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Prof.~David Draper \\
Department of Applied Mathematics and Statistics \\
University of California, Santa Cruz
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\textbf{\large AMS 131: Quiz 3}
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Name: \underline{\hspace*{5.85in}}
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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.
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\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}
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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,
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\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~...).
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\item[(b)]
Given that the chosen person is Female, what's the conditional probability $P ( Y \given F )$ that she responded Yes? Explain briefly.
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\item[(c)]
Given that the chosen person is Male, what's the conditional probability $P ( Y \given M )$ that he responded Yes? Explain briefly.
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(over)
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\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.)
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\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.
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