The Central Limit Theorem

~If the population being sampled is normally distributed, then the sampling distribution of the means is also normally distributed, regardless of the sample size.

~If the population being sampled is not normally distributed, then the sampling distribution of the means also approaches a normal distribution provided that the sample size is relatively large (n greater than or equal to 30 is usually the case).

~The larger the sample size, the closer the distribution of the means is to normal.

~In mathematics, we call such a property, asymptotically normal (approaches the normal curve in the limit)

~This theorem enables us to use the standard normal curve techniques (2nd VARS, menus 2 & 3 on the TI-83) when analyzing the sample means. Be careful not to introduce a round off error by rounding off the z-scores, since square roots are used in the calculation of the standard deviation of the sample means. It is better to use the technique that avoids them. Also, make sure you do not round off square roots that come out irrational. That too will produce a round off error in the result. See lesson #14 under "COURSE LESSONS" for details.

~All we need is the mean of the distribution and its standard deviation.

~Mean of a sample = Mean of the population for any sample size n, Standard deviation of sample = population standard deviation divided by the square root of the sample size (assumes a large enough n).

~Otherwise, we must use the finite population correction. (See lesson #14)

~Here is a review of the important facts to remember:

1) The central limit thm involves two different distributions. The original population & the distribution of the sample means.

2) As the sample size increases, the sampling distribution of the sample means approaches a normal distribution.

3) The mean of all the sample means will equal the population mean.

4)The standard deviation of all the sample means will equal the population standard deviation divided by the square root of the sample size, for a normal population.

5)If the population is not normally distributed, the distribution of the sample means (n>30) can be approximated very well be a normal distribution. The approximation gets better as the sample size increases.

6)If the population is normally distributed, the distribution of the sample means will be normally distributed for any sample size n.

7)When working with the mean for a sample, make sure you use the population standard deviation divided by the square root of the sample size for the sample standard deviation.

8)When working with a one individual value from a normally distributed population, use the population standard deviation alone for the standard deviation for that value.

The Central Limit Theorem is also applied to population proportions & is essential in inferential statistics.