Two groups
Show that in a one-way analysis of variance with k=2 groups, F=T^2, where T is the pooled two-sample t statistic for H_0:\mu_1=\mu_2.
With two groups, the grand mean is \overline Y_{\cdot\cdot}=(n_1\overline Y_1+n_2\overline Y_2)/n, so \overline Y_1-\overline Y_{\cdot\cdot}=n_2(\overline Y_1-\overline Y_2)/n and \overline Y_2-\overline Y_{\cdot\cdot}=-n_1(\overline Y_1-\overline Y_2)/n. Then
\mathrm{SS}_{\mathrm{treatment}}=\frac{n_1n_2^2+n_2n_1^2}{n^2}(\overline Y_1-\overline Y_2)^2=\frac{n_1n_2}{n}(\overline Y_1-\overline Y_2)^2=\frac{(\overline Y_1-\overline Y_2)^2}{1/n_1+1/n_2},
on 1 degree of freedom. The error mean square is \mathrm{SS}_{\mathrm{error}}/(n-2)=s_p^2, the pooled variance. So
F=\frac{(\overline Y_1-\overline Y_2)^2}{s_p^2(1/n_1+1/n_2)}=T^2.