Suppose that observations are drawn from a common normal distribution with mean and standard deviation . Then the sample mean
Central limit theorem. Now we shall drop the assumption of normal distribution for . Instead, we will assume an adequately large sample size . Then the distribution of the sample mean is still approximated by the same normal distribution with the mean and the standard deviation . A general rule for ``adequately large'' is about , 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 (652 = 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?