STA113: Lab 5 (Friday, 15 October, 2004)

The Central Limit Theorem demonstrated on a uniform population

In Lab 4 we learned to do so several different kinds of simulations in MatLab. Then in class you saw how simulations in MatLab can be used to demonstrate the Central Limit Theorem. In this lab you will write your own simulations to deomstrate your understanding of the Central Limit Theorem, and will write a short report about what you learned.

Below is the code used in class for the demonstration of the Central Limit Theorem.

m=1; n=1000;
y=unifrnd(0,1,n,m); % draw random samples from Unif(0,1)
subplot(2,2,1);
histadjusted(y,32);
set(gca, 'XLim', [-0.05,1.05]);

m=2; 
subplot(2,2,2);
y=unifrnd(0,1,n,m);
histadjusted(mean(y,2),32);
set(gca, 'XLim', [-0.05,1.05]);

m=3;subplot(2,2,3)
% The same thing is now done as before.

m=5; subplot(2,2,4);
y=unifrnd(0,1,n,m);
histadjusted(mean(y,2),32);
set(gca, 'XLim', [-0.05,1.05]);
legend('sample size=5');
% The legend can, of course, be put on any of the graphs.

[umean, uvar]=unifstat(0,1);
% (umean, uvar) are the mean and variance of Unif(0,1)
hold on;
x=(0:50)./50;
h=plot(x, normpdf(x, umean, sqrt(uvar/m)));
set(h, 'color', [1 0 0]); % draw a red normal density curve
hold off;

% This was not done in class, but you might of course
% also examine the normality by making a normal probability
% plot:
qqplot(mean(y,2));

Assignment (due 22 October)

Complete the tasks described below and write a short (at most three pages, including graphs) report explaining what you've done and what it demonstrates.

• Your paper's goal should be to explain the Central Limit Theorem to your reader and to demonstrate it.

• Graphs will be an essential part of your paper, of course. These should be titled and clearly labeled.

• Graphs should be referred to at appropriate places in your paper and described clearly. They should be incorporated into the flow of your paper, not dumped in with the reader left to figure out what they mean.

• What a graph shows and what it means are not the same thing. For example, one might make a plot of number of home burglaries in a town versus the lunar phase. A description of what the graph shows would include statements of what the axes represent, the meaning of any odd color coding, etc. An interpretation of the meaning of the graph might include observations of trends, such as: "We note a clear association between number of burglaries and lunar phase that persists over many months. More burglaries tend to occur when the moon is close to new, suggesting that burglers favor dark conditions." The best way to present a graph in a scientific paper is to precede the graph with a description of what the graph shows, and to follow the graph with an interpretation of what the graph means.

• You may work with others while doing the MatLab simulations and you may discuss your results with others. However, you must write and understand your own MatLab code, generate your own graphs, and write your own paper.

• Don't forget to put your name on your paper.

• The paper is to be turned in at the beginning of the lab or recitation section on Friday, 22 October.

• Your paper will be graded in large part on how successfully it communicates and demonstrates the Central Limit Theorem using the example described in the tasks. Pay attention to how you word things, as details can matter. For example, "the average is..." is too vague. The average of what? In this lab, you'll be dealing with three different types of means: The mean of a population, the mean of a sample, and the mean of many sample means. Be clear which mean you mean.

1. Let X be the duration of a telephone call to the Durham Public Library. Suppose, as some studies have suggested is often the case, that X has an exponential distribution. Suppose further that X has a mean duration of two minutes.

2. Simulate 5000 phone calls' durations and make a histogram of them. The MatLab command for simulating from an exponential distribution is exprnd.

3. Suppose you were to take averages of X's in sets of four. That is, you take random samples of four independent phone calls and average their durations together. Simulate 5000 such sample averages and make a histogram of them.

4. Repeat the above for samples of sizes 9 and size 100.

5. For the histogram of sample means of sample size 100, superimpose an appropriate normal pdf on the graph.

6. Make normal probability plots for each of the four simulation results above.

7. For each of the four simulations, find the mean and standard deviation of the sample means. What appears to be the relationship between sample size and the mean of the sample means? Between sample size and the standard deviation of the sample means?