\documentclass[12pt]{article} \addtolength{\textheight}{2.5in} \addtolength{\topmargin}{-0.75in} \addtolength{\textwidth}{1.0in} \addtolength{\evensidemargin}{-0.5in} \addtolength{\oddsidemargin}{-0.5in} \setlength{\parskip}{0.1in} \setlength{\parindent}{0.0in} \newcommand{\given}{\, | \,} \pagestyle{empty} \raggedbottom \begin{document} \begin{flushleft} Prof.~David Draper \\ Department of Applied Mathematics and Statistics \\ University of California, Santa Cruz \end{flushleft} \begin{center} \textbf{\large AMS 131: Quiz 3} \end{center} \bigskip \begin{flushleft} Name: \underline{\hspace*{5.85in}} \end{flushleft} 1.~In the spring of 1993 I taught an introductory statistics class at UCLA. One of the things I did to generate data for analysis in the class was to conduct a (voluntary) survey of the students at the beginning of the quarter: I asked some demographic questions, including gender, and some political questions, including ``Are you in favor of the legalization of marijuana?'' Let's agree to code the gender variable as Female ($F$) or Male ($M$), and the marijuana legalization preference (MLP) variable as Yes ($Y$) or No ($N$). A total of 106 students responded to the survey; the results are summarized in the table below. \begin{center} \begin{tabular}{c|c|c|c} \multicolumn{1}{c}{ } & \multicolumn{2}{c}{MLP} \\ \multicolumn{1}{c}{Gender} & \multicolumn{1}{c}{$Y$} & \multicolumn{1}{c}{$N$} & Total \\ \cline{2-3} $F$ & 29 & 20 & \, 49 \\ \cline{2-3} $M$ & 52 & \, 5 & \, 57 \\ \cline{2-3} \multicolumn{1}{c}{Total} & \multicolumn{1}{c}{81} & \multicolumn{1}{c}{25} & \multicolumn{1}{c}{106} \\ \end{tabular} \end{center} In other words, 29 Female students said Yes (upper left cell), and there were a total of 25 people who said No (second column total). In parts (a), (b) and (c), if a student is chosen at random from these 106 survey participants, \begin{itemize} \item[(a)] What's the probability $P ( Y )$ that the chosen person responded Yes to the MLP question? Explain briefly (for example, the right denominator for this probability is $m$ because~..., and the right numerator is $n$ because~...). \vspace*{0.7in} \item[(b)] Given that the chosen person is Female, what's the conditional probability $P ( Y \given F )$ that she responded Yes? Explain briefly. \vspace*{0.7in} \item[(c)] Given that the chosen person is Male, what's the conditional probability $P ( Y \given M )$ that he responded Yes? Explain briefly. \vspace*{0.7in} \end{itemize} \begin{center} (over) \end{center} \newpage \vspace*{0.05in} \begin{itemize} \item[(d)] Briefly explain why this demonstrates that gender and marijuana legalization preference are (probabilistically) \textit{dependent} in this data set, and briefly describe the nature of the dependence. (\textit{Hint:} What would have been true if these two variables had been \textit{independent}? Think like a Bayesian.) \vspace*{1.0in} \item[(e)] Would you describe the degree of dependence in (d) as weak, moderate or strong? Use your results in parts (a), (b) and (c) to justify your answer. \end{itemize} \end{document}