exercise:3efd1e4d7d: Difference between revisions
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(Created page with "<div class="d-none"><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></div> 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 tha...") |
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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 | |||
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 | 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 | number of problems that a student will get correct. The professor wishes to choose | ||
Latest revision as of 22:20, 14 June 2024
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.