Applying Central Limit Theorem to an exponential distribution - how big should sample size be?

Question

For an exponential distribution, in order for the sampling distribution of its mean to be well approximated by normal distribution (via central limit theorem), how big should a "typical" sample size be? In standard textbooks, sample size of $n = 30$ is often given; is this enough for a strongly skewed exponential distribution?

Answer 1

Here are the densities for the distributions (sample mean from exponential distribution - i.e. a gamma distribution - in black, normal distribution with same mean as the original exponential distribution and standard deviation $\frac1{\sqrt{n}}$ times that in red)

for $n=10$:

and for $n=30$:

and for $n=90$:

It is for you to judge what amounts to "well approximated".

The skewness of $\frac{6}n$ diminishes towards $0$ as $n$ increases, so these charts have skewness $\frac13$ of the previous chart.