The Central Limit Theorem in Practice

What the theorem actually says

The Central Limit Theorem (CLT) is about shape only: whatever the shape of the population, the sampling distribution of $\bar x$ becomes closer to normal as $n$ grows. The centre $\mu_{\bar x}=\mu$ and spread $\sigma_{\bar x}=\sigma/\sqrt n$ are true for every $n$ and do not need the CLT. So a probability question about $\bar x$ has three parts: centre, spread, and a justification for using the normal curve — either "the population is normal" or "$n$ is large enough for the CLT".

How large is large?

There is no single number. If the population is normal, any $n$ works. If it is roughly symmetric with no outliers, $n$ around $15$ is already fine. For strongly skewed populations — incomes, waiting times, insurance claims — the working rule is $n\ge30$. When the problem shows you a skewed dotplot and $n=10$, the correct answer is that the normal approximation should not be used.

A total is a mean in disguise

Questions about a sum convert to questions about the mean: the total of $n$ values exceeds $T$ exactly when $\bar x$ exceeds $T/n$. Example: passenger weights are right-skewed with $\mu=80$ kg, $\sigma=15$ kg. For a lift rated at $4250$ kg carrying $50$ people, the load exceeds the limit when $\bar x\gt85$:

The CLT applies because $n=50\ge30$, even though individual weights are skewed.