If I have two sample variances, and I compute their confidence intervals as $\left[\, \frac{(n-1)s^2}{\chi^2_{n-1,1-\alpha/2}},\ \ \frac{(n-1)s^2}{\chi^2_{n-1,\alpha/2}} \,\right]$, I could reject the null hypothesis if their confidence intervals do not overlap.
Would I do so if and only if I reject the null hypothesis using the F-test on the same sample variances?
If not what is the difference between the two tests?
Yes, you would reject a hypothesis if and only if the hypothesized value is outside of the confidence intervals of the same probability.1,2