Chapter 7 – The Central Limit Theorem

The central limit theorem (CLT) states that the sample mean converges to a standard normal distribution given a large enough sample.

The size of the sample $n$ that is large enough depends on the size of the original population. Sampling must be done with replacement.

The Central Limit Theorem for Sample Means

Suppose $X$ is a random variable with any distribution. Taking a sufficiently large ($n\ge 30$) number of samples $n$, we find that the distribution of sample means $\bar{x}$ of all samples tends to be normally distributed.

• If $n\uparrow$ then $\bar{x}\to \mu$ and ${\mu }_{\bar{x}}\to \mu$ (the law of large numbers)

  • $n\uparrow$ then $\sigma \downarrow$ (notice that $n$ is in the denominator): bigger samples vary less

• ${\mu }_{\bar{x}}=\mu$: the mean of the sample means is the population mean

• ${\sigma }_{\bar{x}}=\frac{\sigma }{\sqrt{n}}$: the stdev of the sample means is the population stdev divided by $\sqrt{n}$

  • ${\sigma }_{\bar{x}}^2=\frac{{\sigma }^2}{n}$

• $z=\frac{\bar{x}-{\mu }_{\bar{x}}}{{\sigma }_{\bar{x}}}$ where ${\sigma }_{\bar{x}}=\frac{\sigma }{\sqrt{n}}$

  • $z=\frac{\bar{x}-\mu }{\sigma /\sqrt{n}}$ by the law of large numbers

• $\bar{x}\sim N(\mu ,\frac{\sigma }{\sqrt{n}})$

  • Recall that $\mu ={\mu }_{\bar{x}}$ by the law of large numbers