Date: Sun, 24 Nov 1996 09:33:29 -0500

From: Alex Bogomolny

Dear Larry:

I hope you haven't asked your question in preparation to a test. Much as I dislike this, my response will be superficial. On your part, being more specific might help me tremendously.

The Central Limit Theorem asserts that under very broad conditions the sum of random variables has asymptotically a normal distribution.

A random variable could be a result of one experiment. After you conduct several of these, the average value is this kind of (now weighted) sum of random variables and is known as the sample mean. So you anticipate the observed sample mean value to be distributed normally. Note that in all likelihood the mean value of a random variable is what one is looking for to estimate running statistical experiments.

The sample mean is known to be an unbiased estimator for the mean value of the random variable which means that the expected value of the sample mean coincides with the expected value of the variable. Knowing that the former is distributed normally leads to the introduction of confidence intervals which is an estimate of goodness of the approximation.

In my view, setting up a statistical experiment and then drawing conclusions from it is more of an art than a science. Examples of abuse and misuse of statistics are abound and well documented. So that when you get sample results you often want to run another series of experiments to check that your conclusions are repeatable. z-test provides a confidence estimate of compatibility of two random samples.

This is the best I can do staying on the conceptual level.

Sincerely,

Alexander