# Central Limit Theorem

4 April 2018

CENTRAL LIMIT THEOREM There are many situations in business where populations are distributed normally; however, this is not always the case. Some examples of distributions that arenâ€™t normal are incomes in a region that are skewed to one side and if you need to are looking at peopleâ€™s ages but need to break them down to for men and women. We need a way to look at the frequency distributions of these examples. We can find them by using the Central Limit Theorem.

The Central Limit Theorem states that random samples taken from a population will have a normal distribution as long as the sample size is sufficiently large. The sample mean will be approximately equal to the population mean. The sampleâ€™s standard deviation will be equal to the populationâ€™s standard deviation. The Central Limit Theorem is so important because with it we will know the shape of the sampling distribution even though we may not know what the population distribution looks like.

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The real key to this entire theorem is the term sufficiently large.

If the sample size isnâ€™t sufficiently large, the frequency distribution for the sample size will not look the same as it does for the population. For populations that are really symmetric, sample sizes of two or three will do. This is due to the fact that symmetric populations tend to have normal distributions already. However, if there is any skewedness at all, you will need a larger sample size to have normal distribution. In these cases, a conservative figure for a sufficiently large sample size is more than thirty.

Here are the steps to finding the probabilities associated with a sampling distribution of x bar. First you need to find the sample mean by dividing the sum of the samples by the number of samples. Next you will need to define the sampling distribution. If you have a sample size that is sufficiently large, this will be approximately normal. The third step is to define the probability statement of interest. The last step is to use the standard normal distribution to find that probability of interest. You do that by finding the z-value and converting it into a probability.

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