Is it correct to apply standard techniques for a confidence interval calculation to the result of a neural network? so I have a binary classier from which I can evaluate on a test set and get a proportion (p) of which the classier has correctly gotten right. I then apply the following function to determine the 95% confidence interval.
$$
 CIBounds = \pm 1.96 * \sqrt{\frac{p(1-p)}{n}}
$$
Although my first initial thought was I believe this assumes the underlying distribution is normal. In my lectures this was never really addressed, how would I go around justifying using this approach (or not using it) for my neural network.

The formula which you provided is used for a binomial (distribution) proportion confidence interval, and as you mentioned, it relies on the normal distribution and is only an approximation. In general the formula is quite accurate for values close to 0.5, but can be quite off for values close to the limits (in fact it will go over the possible limits).
There are exact methods to estimate this CI, although the expression is a little more complicated. In any case, if you use this formula to estimate the CI for a proportion coming from a neural network (although I don't know why), you will be making some assumptions about the underlying distribution. If you are not willing to do that, you are better off using non-parametric approaches.
