It seems that the central limit theorem has many applications, but I just want to know an extremely simple one.
If I have $n$ i.i.d. random variables, $X_1,..., X_n$, and $S=\sum_{i=1}^nX_i$, does the CLT state that as n gets large, $S$ is approximately normal with $\mu=nE(X_i)$ and $\sigma^2=nVar(X_i)$?
Is that why you can approximate $F_{S}(s)$ as: $$\phi\left(\frac{s-nE(X_i)}{\sqrt{nVar(X_i)}}\right)$$Where $\phi$ is the cdf of the standard normal distribution?