Let $(X_1,X_2,...,X_m)$ be a sample of independent $N(\mu_1,\sigma_1^2)$ random variables, and let $(Y_1,Y_2,...,Y_n)$ be a sample of independent $N(\mu_2,\sigma_2^2)$ random variables. All the parameters are unknown, and we wish to test : \begin{equation*} H_0 : \mu_1 = \mu_2 \text{ and } \sigma_1^2 = \sigma_2^2 \\ \text{ against } \\ H_1 : \mu_1 \neq \mu_2 \text{ or } \sigma_1^2 \neq \sigma_2^2 \end{equation*}
The question is to calculate the likelihood ratio $LR(H_1,H_0)$ ( which I think I am doing correctly, but not sure). The main bit is to show that the likelihood ratio test can be expressed as "Reject $H_0$ if $|T| > D$", where $T$ is a function ( which should be found ) of $\bar{X},\bar{Y},S_{XX} $ and $S_{YY}$ where \begin{equation*} \bar{X} = \frac{1}{m} \sum_{i = 1}^{m} X_i \text{ and } S_{XX} = \sum_{i = 1}^{m} (X_i - \bar{X})^2 \end{equation*} and $\bar{Y}$ and $S_{YY}$ defined analogously.