Statement of the Central Limit Theorem

Statement

Let X1, X2,……….,Xn  be a set of n independent random variables having identical distribution with mean and variance , and with X as the sum and as the mean of these random variables. As n becomes large, the central limit theorem states that the distribution of  
approaches the standard normal distribution. 

Example: Antelope Coffee Inc. is considering the possibility of opening a gourmet coffee shop in city. Previous research has indicated that its shops will be successful in cities of this size if the per capita annual income is above $60,000. It is also known that the standard deviation of income is $5,000. A random sample of 36 people was obtained and the mean income was $62,300. Does this sample provide evidence to conclude that a shop should be opened? 

Solution: The distribution of income is known to be skewed, but the central limit theorem enables us to conclude that the sample mean is approximately normally distributed. To answer the questions, we need to determine the probability of obtaining a sample mean at least as high a= 62,300 if the population mean is = 60,000.
First, compute the standardized normal Z-statistic:

From the standard normal table we find that the probability of obtaining a Z value of 2.76 or larger is 0.0029. Because this probability is very small, we can conclude that it is likely that the population mean income is not 60,000 but is a larger value. This result provides strong evidence that the population mean income is higher than $60,000 and that the coffee shop is likely to be a success.