Apologies for the format of this question - I am new to this website. I am having trouble with part (c) of the question below, if anyone could assist that would be great.
Thanks in advance
Suppose that two independent random samples of size $n_1$ and $n_2$ observations are selected from normal populations. Let $X_1,\ldots,X_{n_1}$ and $Y_1,\ldots,Y_{n_2}$ be the two random samples and suppose that $X_i\overset d=N(\mu_1,\sigma^2)$ and $Y_i\overset d= N(\mu_2,\sigma^2).$ Note that we are assuming that the populations have a common variance $\sigma^2$. Define the sample variance from each sample as follows $$S_1^2=\dfrac {\sum_{i=1}^{n_1}(X_i-\overline X)^2} {n_1-1}\quad\text{and}\quad S_2^2=\dfrac {\sum_{i=1}^{n_2}(Y_i-\overline Y)^2} {n_2-1}. $$ Also define two pooled variance estimators $$S_{p_1}^2=\dfrac {(n_1-1)S_1^2+(n_2-1)S^2_2} {n_1+n_2-2}\quad\text{and}\quad S^2_{p_2}=\dfrac {1} {2}\left( S_1^2+S_2^2 \right).$$