[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]

Let [math]S_n[/math] be the number of successes in [math]n[/math] independent

trials. Use the program BinomialProbabilities (Section \ref{sec 3.2}) to compute, for given [math]n[/math], [math]p[/math], and [math]j[/math], the probability

[[math]] P(-j\sqrt{npq} \lt S_n - np \lt j\sqrt{npq})\ . [[/math]]

• Let [math]p = .5[/math], and compute this probability for [math]j = 1[/math], 2, 3 and [math]n = 10[/math], 30, 50. Do the same for [math]p = .2[/math].
• Show that the standardized random variable [math]S_n^* = (S_n - np)/\sqrt{npq}[/math] has expected value 0 and variance 1. What do your results from (a) tell you about this standardized quantity [math]S_n^*[/math]?