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\begin{center}
\bf 36-303: Sampling, Surveys \& Society\\
HW05: Hand in on Blackboard Thu Apr 06, 2012
\end{center}

\noindent\ul{\em Reminders:}
\begin{itemize}
\item Please see Lecture 18, Week 09, for an outline of events for the
  rest of the semester.
\item Reading:
\begin{itemize}
\item Puetchel Handout on Oral Presenations (and another one on making
  attractive numerical tables!)
\item Bem Handout on Writing Papers
\item Examples of past 303 papers
\end{itemize}
These are all under the link ``Guides'' near the ``week09'' link on
http://www.stat.cmu.edu/\~{}brian. 

% \item Exercises \#\ref{taylor} and \#\ref{jackknife} depend on
%   material I will present in lecture Mar 31 and Apr 05.

\end{itemize}

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

\item True/False:
\begin{enumerate}
\item To reduce the margin of error in a stratified sample, we should
  design strata so that the units within each stratum are as similar
  as possible.

\item To reduce the margin of error in a cluster sample, we should
  design clusters so that the units within each cluster are as similar
  as posslbe.
\end{enumerate}

% 3 of 6 situations in lohr, p 118, \#1
\item How would you stratify the population to produce smaller margins
  of error, for each of the following surveys?  Name the stratifying
  variable(s) you would use for each survey.

\begin{enumerate}

\item A political poll to estimate the percentage of registered voters
  in Arizona who approve of the job the governor is doing.

\item A telephone survey of undergraduates at Carnegie Mellon, to
  estimate the total amount of money students spend on textbooks.

\item A sample of prime-time (8pm--11pm Monday through Saturday, and
  7pm--11pm Sunday, in the US Eastern time zone) TV programs on CBS,
  to estimate the average number of promotional announcements (ads for
  other programs on the same network) per hour of broadcast.

\end{enumerate}

% Lohr, p 119, \#3.  Skipped her part b which has an interesting
% followup survey.
\item Hard-shell clams can be sampled using a dredge (a large machie
  that scrapes material off the seabed in shallow waters).  Clams do
  not tend to be uniformly distributed in a body of water however,
  because some areas provide better habitats than others, so taking an
  SRS is likely to result in a large estimated variance for the number
  of clams in the area.  Russell (1972) used stratified random
  sampling to estimate the total number of bushels of hard-shell clams
  in Narragansett Bay, Rhode Island.  The area of interest was divided
  into four strata based on preliminary surveys that identified areas
  in which clams were more, or less, abundant.  Then, $n_h$ dredge
  tows were made in each stratum $h$, for $h=1,2,3,4$.  The acreage
  for each stratum was known, and Russell calculated that
% the area fished during a standard
%  dredge tow was 0.039 acres---that is, 
  it would take 25.6 dredge tows
  to cover one whole acre of seabed.\\
\mbox{}\hfil{\sl [Continued on next page]}

\begin{enumerate}

\item Here are the results from the survey taken before the commercial
  fishing season.  Estimate the total number of bushels of clams in
  the area, and give the standard error of your estimate.  {\em Hint:
    First calculate $N_h$, the number of dredge tows needed to cover
    each stratum $h$, for $h=1,2,3,4$.  }
\begin{center}\def\z{\phantom{0}}
\begin{tabular}{ccccc}
        &         &           & Average Number of & Sample \\
        & Area    & Number of & of Bushels & Variance \\
Stratum & (Acres) & Tows Made & per Tow & for Stratum \\
\hline
\hline
1 & 222.81 & 4 & 0.44 & 0.068 \\
2 &\z49.61 & 6 & 1.17 & 0.042 \\
3 &\z50.25 & 3 & 3.92 & 2.146 \\
4 & 197.81 & 5 & 1.80 & 0.794 \\
\hline
\end{tabular}
\end{center}

% for the SRS variance I took the population-weighted average of 
% the sample stratum variances and doubled it, so students would
% have something to say...
\item The sample variance for an SRS of 18 dredge tows of the area
  represented by these four strata is estimated to be 1.084.  What is
  the design effect in this case?  Was it better to do a stratified
  sample than an SRS?  By how much?

\end{enumerate}

% Lohr, p 172 \#12, modified to be one-stage clustering (orig has 24
% cans per pack and it makes a simple 2-stage clustering problem)...
\item An inspector samples cans from a truckload of canned cream corn
  to estimate the average number of worm fragments per can.  The truck
  contains 580 three-packs (three cans per package).  The inspector
  samples 12 three-packs at random, opens all the cans, and finds the
  following number of worm fragments in each pack:
\begin{center}
\begin{tabular}{ccccccccccccc}
     & \multicolumn{12}{c}{Three-Pack} \\
Can  & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
\hline
1 & 1 & 4 & 0 & 3 & 4 & 0 & 5 & 3 & 7 & 3 & 4 & 0 \\
2 & 5 & 2 & 1 & 6 & 9 & 7 & 5 & 0 & 3 & 1 & 7 & 0 \\
3 & 7 & 4 & 2 & 6 & 8 & 3 & 1 & 2 & 5 & 4 & 9 & 0 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}

\item Explain why this is a cluster sample and not a stratified
  sample.

\item Estimate the mean number of worm fragments per can, along with
  the standard error of your estimate.  {\em Hint: This follows very
    closely the example in the handout on clustered sampling.}

\item Compute the design effect for this cluster sample.

\item Estimate the {\em intraclass correlation}---that is, the
  correlation betwen worm fragment counts in cans from the same
  three-pack---using ideas from class.

\end{enumerate}

\end{enumerate}

\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\item {\bf Cluster Sampling (Lohr, 1999, p. 170).}  An accounting firm
  is interested in estimating the error rate in a compliance audit it
  is conducting.  The population consists of 828 claims and the firm
  audits an SRS of 85 of those claims.  In each of the 85 sampled
  claims, 215 fields are checked for errors.  We can think of the
  claims as psu's (clusters) and the fields as ssu's (units), in a
  single-stage cluster sample.

Several clusters had the same number of errors, so the results of this
single-stage cluster sample can be summarized as follows:
\begin{center}
\def\z{\phantom{0}}
\begin{tabular}{ccc}
Number of claims & Number of errors & $\bar y_i$ \\
(clusters)       & out of 215 fields & $=$ (\#errors)/(\#fields) \\
\hline
\hline
\z1  & 4  & 0.01860\\
\z1  & 3  & 0.01395\\ 
\z4  & 2  & 0.00930\\ 
22 & 1  & 0.00465\\
57 & 0  & 0.00000\\
\hline
\end{tabular}
\end{center}
(data courtesy of Fritz Scheuren).  
\begin{enumerate}
\item Use the ideas about cluster sampling discussed in class and in
  the handout on cluster sampling, to estimate the {\em proportion} of
  all 828$\times$215=178,020 fields with errors in them.  Give a
  standard error for your estimate.

\item Use your answer in part (a) to estimate, and give a standard
  error for, the {\em total} number of errors in all 178,020 fields.

\item Suppose that, instead of taking a cluster sample, the firm takes
  an SRS of $(85)\cdot(215)=18,275$ fields from the $178,020$ fields
  in the population.  If the estimated error rate is the same as in
  part (a), what would the estimated variance be?  How does this
  compare to the estimated variance in part (a)?

\end{enumerate}




\item {\bf Taylor Series Variance Estimates.}\label{taylor} 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{enumerate}
\item Estimate the mean number of hours/wk students study at this
  university, and give the standard error.

\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 by combining sex and college of the students,
compute post-stratum weights to make the sample more representative of
the population, and compute the weighted average number of hours/wk
students study.

\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 $n=20$ 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}

}

