I apologize for the image I am posting below. I am new to StackExchange and I am not yet familiar with the MathJaX equations, so I took a screenshot.
Here is my question:
Let the independent random variables $Y_1, \ldots , Y_n$ have the following joint pdf where $x_1, \ldots , x_n$ are not all equal. We have the null hypothesis specified below and we want to use a likelihood ratio test to test the null hypothesis against all possible alternative hypotheses. $$ L(\alpha, \beta, \sigma ^ 2) = \left(\frac{1}{2 \pi \sigma ^ 2} \right) ^ {n / 2} \exp\left\{-\frac{1}{2 \sigma ^ 2} \sum_{i = 1}^n \left[y_i - \alpha - \beta (x_i - \bar x)\right] ^ 2 \right\} $$ $$ H_0: \beta = 0 \text{ ($\alpha$ and $\sigma^2$) unknown} $$
The question asks to find a the likelihood test statistic and check to see if it can be based on a familiar test statistics.
So far, all I know is that I believe it will be based on a $T$-statistic, but I do not know how to show this.