 Solution

For an independent second sample, letUnder $H_0:{\sigma }^2={\tilde{\sigma }}^2$, independence and the two chi-squared laws give the F-distributionFor the one-sided alternative ${\sigma }^2>{\tilde{\sigma }}^2$, reject for large $F$, using the upper $\alpha$ critical value.

No values of the unknown means enter this statistic: centering by the sample means removes them. If both means are known, one may instead use sums about the known means; the corresponding degrees of freedom are $n$ and $\tilde{n}$, rather than $n-1$ and $\tilde{n}-1$. Thus knowledge of the means changes the degrees of freedom and critical value, while the variance-ratio principle is unchanged.