Fall Semester 2003

This assignment has to be turned in on paper (Write up, print-out).

• Note though that you might be getting questions about it in lab.
• So please be prepared to quickly do something similar, if asked to.

Serendipity has produced many breakthroughs in science, and one of them is the normal distribution. Abraham de Moivre (1667-1754), a mathematics tutor (at the time), was searching for a shortcut method of computing probabilities for binomial random variables and in the process derived the function rule for the ubiquitous normal distribution.

We toss a number of coins (let's say 20) and we count the number of times we observe 0 heads, 1 head, and so on, all the way to observing 20 heads. If we do this experiement a number of times (for example 1000 times) the histogram of this experiment would be a good fit for the normal curve. If the number of coins were increased indefinitely, the number of bars in the histogram would increase, and their outline would eventually coincide with that of the normal curve. De Moivre derived the function rule giving the height of the curve for any value of a random variable. (He was pretty good. Can you do that? You are not expected to).

Here's your task:

Simulate the experiment with 20 coins, 1000 times.

Prove or disprove De Moivre's finding.

Be persuasive in your argument.

Grading

The AI will provide you with feedback in a week. Grades will be posted on-line.

Here's some help getting started with this assignment:

Experiments and the normal curve (II)

1. Start Excel, in cell A5 write this formula:

   =rand()

2. Press F9 a number of times to see the number change.

3. Now create the following table in cells B1:C2:

     0 0
   0.5 1

4. Change the formula in A5 to be

   =vlookup(rand(), $b$1:$c$2, 2)

   (Note: highlight the table, press F4.)

5. Press F9 a few times to see the number generated taking values of 0 and 1.

   This is a coin, thrown once.

6. Now simulate 20 coins being thrown 20 times.

   (Number the columns 1-20 up to column T. Then resize the columns. Paste the formula in the cells on the first row. Then resize, as needed. Having this experiment run 20 times means copying the line A5:T5 all the way down to row 25, including.)

7. In column U (U5:U25) calculate the result of the experiment: sum up the values in columns A through T. This is the formula for U5 (please don't start in U4).

   -sum(a5:t5)

   Paste this formula in cells U6:U25.

8. Press F9 a few times, for the fun of it.

9. Prepare your bins. How many?

   For 20 coins you get 21 bins (0-20).

   You can put them in cells W5:W26, and here's how.

   Enter a 0 in W5. Then choose Edit->Fill->Series from the main menu.

   Choose Series in Column, Step value 1, Stop value: 20. Then click OK.

10. Now prepare your histogram.

    Let's do that in column X. Select X5:X26. Then click the fx button.

    The fx function is called the Paste Function button, so you can identify it easily.

    Choose function category: ALL.

    Under function name browse for: FREQUENCY.

    Click once, read the message that describes the function.

    Click OK. Type U5:U25 in the Data_array field.

    Type W5:W25 in the Bins_array field.

    Then Click OK. Well, oops! Freeze, don't touch anything now. Oh boy!

    As you can see that was wrong. (Don't touch any key, or the mouse please).

    Something was wrong, but what? Can we fix it? (What is going on?)

    (Where in Berk and Carey can we read about this?)

    So take your mouse (slowly, slowly) and click in the formula field at the top of the screen next to the equals sign. Once you click there press control shift and enter at the same time. (First press control and shift and when they are both pressed press Enter).

    All this time the cells X5:X25 should have been selected. The formula changes (it has curly braces around it now). The distribution does change to show a certain concentration in the middle. Plot it now to see what it looks like.

11. To do that, while X5:X25 is selected click Chart Wizard button.

    Choose Area, click Finish. That would be good enough for now.

12. Press F9 a few times, see the picture change.

13. Now the question is: would more trials make a difference?

14. Let's see.

15. Take the last line (select it) and paste it through line 1005.

16. Select the U25 cell and then double-click on the lower right corner of it. All sums are calculated to U1005.

17. Select cells X5:X25. Write this formula

    =frequency(u5:u1005,w5:w25)

    Then BEFORE PRESSING ENTER press the control and shift keys.

    Once they are both pressed press Enter.

18. Plot X5:X25 like before and think whether it's any different.

19. Why or why not?