[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

In general, the Central Limit Theorem gives a better estimate than

Chebyshev's inequality for the average of a sum. To see this, let [math]A_{25}[/math] be the average calculated in Exercise Exercise, and let [math]N[/math] be the normal approximation for [math]A_{25}[/math]. Modify your program in Exercise Exercise to provide a table of the function [math]F(x) = P(|A_{25} - 10| \geq x) = {}[/math] fraction of the total of 1000 trials for which [math]|A_{25} - 10| \geq x[/math]. Do the same for the function [math]f(x) = P(|N - 10| \geq x)[/math]. (You can use the normal table, Table \ref{tabl 9.1}, or the procedure NormalArea for this.) Now plot on the same axes the graphs of [math]F(x)[/math], [math]f(x)[/math], and the Chebyshev function [math]g(x) = 4/(3x^2)[/math]. How do [math]f(x)[/math] and [math]g(x)[/math] compare as estimates for [math]F(x)[/math]?