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\begin{center}
\bf 36-303: Sampling, Surveys \& Society\\
HW06: Hand in on Blackboard Thu Apr 12, 2012
\end{center}

\noindent\ul{\em Reminders \& Announcements:} (dates corrected)
\begin{itemize}
\item Fri Apr 6: 
\begin{itemize}
\item Drafty draft final reports due 1 pdf or 
doc file per team - email or submit on Blackboard)
\item HW 05 due - Submit on Blackboard
\end{itemize}
\item Thu Apr 12: 
\begin{itemize}
\item Midterm Review
\item HW 06 (last one!) due - Submit on Blackboard 
\end{itemize}
\item Tue Apr 17: Final Midterm Exam (in class; cumulative)
\item Thu Apr 19: Carnival! No Class
\item Apr 24, Apr 26: TBA
\item May 1, 3: Final Project Presentations
\item Fri May 4: 
\begin{itemize}
\item Final Papers due (1
  pdf or doc file per group - email or submit on Blackboard)
\item Second set of Peer Evaluations due (1 email to me, per class
  member, containing peer evaluations for each member of your group,
  as attachments)
\end{itemize}
\item May 9: Meeting of the Minds (for those participating)
\end{itemize}


\noindent\ul{\em Exercises to Turn In (there are 2 exercises):}
\begin{enumerate}

\comment{

% THere is a good raking exercise here but I am too tired to put it
% together.


% should be like an example in Lohr 2nd ed. but I left the book at
% school; I will use the college work hrs data from the handout on 
% post-stratification...
\item {\bf Raking.}\label{raking} At a
  certain university there are 1998 undergraduates.  An SRS without
  replacement, of size $n=20$ is taken, to find out what is the
  average number of hours per week the students study.  Some
  demographic data was collected also.  Here is the data from 20
  respondents in the survey:

\begin{center}
\begin{tabular}{@{}c@{\hspace*{4em}}c@{}}
\begin{tabular}{ccc}
Sex & College & Hrs/Wk \\
\hline
\hline
M & Eng & 28 \\
M & Eng & 29 \\
M & Eng & 23 \\
M & Eng & 35 \\
M & Eng & 29 \\
M & Eng & 30 \\
M & Eng & 34 \\
M & Eng & 31 \\
F & Eng & 30 \\
F & Eng & 31 \\
\hline
\end{tabular} 
&
\begin{tabular}{ccc}
Sex & College & Hrs/Wk \\
\hline
\hline
F & Eng & 36 \\
F & Eng & 33 \\
M & Lib & 27 \\
M & Lib & 28 \\
F & Lib & 29 \\
F & Lib & 30 \\
F & Lib & 28 \\
F & Lib & 28 \\
F & Lib & 32 \\
F & Lib & 30 \\
\hline
\end{tabular}
\end{tabular}
\end{center}

\begin{itemize}

\item Administrative records show that the total undergrad enrollment
  at the university is
\begin{center}
\begin{tabular}{c|cc}
Sex & Eng & Lib \\
\hline
M & 617 & 380 \\
F & 450 & 551 \\
\end{tabular}
\end{center}
Make four post-strata for the sample by combining sex and college of
the students, compute post-stratum weights.

\item 

\end{itemize}

}


% taken from an example in Ch 4 of Lohr 2nd ed.
\item {\bf Taylor Series Variance Estimates.}\label{taylor} An SRS
  without replacement of 25 households in a particular region of the
  city reveals the following data:
\begin{center}
\def\z{\phantom{0}}
\begin{tabular}{ccp{1.5in}}
Number of persons & Number of households & Household \\
in household      & in SRS               & income (in \$) \\
\hline
\hline
1                 & \z3                  & 48111 36582 38245 \\
2                 & \z6                  & 46246 50309 58373 58800
                                           58244 36898 \\
3                 & \z7                  & 63274 60458 73379 61650
                                           73983 60696 62211 \\
4                 & \z7                  & 67884 61633 88080 69376
                                           68719 55496 77201 \\
5 or more         & \z2                  & 92149 78404 \\
\hline
Total             & 25                   & % 1546402

\end{tabular}
\end{center}

\begin{enumerate}
\item Estimate the mean household income from this SRS, and give the
  standard error.

\item From US Census data, the distribution of household size in the region
is known:
\begin{center}
\begin{tabular}{cc}
Number of persons & Percentage of  \\
in household      & Households     \\
\hline
\hline
1 & 25.75 \\
2 & 31.17 \\
3 & 17.50 \\
4 & 15.58 \\
5 or more & 10.00 \\
\hline
\end{tabular}
\end{center}
Using the household sizes as post-strata, compute post-stratum weights
to make the sample more representative of the population, and compute
the weighted average household income.

\item Use the Taylor Series method discussed in class to compute a
  standard error for your weighted estimate in part (b).  Compare to
  the standard error you computed in part (a).

\end{enumerate}




\item {\bf Jackknife Variance Estimates.}\label{jackknife} Use the
  jackknife method discussed in class to compute a standard error for
  the weighted estimate in problem \ref{taylor}(b).  Compare to the standard
  errors you calculated in problems \ref{taylor}(a) and \ref{taylor}(c).

\item[]
\ul{\em Notes on \#\ref{taylor} and \#\ref{jackknife}} {\em: The
  sample size $n=25$ is small enough that you can do all the
  computations ``by hand'' using Minitab, R, Excel, etc.  If you know
  R and you want to compare your answers with the results you get from
  the R functions I introduced in class, please feel free to do so.}

\comment{

% A quesion not asked (but would be good)  [might need a little
% attention to notation and wording...

\item {\em Jackknifing a simple sample.}  Suppose that we have only
  one post-stratum, with sample size $n$, and suppose the sampled
  observations $y_1, \ldots, y_n$ are an SRS {\bf with} replacement,
  so we can ignore fpc's.
\begin{enumerate}
\item Show that for each leave-one-out sample of size $n-1$, the $n-1$
  post-stratification weights $w_i^{(r)}$ are all equal---and so they
  might as well be equal to 1 in the weighted average:
\[
    \bar y^{(r)}_w = \frac{\sum_{i\neq r} w_i^{(r)} y_i}{\sum_{i\neq r} w_i^{(r)}}
\]
\item Show that in this one-stratum case, the jackknife estimate of the
  population mean equals the SRS estimate:
\[
    \bar y_{JK} = \frac1n\sum_{r=1}^n \bar y^{(r)}_w = \frac1n
    \sum_{i=1}^n y_i = \bar y_{srs}
\]
\item Show that in this one-stratum case, the jackknife estimate of
  the variance equals the SRS estimate:
\[
    s^2_{JK} = \frac{n-1}n \sum_{r=1}^n (\bar y^{(r)}_w - \bar
    y_{jk})^2 = \frac1{n-1} \sum_{i=1}^n (y_i - \bar y_{srs})^2 = s^2_{srs}
\]

\end{enumerate}

}

\end{enumerate}

\end{document}
