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My book has asked to calculate the variance of the pooled estimator. I understand everything until the second to last step but I am having trouble calculating the variance from the given information.
So far I have that $S_p^2 = \dfrac{n_1S_1^2 + n_2S_2^2 - S_1^2 - S_2^2}{n_1 + n_2 - 2}.$
Some how my book goes from this to the answer. Any help in solving this last step would be appreciated.
The variance of the $\chi^2_{n_1+n_2-2}$ distribution is $2(n_1+n_2-2)$.
Since your estimator is obtained by multiplying that $\chi^2_{n_1+n_2-2}$ random variable by $\frac{\sigma^2}{n_1+n_2-2}$, the variance is $$2(n_1+n_2-2) \cdot \left(\frac{\sigma^2}{n_1+n_2-2}\right)^2=\frac{2\sigma^4}{n_1+n_2-2}.$$ [This is using the fact that $\operatorname{Var}(aX) = a^2 \operatorname{Var}(X)$.]
