I have the following which is based on a normal distribution having mean $\mu$ and variance $\sigma^2$.
I tried to find out the distribution of this: $\frac{\sqrt{N}(\bar{X_N}- \mu)}{\frac{\sum_{n=1}^{N}(X_n - \bar{X_N})^2}{(N-1)} }$ In my opinion it could be normally distributed, asymptotically since it is similar to $\frac{\sqrt{N}[\bar{Y_N}-E(Y)]}{\sqrt{V(Y)}}$ but I'm not sure. Would you agree?
Thanks
This is a very poorly worded question, as you haven't written down what any of the random variables are supposed to be and what assumptions you make on them.
Guessing that you have some independent Gaussian random variables and $\bar{X}_n$ is supposed to be their (empirical) mean, the answer is no. The distribution is a t-distribution.