CENTRAL LIMIT THEOREM

The Central Limit Theorem is one of the most important concepts in statistics. Everything else depends on this fundamental concept. The central limit theorem states that the means of a sample obtained from any population will always have a normal distribution and it does not matter which distribution the population follows. The sample means, follow a more proper normal distribution as the size of the sample will keep on increasing. Therefore, the sample mean will be always normally distributed and specially distributed when the sample size is above 30, no matter if the population is binomial, negatively skewed or positively skewed.

RANDOM SAMPLE 1

The central limit theorem says that the mean of the sample is equal to the population mean. Also the standard deviation of the sample mean is equal to the standard deviation of the population mean. The average of all the sample means is around 4.997 and its standard deviation from the mean is 0.507. The sample size is 100 and the lambda parameter used here is 0.2. The number of observations in each sample is 100. The histogram shows that the mean of the sample is approximately normally distributed. Questions 1-5 all verified the properties of central limit theorem as mentioned above.

RAMDOM SAMPLE 2

The sample distribution with regard to the central limit theorem has been performed again and this time the value of lambda has been increased from 0.2 to 0.4. As the value of lambda has been doubled, so the mean of the sample means and the standard deviation of the sample means has been decreased by half. The normal distribution curve for this part is normal but the previous histogram was more appropriately showing the normal distribution of the means.

RANDOM SAMPLE 3

The same exponential distributions have been performed again. This time the lambda has been reduced to 0.2 again but the sample size for each sample has been increased from 100 to 225. It is the rule of the Central Limit Theorem that as the sample size is increased, the normal distribution curve for the sample is perfectly formed. Therefore, the mean and standard deviation have been calculated for these conditions. The histogram shows that as the sample size has been increased; and the mean of the sample is following a perfectly normal distribution. The graph shows a good normal distribution curve. This proves all the conditions of the Central Limit Theorem...............................

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