Project : Central Limit Theorem Assignment : Design and execute an experiment th

Question

Project: Central Limit Theorem

Assignment:

Design and execute an experiment that demonstrates the Central Limit Theorem. For example, consider the following.

Randomly and independently draw 6 samples with X observations each from the same non-normal population

Compute the sample means for each of the 6 samples

Treat the 6 sample means as a distribution and determine their (the sample means’) mean, standard deviation, and distribution

Repeat the preceding three steps in their entirety 4 times, once each with sample sizes X = 7, 14, 21, and 28

Analyze what happens to the preceding as X becomes larger

Deliverables:

Electronically submit a one page executive summary that describes your experiment in a step-by-step manner and summarizes and defends your findings; also, as attachments, include graphs, data, narrative, etc that substantiates your findings and conclusions.

In the executive summary, answer the following.

What is the Central Limit Theorem, e.g., what does it yield, why is it important, what assumptions underlie it, when does it apply, etc?

How specifically does your experiment substantiate or not substantiate the Central Limit Theorem with respect to values of the mean and standard deviation and to the shape of the distribution?

Explanation / Answer

Suppose that you have a large population of data whose mean is 'm' and whose standard deviation is 'S'

The SD of this population 'S' is assumed to exist, that is, it is finite.

Next, suppose you start collecting samples from this population of size 'n'.

Also suppose that you collect many many samples of this size from the population, and calculate the means of all these samples. Thus you have large number of sample means.

Next, when you plot these sample means, the distribution which will be obtained will also have a particular shape.

Central Limit Theorem (CLT) talks about this distribution of sample means, and has the following main points:

(i) The distribution of sample means ( also called as the 'sampling distribution of sample means' ) will always have a normal distribution, irrespective of the distribution of the original population. This means that if the original population has a random distribution, its samplingdistribution will have a normal distribution.

(ii) The mean of this sampling distributiob will be approximately equal to the mean of the original population.

(iii) The standard deviation of this sampling distribution, denoted by S', is equal to:

S' = S/n0.5

So a smalled sample size means that the sampling distribution will have a flatter top, while a large sample size 'n' means that the sampling distribution will have a pointed top.

These are the important points of the CLT.

Hope this helps !

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