e-Statistics

Central Limit Theorem

Suppose that observations $X_1,X_2,\ldots,X_n$ are drawn from a common normal distribution with mean $\mu$ and standard deviation $\sigma =$ . Then the average

$\displaystyle \bar{X} = \frac{X_1 + X_2 + \cdots + X_n}{n}$

has the normal distribution with the same mean $\mu$ and a distinctly smaller standard deviation $\displaystyle\frac{\sigma}{\sqrt{n}} =$ .

Central limit theorem. Now we shall drop the assumption of normal distribution for $X_1,X_2,\ldots,X_n$ . Instead, we will assume an adequately large sample size . Then the distribution of the average $\bar{X}$ is still approximated by the same normal distribution with the mean $\mu$ and the standard deviation $\displaystyle\frac{\sigma}{\sqrt{n}}$ . A general rule for “adequately large” is about $n \ge 30$ , but it is often good for much smaller .

Example. The daily sales of a farmer's market vary from day to day, but it is normally distributed with mean $900 and standard deviation $300. The market is open six days a week. (a) How much variability do you expect in the average sales in a week? (b) How many days in a year (6 $\times$ 52 = 312 days) do you expect the sales less than $600? (c) How many weeks in a year (52 weeks) do you expect the weekly average sales less than $600?