Surrounding the central limit theorem there exist several heuristics which say when a normal distribution is a reasonable approximation to the mean $\frac{X_1 + \cdots + X_N}{N}$ of $N$ independent (and possibly identically distributed) random variables $X_1, \dots, X_N$. For example:
- $N \geq 30$ as long as $X_i$ "isn't too weird";
- for binomial distributions with $n$ trials and probability of trial success $p$, a normal approximation is valid when $\min\{np, n(1-p)\} \geq 5$.
I've come accross these heuristics many times, but never with accompanying justification. My question is: how are these results derived?
After a quick search, I found the Berry-Esseen theorem, but it doesn't appear that this is what's used to derive the above (or similar) estimates on normal approximations.