= Solution
For an independent second sample, let
$$
\widetilde S_{XX}=\sum_{j=1}^{\widetilde n}
(\widetilde X_j-\overline{\widetilde X})^2.
$$
Under $H_0:\sigma^2=\widetilde\sigma^2$, independence and the two chi-squared laws give the <F-distribution>
$$
\boxed{F=
\frac{S_{XX}/(n-1)}{\widetilde S_{XX}/(\widetilde n-1)}
\sim F_{n-1,\widetilde n-1}}.
$$
For the one-sided alternative $\sigma^2>\widetilde\sigma^2$, reject for large $F$, using the upper $\alpha$ critical value.
\b[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 $\widetilde n$, rather than $n-1$ and $\widetilde n-1$. Thus knowledge of the means changes the degrees of freedom and critical value, while the variance-ratio principle is unchanged.
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