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\bf 36-303: Sampling, Surveys \& Society\\
HW02: Due Tue Feb 8, 2011 in class
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\noindent\ul{\em Reminders:}
\begin{itemize}
\item Things to do:
\begin{description}
\item[Also Due Tues Feb 8:] Team Assig. I.3
  \begin{itemize}
    \item Choose single project to do based on my feedback for I.2
    \item Turn in revised versions of A, B, C, D, E, F, for the single
      project you will do this semester.  Details in
      ``project-schedule.pdf'' handout (available at the class
      website,\\ \verb|http://www.stat.cmu.edu/~brian/303|).
    \item Start on IRB application (but don't turn it in yet).  See
      irb materials at\\ \verb|http://www.stat.cmu.edu/~brian/303|.
   \end{itemize}
\item[Get started on...] Team Working Agreement, Due Feb 15.
\begin{itemize}
\item Read and do the exercises in the twa.pdf file available at
  \verb|http://www.stat.cmu.edu/~brian/303|. 
\item Turn in a completed team working agreement
  \begin{itemize}
  \item On paper and signed by everyone on your team.  Every team
    member should also have a signed copy.
  \item In email to me (perhaps not signed).
  \end{itemize}
\end{itemize}
\end{description}
\item Things to read:
\begin{itemize}
\item You should be reading Lohr, Appendix B (handout, also available
  at class website).
\item For next week: Groves, Ch's 7 \& 8.
\end{itemize}
\item Clear, careful writing and interpretation of results is an
  important part of both weekly homeworks and the projects.  {\em I
    always expect neatly typed or neatly handwritten work.}
\item Always be judicious about including computer output and graphs:
  show enough that we can clearly see what you are doing, but not so
  much that we will get lost or bored leafing through your work!
\item Mostly homeworks will be submitted on paper in class.

\end{itemize}


\noindent\ul{\em Exercises to Turn In (there are 4 excercises):}
\begin{enumerate}


\item Which of the following modes of data collection is most likely
  to produce question-order effects:
  % like Groves Ch 5 p 179 \#4, but added web and reverse-coded
\begin{itemize}
\item Face-to-face
\item telephone
\item Mail-out/mail-back
\item Web-based survey (like surveymonkey.com)
\end{itemize}
Why?

\item Let $X$ and $Y$ be discrete random variables with finite sample
  spaces $\{x_1,\ldots, x_K\}$ and $\{y_1,\ldots, y_K\}$, and let
  $p_{ij} = P[X=x_i ~\mbox{\em and}~ Y=Y_j]$. Use the definitions of
  $E[~]$, $Var(~)$, and summation notation ($\sum_{i=1}^K$), to show
\begin{enumerate}
\item $E[aX + bY + c] ~ = ~ aE[X] + bE[Y] + c$
\item $Var(aX + bY + c) = a^2Var(X) + 2abCov(X,Y) + b^2Var(Y)$
\item If $X$ and $Y$ are independent, then $E[X|Y=y]=E[X]$, for any
  $y$.
\end{enumerate}

% \clearpage

\item Recall the ``Randomized response'' model, from lecture, for the
  question:
\begin{center}
\fbox{\parbox{4in}{
Flip a coin, but don’t tell me whether it’s heads or tails.
\begin{itemize}
\item If heads, answer truthfully: have you ever cheated in a CMU class?
\item If tails, answer truthfully: is the last digit of your SSN odd?
\end{itemize}
}
}
\end{center}
Recall from lecture that 
\[
   \pi = \frac{\lambda - (1/2)\cdot(1-p)}{p}
\]
where $p=P[Heads]$; $\pi=P[Cheat]$; and $\lambda=P[Yes]$.  Consider a
SRS of $n$ students \ul{\em with replacement}\footnote{Surveys are
  seldom conducted this way, but it is easier for the math.}.  Let
$\hat\lambda$ be the fraction of ``Yes'' answers in the survey, and
let $\hat\pi = (\hat\lambda - \frac12(1-p))/p$.
\begin{enumerate}
\item Show that $E[\hat \pi]=\pi$.
\item Express $Var(\hat\pi)$ in terms of $Var(\hat\lambda)$ and show
  that, as $p$ gets closer and closer to 1, $Var(\hat\pi)$ gets closer
  and closer to $Var(\hat\lambda)$.
\item Suppose you use a fair coin, so that $p=\frac12$, and you think
  the true rate of cheating on campus is around 0.10.  How large a
  sample would you need, so that a 95\% confidence interval for $\pi$
  would be only 0.02 wide?
\end{enumerate}



% \item Groves, Ch 2, pp. 64--65. \#1. % \#5.

\item For the following, do not provide responses based on politics,
  social desirability, etc.  Instead, provide analytic responses based
  on the principles outlined in Groves, Ch 7, and/or in class.
\begin{enumerate}
\item Groves, Ch 7, p. 255 \#1.

\item Groves, Ch 7, pp. 255--256, \#5.

\item Groves, Ch 7, p. 256, \#6.

% \item Groves, Ch 7, pp 256--257, \#8.  
\end{enumerate}


\end{enumerate}

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