Central Limit Theorem

central limit theorem : Consider a sample of $n$ observations selected from any population with a mean $\mu$ and standard deviation $\sigma$. Then, when $n$ is sufficiently large, the sampling distribution of $\bar{x}$ will be approximately a normal distribution with mean ${\mu }_{\bar{x}}=\mu$ and standard deviation ${\sigma }_{\bar{x}}=\frac{\sigma }{\sqrt{n}}$. The larger the sample size, the better the will be the normal approximation to the sampling distribution of $\bar{x}$. Rephrased: If you have a distribution, you can take the mean of a bunch of different samples and those means will form a normal distribution.

How big $n$ needs to be depends on the skew of the distribution. The bigger the skew, the more you need. The book says $n\ge 30$ is sufficient for most cases.