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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 206: Quiz 1 \textit{[40 total points]}}

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Name: \underline{\hspace*{5.85in}}

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Here is Your background information, translatable into $\mathcal{ B }$, for this problem.

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\item

\textit{(Fact 1)} As a broad generalization (which you can verify empirically), statisticians tend to have shy personalities more often than economists do --- let's quantify this observation by assuming that 80\% of statisticians are shy but the corresponding percentage among economists is only 15\%. 

\item

\textit{(Fact 2)} Conferences on the topic of \textit{econometrics} are almost exclusively attended by economists and statisticians, with the majority of participants being economists --- let's quantify this fact by assuming that 90\% of the attendees are economists (and the rest statisticians). 

\end{itemize}

Suppose that you (a physicist, say) go to an econometrics conference --- you strike up a conversation with the first person you (haphazardly) meet, and find that this person is shy. The point of this problem is to show that the (conditional) probability $p$ that you're talking to a statistician, given this data and the above background information, is only about 37\%, which most people find surprisingly low, and to understand why this is the right answer. Let $St$ = (person is statistician), $E$ = (person is economist), and $Sh$ = (person is shy).

\begin{itemize}

\item[(a)]

Identify (in the form of a proposition $B_1$, one of the elements of $\mathcal{ B }$) the most important assumption needed in this problem to permit 
its solution to be probabilistic; expain briefly. \textit{[5 points]}

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\item[(b)]

Using the $St$, $E$ and $Sh$ notation, express the three numbers (80\%, 15\%, 90\%) above, and the probability we're solving for, in conditional probability terms, remembering to condition appropriately on $\mathcal{ B }$. \textit{[5 points]}


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\item[(c)]

Briefly explain why calculating the desired probability is a good job for Bayes's Theorem. \textit{[5 points]}

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(over)

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\item[(d)]

Briefly explain why the following expression is a correct use of Bayes's Theorem in odds form in this problem. \textit{[5 points]} \vspace*{-0.1in}

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\Large

\[
\begin{array}{ccccc}
\left[ \frac{ P ( St \given Sh , \, \mathcal{ B } ) }{ P ( E \given Sh , \, \mathcal{ B }) } \right] & = & \left[ \frac{ P ( St \given \mathcal{ B }) }{ P ( E \given \mathcal{ B } ) } \right] & \cdot & \left[ \frac{ P ( Sh \given St , \, \mathcal{ B }) }{ P ( Sh \given E , \, \mathcal{ B }) } \right] \\
( 1 ) & = & ( 2 ) & \cdot & ( 3 )
\end{array}
\]

\normalsize

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\item[(e)]

Here are three terms that are relevant to the quantities in part (d) above:

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\item

(Prior odds in favor of $St$ over $E$ given $\mathcal{ B }$)

\item

(Bayes factor in favor of $St$ over $E$ given the data and $\mathcal{ B }$)

\item

(Posterior odds in favor of $St$ over $E$ given the data and $\mathcal{ B }$)

\end{itemize}

Match these three terms with the numbers $( 1 ), ( 2 ), ( 3 )$ in the second line of the equation in part (d). \textit{[5 points]}

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\item[(f)]

Compute the three odds values in part (e), briefly explaining your reasoning, thereby demonstrating that the posterior odds value $o$ in favor of $St$ over $E$, given the data and $\mathcal{ B }$, is $o = \frac{ 16 }{ 27 } \doteq 0.593$. \textit{[5 points]}

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\item[(g)]

Use the expression $p = \frac{ o }{ 1 + o }$ to show that the desired probability in this problem --- the conditional probability that you're talking to a statistician, given the data and the background information --- is $p = \frac{ 16 }{ 43 } \doteq 0.372$. \textit{[5 points]}

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\item[(h)]

Someone says, ``That probability can't be right: 80\% of statisticians are shy, versus 15\% for economists, so your probability of talking to a statistician has to be over 50\%.'' Briefly explain why this line of reasoning is wrong, and why $p$ should indeed be less than 50\%. \textit{[5 points]}

\end{itemize}

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