[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]

A professor wishes to make up a true-false exam

with [math]n[/math] questions. She assumes that she can design the problems in such a way that a student will answer the [math]j[/math]th problem correctly with probability [math]p_j[/math], and that the answers to the various problems may be considered independent experiments. Let [math]S_n[/math] be the number of problems that a student will get correct. The professor wishes to choose [math]p_j[/math] so that [math]E(S_n) = .7n[/math] and so that the variance of [math]S_n[/math] is as large as possible. Show that, to achieve this, she should choose [math]p_j = .7[/math] for all [math]j[/math]; that is, she should make all the problems have the same difficulty.