Simulation assignments and random numbers

You will be asked to do several simulation assignments. To do those assignments you must generate random numbers in Excel. For example, using the Randbetween(0,9) function.

Because each time Excel calculates the spreadsheet the numbers change, it may be frustrating to generate statistics based on those numbers. For example if you calculate the mean of the numbers you get a slightly different result each time you try it.

To avoid this problem I suggest that after you generate the random numbers the first time, copy them (i.e., highlight them, right click and choose copy) and then paste them to another part of the spreadsheet by using the PASTE SPECIAL option. In that option choose "values" and then "ok."

If anyone has a better way of doing this please let me know (for extra credit, of course).

There are 8 parts to this assignment. Please label them in your report to me as Part 1, Part 2, etc.

In chapter 5 you will learn about the binomial random variable.  Among other applications, this predicts the probability of flipping four fair coins and getting 0, 1, 2, 3 or 4 heads. In the terminology of Chapter 5, P(X=0), P(=1), etc.

Part 1:

After learning the binomial probability formula I want you to calculate these 5 probabilities. Make sure they add up to 1.00 as all possibilities are represented in 0, 1, 2, 3, 4 and 5. Express the results as a probability distribution.

Part 2:

After learning the other main formula of chapter 5, E(X), calculate the Expected value of this random variable. Common sense dictates that the answer is 2 because when you flip 4 coins you ‘expect’ to get 2 heads and 2 tails on the average. Verify it by the E(X) formula AND by the ‘shortcut’ E(X) = np formula

Part 3:

Calculate the standard deviation of the theoretical distribution, σ = Σ(x-u) ² P(X). Also do this by the ‘shortcut’ binomial formula for σ = √np(1-p). The answer in both cases should be exactly 1.

Part 4:

Then simulate flipping 4 fair coins 100 times. You can do this by physically flipping the coins or, as I want you to do it, by using Excel.

In Excel there is a function which generates random integers between any two points. If you type into the A1 cell the Excel formula (by using "=") Randbetween(0,1) the computer will fill that cell with a 0 or 1 with 50/50 probability. By copying that formula into the cells A1, B1, C1 and D1 you should see: 0 0 1 0, for example, or 0 1 0 1.

If you interpret a 0=tails and 1=heads the computer can do the dirty work of flipping 4 coins. Then put in the E1 cell the sum of A1, B1, C1 and D1. This represents how many heads were observed. In the first example you would see a "1" and in the second example a "2." If 1 1 1 1 were seen that represents getting 4 heads.

Copy (by dragging the corner of the cells) the first row 100 times. When you are finished the E column should contain the numbers 0 or 1 or 2 or 3 or 4 representing the number of heads flipped by the computer in that row.

Show me the results of this 100x5 table

Part 5:

Since the probability of getting all 4 heads, according to the binomial formula, is 1/16 or 6.25%, approximately 6 out of 100 rows should contain a "4." Create a brief table that compares the theoretical binomial probability to what was actually ‘flipped’ by the computer for 0 heads, 1 head, 2 heads, 3 heads and 4 heads.

Part 6:

Calculate the actual mean or average of column E and compare this number to the theoretical E(X).

Part 7:

Calculate the actual standard deviation of column E and compare this number to the theoretical σ.

Part 8:

Finally, explain any discrepancy between the theoretical and actual results for the probabilities and E(X) and tell me how you would modify the assignment so that the discrepancy is smaller.

Price: $19.16

Solution: The downloadable solution consists of 10 pages, 916 words and 5 charts.
Deliverable: Word Document