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\bf 36-303: Sampling, Surveys \& Society\\
HW02: Due Thursday Feb 11, 2010 in class
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\noindent\ul{\em Reminders:}
\begin{itemize}
\item Things to turn in:
\begin{itemize}
\item Due Tonight (Feb 4) by Midnight: Team Assig. I.2 (revising I.1,
  especially target pop, frame, mode)  -- email to brian@stat.cmu.edu
\item Due Mon Feb 8: Team Assig. I.3 (choosing which project to do) --
  email to brian@stat.cmu.edu
\begin{itemize}
\item {\bf Please wait for email feedback from me on I.2 (and instructions on
  I.3) before starting to work on I.3.}
\end{itemize}
\end{itemize}
\item Things to read:
\begin{itemize}
\item Done already: Groves, Ch 1, 2, 4 (first part), 5, 11 (first
  part); and Lohr Appx B
\item For next week (and this hw assignment): Groves Ch 7; after that,
  parts of 8 and 9
\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 You are operating under the same ethical guidelines as any
  intellectual worker: {\em beware of falsification, fabrication and
    plagiarism.}  Citing the source (person, book, etc) of material
  you use usually protects you against plagiarism claims.
\end{itemize}

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

\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:
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\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}
}
}
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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, pp. 238--239, \#4.  % \#1. 

\item Groves, Ch 7, p. 239, \#5. % \#4.

\item Groves, Ch 7, p. 240, \#6.  
\end{enumerate}

\end{enumerate}

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