{1}{F_{(1-\alpha/2,n_1-1,n_2-1)}}\bigg)$

where,

• $s_1^2$ and $s_2^2$ are the sample variances for the first and second sample respectively,
• $1-\alpha$ is the confidence coefficient,
• $F_{(\alpha/2,n_1-1,n_2-1)}$ and $F_{(1-\alpha/2,n_1-1,n_2-1)}$ are the table values of $F$.