Normal Approximation for Binomial Distributions

The Exercise:

• Let X be a binomial random variable with parameter n = 10 and p = 0.9 .

To study the distribution of X , first use simulation to generate one DataDesk variable containing 1000 realizations of X (Computer Hint: this is done using the menu entry Generate Random Numbers... under the Manip menu. Specify 1 variable with 1000 cases, and select binomial experiments. Input the number of Bernoulli trials per experiment n = 10, and probability of success p = 0.9).

1. Plot the histogram of X ‡. is this picture symmetrical †?
2. Now let the parameter n be 20 and keep p = 0.9 and generate a new binomial random variable with 1000 cases. Plot the histogram ‡.
3. Repeat (2.) with n = 50, n = 100, and n = 2000 without changing p.
   1. Plot histograms ‡ (print all of the histograms you generate in a layout) and carefully describe what happens to the shape, center and spread of the histograms †.
   2. What are the theoretical values of the mean and standard deviation of X if X is a binomial random variable with parameters n and p †?
   3. Do your histograms reflect these theoretical values †?

Many of the results here illustrate the important theorem which lies at the center of work in statistics: The Central Limit Theorem coming soon to a classroom near you!

To Turn in:

• Please print out the results marked with ‡.
• Answer all questions marked with †.
• Hand in your completed assignment when your TA asks for it during lab next week.