Ph 490

3/2/2004 11:52 AM

Problem Set 4, due Tuesday March 8, 2004

Problem 1. The goal of this problem is to carry out a test of the Central Limit Theorem by showing that the sum of a few errors, generated from non-Gaussian distributions, is a good approximation to a Gaussian distribution. For the properties of the Gaussian, refer to Bevington and Robinson, Ch. 2, or the tables at the end of the write-up for Lab B2.

The test will be carried out using Excel to generate and process a fairly large number of uniformly distributed random numbers. We will make histograms of individual random numbers and of sums of two, four, and nine random numbers. The histogram of the sum of nine random numbers will be tested to see if it has become a good approximation to a Gaussian. (Note that adding random numbers is a way of simulating a "random walk.")

(a) Start by opening an Excel spreadsheet. In cell B11, enter

rand()-0.5

This generates a random number between -0.5 and 0.5, whose average value is 0. Drag this formula over to cell J11 and down to cell 1010. Label columns B through J as N1 through N9. These represent nine different samples of 1000 random numbers.

Now in cell enter

in cell K11: =sum(B11:C11)/sqrt(2)

in cell L11: =sum(B11E11)/sqrt(4)

in cell M11: =sum(B11:J11)/sqrt(9)

and drag these three columns all the way down. These three columns represent sums of 2, 4, and 9 numbers. Label them S2, S4, and S9.

Next, go to sheet 2 and make some histograms. First enter the numbers -1., -.9, -.8, . . . .9, 1. in cells B11 to B31. These are the limits for the bins of the histograms. Now make histograms of N1, S2, S4, and S9, as follows. Click on Tools, Data Analysis, and Histogram. [If Data Analysis is not there, go to Tools, Add Ins, and add in the Analysis Toolpack.] In the Histogram dialogue box, enter the following information:

Data: sheet1!B11:B1010

Bin Range: B11:b31

Output Range: C10

check Chart Output

Then click OK. You should get a histogram of numbers running from -.5 to .5, with a flat distribution.

Next repeat for:

S2: data in K11-K1010, output in E10

S4: data in K11-L1010, output in G10

S9: data in M11-M1010, output in I10

(b) Visually evaluate whether this series seems to be converging to a Gaussian.

(c) Make a numerical test of the Gaussian approximation for the four distributions by calculating the average value and standard deviation of the mean (Average and Stdev, for Excel), and then seeing if the expected number of entries lies outside of the interval

[-s,s].