Show that, if $\sigma$ is unknown, the likelihood ratio statistic for testing a value of $\alpha$ is given by $D = n \log\left(1 + \frac{1}{n-1}T^2\right)\;,$ where $T = \frac{\hat{α} -\alpha}{\sqrt{s^2/n}}$
So far, I have the following: $\hat\alpha=\overline{y}$ and $\hat\sigma=\sqrt{\frac{\sum \left ( y_i-\bar{y} \right )^2}{n-1}}$.
Now, when I plug this in to my ratio, I have: $D=2\left [ l(\hat{\mu}, \hat{\sigma})-l(\mu_0,\hat{\sigma}) \right ]=\frac{n(\bar{y}-\mu_0)^2}{\hat{\sigma}}=\frac{(\bar{y}-\mu_0)^2}{c\hat{\sigma}}\text{ where }c=\frac{1}{n}$
So, just to clarify the following things:
I accidentally typed $n-1$. I meant $n$ in the denominator for $\sigma$.
My log-likelihood function is: $l(\mu, \sigma)=-n\log(\sigma)-\frac{\sum{(y_i-\bar{y})^2}}{2\sigma^2}$
When I expand my ratio statistic and simplify, the logs cancel because they are identical, and I am left with the equivalent expression of $T^2$. I don't understand how I am supposed to get the expression that I am asked for. Any ideas of what I am doing wrong?