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### Computer Exercises: Central Limit Theorem

(This exercise was designed to work with SPSS. It may be modified to work with other programs, and it may have to be modified to work with the current version of SPSS.)

### Simulating the Kerrich Experiment

A mathematician named John Kerrich tossed a coin 10,000 times and recorded all of the results. You might wonder why someone would do something like this, and in the case of Kerrich it was because he had time to kill—he was in a POW camp for several years. In lab today we will try to replicate Kerrich's experiment in less than an hour.

Open SPSS 15 for Windows and start with an empty sheet. Holding down the <enter> key, scroll down to line 10,000. (This will take several minutes.) Place a number at 10,000. You should see that the count above your entry is now black. (If you mess this up, you may have to redo it, and you really do not want to repeat it.)

Pull down Transform-->Compute Variables. You will get a dialog box. Place the name temp in the target variable and the number 1 in the box for numeric expression. Press OK. This will create a column of 10,000 ones.

Pull down Transform-->Create Time Series. In the dialog box, pull down the function menu to Cumulative Sum. Then click over temp to new variable. (It will put an equation in the box—that is OK.) Click OK. You should get a column in your data sheet numbered 1 to 10000. (Ignore what happens in the output window.) Rename this column to count.

Pull down Transform—Random Number Generators. Check the "Set Starting Point" box and make sure the Random radio button is on. Click OK. (If you do not do this, you may get a preset set of random numbers. I prefer everyone get his or her own unique random results.)

Pull down Transform--> Compute Variables. In the target variable put the name toss. In the numeric expression box put trunc(rv.uniform(0,2)). What this will do is create a series of random numbers between 0 and 2, and then chop off all the decimals. Anything between 0 and 1 will become a zero, and anything between 1 and 2 will become a 1. So you will get a column of zeros and ones once you click the OK button. (Chopping off the decimals is called truncation. That is the trunc part of the expression. The rv.uniform(0,2) creates random numbers between 0 and 2.)

4. Just to make life a bit more difficult, let us change the zeros to -1. The easiest way to do this is to go back to Transform-->Compute Variables. In the target variable leave the name toss. In the numeric expression box put 2*toss-1. Call this new variable toss1. Click OK. Can you explain why this will leave the 1s as 1s and make the 0s into -1s?

You have just simulated tossing a coin 10000 times. Let us see what we get.

We want to look at the sum of the tosses as we continue to toss. We can do this with the Transform-->Create Time Series. In the dialog box, pull down the function menu to Cumulative Sum. Then click over toss to new variable. Click OK. (Ignore what happens in the output window.)

Change the name of this new column to sum.

5. What does a positive number in this column mean? What does a negative number mean?

6. What do you think will happen to the sum as the number of tosses gets bigger? Will it get closer and closer to zero, or will it get further and further from zero? We can look at the numbers, or we can look at a graph. Let us look at the graph. Pull down Graphs-->Scatter. Click Define. Put sum on the Y axis and count on the X axis. Click OK. Describe what you find.

7. In an imaginary dialog with Kerrich, an assistant says, "In the long run the number of heads and the number of tails even out." Kerrich says that is not true. He says that the more you toss, the more you are likely to be away from zero. Is that what happens in your case or not?

Kerrich does say that the average should approach 0. We can see if this happens by dividing sum by count. You can do this using Transform-->Compute. (You can figure out what you need to do. Call the result average.)

8. Do a scatter diagram with count on the x axis and average on the y axis. Does it tend to approach zero?

9. In the next week or so we will show that when you have tossed the coin 100 times using these numbers of 1 and -1, you will usually have a rest that is between -20 and +20. Do you? At the 900 toss count, your result is expected to be between -60 and + 60. Is yours? At 3600 your result is expected to between -120 and 120. Is yours? And at 10,000, your result is expected to be between -200 and + 200. Is yours?

If you have time, repeat the experiment ten times, (hint—call your variables toss1, toss2, toss3, etc.) Do all the results look pretty similar or not? (Hint—do all of each step at the same time—it goes very fast that way.)

The chance error with 10,000 tosses is likely to be about 50. That means that most of the time your column called sum is unlikely to be more than 200 or less than -200. Of your eleven trials, how many had bigger sums at 10000?

10. Record all the sums that you had at 100, 1000, and 10000. Suppose you had to pay the house 1 cent every time you played. You win \$1 with a heads and lose \$1 with a tail. What percentage of times would you be ahead at 100 tosses? At 1000? At 5000? At 10000?

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