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\bf 36-303: Sampling, Surveys \& Society\\
HW03: Due Tue Feb 14, 2012 on Blackboard
\end{center}

\noindent\ul{\em Reminders:}
\begin{itemize}
\item Things coming up:
\begin{description}
\item[Due Thu Feb 16:] Team Assig. I.3
  \begin{itemize}
    \item Choose single project to do, {\em  based on my feedback for I.2}.
    \item Turn in revised versions of A, B, C, D, E, F, and G 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[Due Thu Feb 23:] Team Working Agreement.
\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 Submit a copy on Blackboard to me.
  \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 Blackboard.

\end{itemize}

\noindent\ul{\em Exercises to Turn In (there are 5 exercises):}
\begin{enumerate}

% \item Please answer the following questions:
% \begin{enumerate}


\item Consider a set of $n$ trials involving independent and
identically distributed random variables $X_1, X_2,\ldots, X_n,$ where
the outcome of $1$ corresponds to success and the outcome of $0$
corresponds to failure,
and the probability of success on each trial is $p$.
\begin{enumerate}  \parskip=0pt
\item Show that $E(X_i)=p$, and $\Var(X_i)=p(1-p)$ for $i=1,2,\ldots,n$.

\item Let $Y= X_1+ X_2 +\ldots + X_n.$ Show that $E(Y)=np$, and
$\Var(Y)=np(1-p)$.

\item Let $\hat{p}=Y/n$.  Show that $E(\hat{p})=p$, and
$\Var(\hat{p})=p(1-p)/n$.
\end{enumerate}
Use facts from lecture notes and/or the Lohr Appendix B handout to justify your 
reasoning for these problems.

\clearpage

\item We say that two discrete random variables are {\em independent}
  if $P[X=x,Y=y] = P[X=x]\times P[Y=y]$ for all specific values $x$
  and $y$ that $X$ and $Y$ could be.
\begin{enumerate}

\item The table below shows the distribution for two random variables
  $X$ abd $Y$.  For example, \\ $P[X=1, Y=3] = 3/8$.
\begin{center}\spacing{1.5}
\begin{tabular}{ccc}
$x$ & $y$ &  $P[X=x,Y=y]$ \\
\hline
 1 & 4 &  $\frac18$ \\
 1 & 3 &  $\frac38$ \\
 2 & 4 &  $\frac38$ \\
 2 & 3 &  $\frac18$
\end{tabular}
\end{center}
Use the information in the table to show whether $X$ and $Y$ are
independent, or not.

\item For any two discrete random variables $X$ and $Y$ (not
  necessarily the ones in part (a)), show that \underline{\bf If} $X$
  and $Y$ are independent, \underline{\bf then} $P[X=x|Y=y] = P[X=x]$,
  for any specific $x$ and $y$ that $X$ and $Y$ could be.
  
  

\end{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

\comment{

\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 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?



\comment{

\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}

}

\item The Current Population Survey (CPS, http://www.census.gov/cps/), 
sponsored jointly by the U.S. Census Bureau and the U.S. Bureau 
of Labor Statistics (BLS), is the primary source of labor force 
statistics for the population of the United States. The CPS is 
the source of numerous high-profile economic statistics, including 
the national unemployment rate, and provides data on a wide range 
of issues relating to employment and earnings. 
\begin{enumerate}
\item Answer items E, and F from the class project outline, for
the CPS.  Be particularly clear about the target population, and 
use terms from Groves Ch 5 to specify the mode of data collection 
as carefully as possible.
\item The unemployment rate is one of the key variables that the 
CPS produces.  By exploring the CPS website (or other websites), 
find the exact definition of the unemployment rate that the CPS
uses, including who is eligible to be included in this calculation, 
and who is not.  Comment on whether this is a suitable measure of 
unemployment.  Give a list or references or websites you used for
this exercise.

\end{enumerate}


\end{enumerate}


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