The Central Limit Theorem is one of the most important theorems in statistics.

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The Central Limit Theorem is one of the most important theorems in statistics. The statement of the theorem means that for any population distribution, regardless of shape, if we repeatedly take samples of a large enough size and record the sample means, then the distribution of those means will approach a normal distribution with a mean the same as the population mean, and a standard deviation equal to the population standard deviation divided by the square root of the sample size. We will now look at the sampling distribution of the mean. Your worksheet showing simulations will be long. Submit ONLY the FIRST page of each simulation including the results of at least the first 30 simulations. Include also all your calculations and plots which must be done in R. 

a. Generate 1000 realizations of a random variable from the uniform (0,1) population, call the data by X1 and construct a well-labeled histogram of X1. 

b. Find the mean of the data in X1. The theoretical mean is 0.50. 

c. Find the standard deviation of the data in X1. 

d. Generate two more realizations each with 1000 uniform (0,1) random variables and call these realizations by X2, X3 respectively. Then take the average of the corresponding cells in X1, X2, X3 and call it by "average". Construct a well-labeled histogram of the 1000 values in "average". 

e. In one sentence compare this histogram with the one you obtained in (a).

f. Find the mean of the data in "average". The theoretical mean is also 0.50. g. Find the standard deviation of the data in "average". In one sentence compare your answer with (c). 

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