Checking out the Central Limit Theorem on your own

In recent lectures, a major focus point has been the Central Limit Theorem (CLT). Remember, the CLT states that if the population is normal or if the sample size is large enough, the sampling distribution of the the sample mean is approximately normally distributed with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

The purpose of this lab is "to see the CLT in action for yourself". How large does the sample size have to be in order for the sampling distribution to approach normality? How does this differ for various populations? We will use a computer program (written in C) to take some "samples" from a larger "population" and to calculate the corresponding sample means. Then, we will graph the results using SAS/INSIGHT.

Preliminaries

In the first lab, you should have created a directory to store the data sets and other files related to this class. You probably called it "sta110b". To see if it's there, you can issue the command "ls" at the UNIX prompt to list all the files and subdirectories stemming from your home directory. As a reminder, it looks like this (with the prompt and command all together, and with your username and machine name substituted for mine):

jls11@teer17% ls

cd sta110b

cp ~jls11/public/sample.exe sample.exe
cp ~jls11/public/normal_population normal_population
cp ~jls11/public/rtskew_population rtskew_population
cp ~jls11/public/bimodal_population bimodal_population

Start SAS

sas &

Read the "population" files into SAS

One way to get an idea of what is in the "population" files (you'd probably like to see the shapes of the population histograms before you use the sampling program) is to read them into SAS. This requires some code to be entered in the SAS program editor window. You can copy and paste this code from Netscape directly into the program editor. Highlight the text to be copied using the mouse's left button. Then, position the cursor at the spot in the program editor where you want the text to go. In this case, you want the text to begin 2 spaces to the right of 00001. Click the middle button to paste the text. (The numbers 00001, 00002, etc. are there primarily for mainframe SAS users; you can just ignore them.) Now we tell SAS in which SAS library (for us, this amounts to which directory) datasets can be placed.

 libname sta110b '~/sta110b';

To read the data set into SAS, submit the following code. As you can see from the first line, the data set will be named ex1 and will be stored in the directory refered to by the libref sta110b (which is your newly created sta110b directory).

data sta110b.populn;
  infile '~/sta110b/normal_population';  /* reads data from file "normal_population" */
  input normal;                          /* one variable named "normal" in file "normal_population" */
  infile '~/sta110b/rtskew_population';  /* reads data from file "rtskew_population" */
  input rtskew;                          /* one variable named "rtskew" in file "rtskew_population" */
  infile '~/sta110b/bimodal_population'; /* reads data from file "bimodal_population" */
  input bimodal;                         /* one variable named "bimodal" in file "bimodal_population" */
run;

Start INSIGHT

Go the command line in the toolbox window, type insight, followed by return. In the next window, choose the dataset you created; the data window containing your data set should appear. Notice that the dataset is quite large; in the upper left of the "spreadsheet" we can reaffirm that there are 5000 rows and 3 columns.

Create a histogram of all the data

Under the Analyze menu, choose the option Distribution. Click on each variable, and then click on the button labeled Y to add each to the list of variable to be graphed. (You can add all three at once if you hold hold down the control key when you click on each variable in turn.) All three variables will be treated as response variables. After you "OK" your selections, another window will appear with the histograms and summary statistics. From this we can see that each of the histograms has a different shape (explaining the variable names chosen for each). Whenever you want to, you can close the histogram window by selecting End under the File menu.

Use the C program to generate random samples from each distribution

Go back to the xterm window where you issued the "ls" and "cp" commands before. (We're going to ignore SAS for a few minutes). Before starting the sampling program, type (at the prompt)

limit coredumpsize 0

./sample.exe

When the program is done, the prompt will reappear. If you issue the list "ls" command again, you should see the name of the file you chose as the output file name. Run the program a couple of times with different sample sizes (remember to save them to different output files), and then use the following instructions to compare them using SAS.

Read in the sample means as a SAS dataset

data sta110b.normsamp; /* change "normsamp" to another name when dealing with bimodal or rightskewed distribtuions (to avoid overwriting "normsamp") */
  infile '~/sta110b/your_outputfile1'; /* first output file to read */
  input samp1;                         /* name of variable in first output file */
  infile '~/sta110b/your_outputfile2'; /* second output file to read */
  input samp2;                         /* name of variable in second output file */

  ...  /* add more outputfile results here as necessary (according to the pattern) */

run;

Imposing the normal curve over a histogram

Running the program with your own distribution

To end the SAS program

To logout from ACPUB

While the cursor is on the background (not in the xterm windows, netscape, or the windows corresponding to any other program), click the left button. From the menu that appears, select Logout.

To preserve the security of your account (including your files and password), you must logout!