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\begin{center}
\large\bf
36-303: Sampling, Surveys and Society\\
Exam 2\\
Tue Apr 13, 2010
\end{center}

\vspace*{0.5in}

\begin{itemize}\parsep=0pt\parskip=0pt\itemsep=0pt
\item You have 80 minutes for this exam.
\item The exam is closed-book, closed notes.
\item A calculator is allowed.
\item {\bf Two formula sheets are provided  for your
  convenience.}
\item Please write all your answers on the exam itself; your work must
  be your own.
\end{itemize}

\vspace*{0.1in}


\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\bf Question & \bf Points Possible & \bf Points Earned \\ \hline
1 & 20 & \\ 
  &    & \\ \hline
2 & 20 & \\ 
  &    & \\ \hline
3 & 18 & \\ 
  &    & \\ \hline
4 & 24 & \\ 
  &    & \\ \hline
5 & 18 & \\ 
  &    & \\ \hline
Total & 100 & \\ 
  & & \\ \hline
\end{tabular}
\end{center}

\vspace*{\fill}

\noindent
Name: ~~~~~\rule{4.25in}{0.1pt}

\vspace*{\fill}

\noindent
Signature: \rule{4.25in}{0.1pt}

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\begin{center}
\large\bf Some Useful Formulas From the Statistics of Survey Sampling, I
% \\
% \normalsize Do not bring this to the exam.  A copy will be provided for you.
\end{center}

\noindent
{\bf Equally-Likely Outcomes \& Counting}

\begin{itemize}

\item If $K$ outcomes $O_1, \ldots, O_K$ are equally likely, then the
  probability of any one of them is $1/K$. 

\item Consider taking a sample of $n$ objects from a population of $N$
  objects. 
\begin{itemize}
\item Sampling with replacement, there are $N^n$ possible samples of
  size $n$; the probability of any one of them is $1/N^n$.
\item Sampling without replacement, there are 
$
	{N \choose n} = \frac{N!}{n!(N-n)!}
$ 
possible samples of size $n$ [where
$N!=N\cdot(N-1)\cdot(N-2)\cdots3\cdot2\cdot1$], so the probability of
any one of them is $1\left/{N\choose n}\right.$.
\end{itemize}

\end{itemize}

\noindent
{\bf Discrete Random Variables}

Let $X$ and $Y$ be random variables with sample spaces
$\{x_1,\ldots,x_K\}$ and $\{y_1,\ldots,y_K\}$ and distributions
\begin{eqnarray*}
  P[X=x_i,Y=y_j] ~ = ~ p_{ij} ~~,~~~
  P[X=x_i]      ~  = ~ p_{i\cdot} ~ = ~ \sum_{j=1}^K p_{ij} ~~,~~~
  P[Y=y_j]       ~ = ~ p_{\cdot j} ~ = ~ \sum_{i=1}^K p_{ij}
\end{eqnarray*}
Then, for example
\begin{eqnarray*}
  E[X] ~ = ~ \sum_{i=1}^K x_i p_{i\cdot} ~~,~~~
  Var(X) ~ = ~ \sum_{i=1}^K (x_i  - E[X])^2p_{i\cdot} &,~&
  Cov(X,Y) ~ = ~ \sum_{i=1}^K (x_i  - E[X])(y_i-E[Y])p_{ij}\\
  P[X=x_i|Y=y_j] ~ = ~ p_{ij}/p_{\cdot j}~~,~~~
  E[X|Y=y_j] ~ = ~ \sum_{i=1}^K x_i P[X=x_i|Y=y_j] &,~&
  E[aX + bY + c] ~ = ~ aE[X] + bE[Y] + c
\end{eqnarray*}

\noindent
{\bf Random Sampling From a Finite Population}

Consider a population of size $N$ and a sample of size $n$.  Let $y_i$
be the (fixed) values of some variable of interest in the population
(such as a person's age, or whether they would vote for Obama).  Let
\[
	Z_i = \left\{\begin{array}{l}
        1, \mbox{~if $i$ is in the sample}\\
        0, \mbox{~else}
        \end{array}\right.
\]
be the random sample inclusion indicators, and let $Y_i$ be the random
observations in the sample.  Then the sample average can be written
% \def\cond{\parbox{2.9em}{\scriptsize\em $i$ is in the sample}}
\[
    \bar Y ~ = ~ \frac1n\sum_{i=1}^n Y_i 
    ~ = ~ \frac1n\sum_{i=1}^N Z_i y_i
\]
The $Z_i$'s are Bernoulli random variables with
\begin{eqnarray*}
  E[Z_i] ~ = ~ \frac{n}{N} ~~,~~~ Var(Z_i) =
  \frac{n}{N}\left(1-\frac{n}{N}\right) 
  ~~,~~~ Cov(Z_i,Z_j) = - \frac{1}{N-1} \frac{n}{N}\left(1-\frac{n}{N}\right) 
\end{eqnarray*}

\noindent
{\bf Confidence Intervals and Sample Size}

\begin{itemize}

\item[(a)] A CLT-based $100(1-\alpha)\%$ confidence interval for the
  population mean is $(\bar Y - z_{\alpha/2}SE~,~~\bar Y +
  z_{\alpha/2}SE)$.
\item[(b)] For sampling with replacement from an infinite population, $SE =
  SD/\sqrt{n}$.
\item[(c)] For sampling without replacement from a finite population, the
  SE has to be multiplied by the finite population correction (FPC).
\item[(d)] For a given margin of error (ME, half the width of the CI) and
  confidence level $1-\alpha$, we can find the sample size by solving
\[
      z_{\alpha/2}SE < ME
\]
for $n$.  The same approach works for both SRS with replacement (using
the SE in (b)) and SRS without replacement (using the SE in (c)).

\end{itemize}
% {\em You should memorize the formula for the FPC, and the two formulas
%   for sample size calculations (one for sampling with replacement, one
%   for sampling without replacement).}

}

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\begin{center}
\large\bf Some Useful Formulas From the Statistics of Survey Sampling, II
% \\
% \normalsize Do not bring this to the exam.  A copy will be provided for you.
\end{center}

\noindent
{\bf Stratified Sampling}

\noindent
Consider $H$ strata with population counts $N = \sum_{h=1}^H N_h$ and
sample counts $n=\sum_{h=1}^H n_h$.  Let $f_h = n_h/N_h$; $W_h =
N_h/N$; and $\bar y_h = \frac{1}{n_h}\sum_{i=1}^{n_h} y_{ih}$ in each
stratum, and let $s_h^2=\frac{1}{n_h-1}\sum_i(y_{ih}-\bar y_h)^2$ be the
sample variance in each stratum.  Then
\begin{eqnarray*}
\overline y_{st} ~ = ~ \sum_{h=1}^H W_h \overline y_h ~~,~~~~
\Var(\overline y_{st}) 
	~ \approx ~  \sum_{h=1}^H W_h^2 (1-f_h) \frac{s_h^2}{n_h} ~~,~~~~
DEFF ~ = ~ \frac{\Var(\overline y_{st})}{\Var(\overline y_{srs})}
	=\frac{\sum_{h=1}^H W_h^2 (1-f_h) \frac{s_h^2}{n_h}}
	{(1-f)\frac{s^2}{n}}
\end{eqnarray*}



\noindent
{\bf Cluster Sampling}

\noindent
Consider a population of $N$ clusters.  We take an SRS ${\cal S}$ of
$n$ clusters, and all units within each sampled cluster (one-stage
clustering).  Assume clusters all have same size $M$.  Let $\bar y_i
= \frac{1}{M}\sum_{j=1}^M y_{ij}$ in each cluster.  Then
\begin{eqnarray*}
  \bar y_{cl} ~ = ~ \frac{1}{n} \sum_{i\in {\cal S}} \bar y_i
~~,~~~~
   \Var(\bar y_{cl}) 
  ~ \approx ~  \left(1-\frac{n}{N}\right) \frac1n s^2_{\bar y_i}
  ~ = ~ \left(1-\frac{n}{N}\right) \frac1n  \left[\frac1{n-1} 
     \sum_{i\in{\cal S}} (\bar y_i - \bar y_{cl})^2\right]
\end{eqnarray*}
and 
\[
 DEFF = \frac{\Var(\bar y_{cl})}{\Var(\bar y_{srs})}
    =  \frac{ M s^2_{\bar y_i}}{s^2_{y_{ij}}}
    \approx 1 + (M-1)\rho 
\]
where $s^2_{\bar y_i}$ is the sample varance of the cluster means,
$s^2_{y_{ij}}$ is the sample variance of the individual observations,
and $\rho$ is the intraclass (intracluster) correlation, or ICC.

\noindent 
{\bf Post-Stratification Weights and Means}

\noindent
As part of survey data collection it is a good idea to get general
demographic information (e.g. in our surveys: sex, age, class, major,
hometown, etc.).  After data collection we compare the proportions in
each of these categories in our sample with the same proportions in
the population. If they agree, great.  If not, calculate
\[
    w_i = (N_h/N)/(n_h/n) \mbox{~for each $i$ in post-stratum $h$}~~,~~~
    \mbox{\em and ~~} \overline y_w = \frac{\sum_i w_i y_i}{\sum_i w_i} 
\]


\noindent
{\bf Post-Stratification Variance Calculations}

\noindent
\ul{\em Taylor series}:
\[
\Var_{TS}(\overline y_w) \approx 
	\frac{1}{\left(\sum_i w_i\right)^2}
	\left[
	\Var(\sum_i w_i y_i) - 2\overline y_w \Cov(\sum_i w_i y_i, \sum_i w_i)
	+ (\overline y_w)^2\Var(\sum_i w_i)
	\right]
\]
where $\bar y_w$ is as above, $\bar w=\frac1n\sum_i w_i$,~~
$\bar{wy}=\frac1n\sum_i w_i y_i$,~~\\
\begin{center}
$\Var(\sum_{i=1}^n w_i) 
	 \approx  n  \cdot \frac{1}{n-1}\sum_{i=1}^n (w_i - \bar w)^2$,~~
$\Var(\sum_{i=1}^n  y_i w_i) 
	~ \approx ~ n \cdot \frac{1}{n-1} \sum_{i=1}^n (w_i y_i - \bar{wy})^2$,~~\\[1em]
% \begin{eqnarray*}
%     \Var(\sum_{i=1}^n w_i) 
%	~ \approx ~ n  \cdot \frac{1}{n-1}\sum_{i=1}^n (w_i - \bar w)^2 
%~~,~~~
%	\Var(\sum_{i=1}^n  y_i w_i) 
%	~ \approx ~ n \cdot \frac{1}{n-1} \sum_{i=1}^n (w_i y_i - \bar{wy})^2
%\\
$	\Cov(\sum_{i=1}^n y_i w_i, \sum_{i=1}^n w_i) 
	 \approx 
	n \cdot \frac{1}{n-1} \sum_{i=1}^n (w_i y_i - \bar{wy}) (w_i - \bar w)
$
%\end{eqnarray*}
\end{center}
\noindent
\ul{\em Jackknife}:
\begin{itemize}
\item Replicate $n$ times (by removing one obs. each time and
  recalculating weights):
\[
	\overline y_w^{(r)} = \frac{\sum_{i=1}^n w_i^{(r)}
          y_i^{(r)}}{\sum_{i=1}^n w_i^{(r)}} 
\]
\item Calculate
\begin{eqnarray*}
	\bar y_{JK} ~ = ~ \frac1n\sum_{r=1}^n \bar y_w^{(r)}
~~,~~
	Var_{JK}(\bar y_w) 
	~ \approx ~ \frac{n-1}{n} \sum_{r=1}^n (\bar y_w^{(r)} - \bar y_{jk})^2
\end{eqnarray*}
\end{itemize}

}


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\clearpage

\setcounter{page}{1}

\begin{enumerate}

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\clearpage

\item\pts{20} REVISE. {\em Multiple Choice (4 parts).  For each part, circle the
  roman numeral of the one best answer.}

\begin{enumerate}

\item\pts{5} Let $Y_i$ be the number of monthly neighborhood watch
  meetings attended by the $i^{th}$ resident in a neighborhood, in the
  last year.  You are going to conduct a survey, using a SRS to
  estimate $\bar Y_{pop}$, the population mean number of meetings
  attended by neighborhood residents.  Among the $N_R$ residents who
  {\em would respond} to your survey, the mean number of meetings is
  $\bar Y_R$, and among the $N_M$ number of residents who would {\em
    not respond}, the mean number of meetings attended is $\bar Y_M$.
  In class we showed that the {\em bias} between $\bar Y_R$ and $\bar
  Y_{pop}$, due to missing responses, is
\[
    \bar Y_R - \bar Y_{pop} = \frac{N_M}{N} (\bar Y_R - \bar Y_M)
\]
where $N = N_R + N_M$.\\[-0.75em]

Which statement below is false (or, circle iv. if all are OK)?
\begin{enumerate}\small
\item All other things being equal, the more people who would {\em not
  respond}, the smaller the bias due to missing responses.
\item All other things being equal, the bigger the difference between
  mean number of meetings attended by nonresponders, vs the mean
  number attended by responders, the bigger the bias due to
  missing responses.
\item The larger your SRS, the better you can estimate $\bar Y_R$.
\item All of the above statements are true.
\end{enumerate}

\item\pts{5} In one-stage clustered sampling, the ICC $\rho$ measures
\begin{enumerate}\small
\item The correlation between observations in different clusters.
\item The correlation between the cluster means of different clusters.
\item The correlation between observations in the same cluster.
\item The correlation between the cluster mean and the individual
  observations in the cluster.
\end{enumerate}

\item\pts{5} A 36-303 survey project constructs an SRS of
  undergraduates and directs them to a website where they can answer
  the survey questions.  However some respondents are never presented
  with the last page of the survey, due to a bug in the website
  software (the error can happen at any time to any respondent, but
  only a few respondents fail to complete the survey for this
  reason). This is most likely an example of
\begin{enumerate}\small
\item Unit nonresponse, Missing At Random (MAR)
\item Item nonresponse, Missing Completely At Random (MCAR)
\item Unit nonresponse, Missing Completely At Random (MCAR)
\item Item nonresponse, Missing At Random (MAR)
\end{enumerate}

\item\pts{5} Suppose we divide a sampling frame into groups, which we
  may treat as either strata for stratified sampling, or clusters for
  cluster sampling.  If we make the groups so that {\em observations
    {\bf within groups} are more {\bf similar} to each other}, and
  {\em observations {\bf between groups} are more {\bf different} from
    each other}, then, all other things being equal, we expect
\begin{enumerate}\small
\item The variance of the stratified sample mean $\bar y_{st}$ will go
  {\bf up} and the variance of the cluster sample mean $\bar y_{cl}$
  will go {\bf down}.
\item The variance of the stratified sample mean $\bar y_{st}$ will go
  {\bf down} and the variance of the cluster sample mean $\bar y_{cl}$
  will go {\bf up}.
\item Both variances will go {\bf up}.
\item Both variances will go {\bf down}.
\end{enumerate}

\end{enumerate}

\vspace*{\fill}

\begin{center}
\bf[This space intentionally left blank]
\end{center}

\vspace*{\fill}

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\clearpage

\item\pts{20} {\em Stratified Sampling (3 parts).}
% From Dick Scheaffer book, pp 61 ff

Most local advertisers have a choice of TV, print or radio
advertisements.  In Jefferson County there are two towns: Alpha is
built around a factory, and most of the households contain factory
workers with school-age children; Berea is an exclusive suburb of the
city of Danville in the next county, and contains mostly older people
with few children at home.  To see if it is worthwhile to advertise on
TV, a stratified sample of households with three strata---Alpha,
Berea, and the remaining rural area in the county---is taken, and
television viewing time, in hours per week, is recorded for each
household in the sample.

The data are as follows:
\begin{center}
\begin{tabular}{rccc}
\hline
                       & \em Stratum 1 & \em Stratum 2 & \em Stratum 3 \\
\hfill                 & \em Alpha     & \em Berea     & \em Rural Area\\
\hline
\hline
\hfill \em Number of households  & $N_1=155$ & $N_2=62$  & $N_3=93$\\
\hfill \em Stratum sample size   & $n_1=20$  & $n_2=8$   & $n_3=12$\\
\hline
\em TV viewing time, hr/wk & 
\begin{tabular}[t]{r@{~~}r@{~~}r@{~~}r}
35 & 28 & 26 & 41 \\
43 & 29 & 32 & 37  \\
36 & 25 & 29 & 31  \\
39 & 38 & 40 & 45 \\
28 & 27 & 35 & 34 
\end{tabular}
&
\begin{tabular}[t]{r@{~~}r@{~~}r@{~~}r}
27 &  4 & 49 & 10 \\
25 & 41 & 25 & 30 
\end{tabular}
&
\begin{tabular}[t]{r@{~~}r@{~~}r@{~~}r}
 8 & 15 & 21 &  7 \\
14 & 30 & 20 & 11 \\
12 & 32 & 34 & 24 
\end{tabular}
\\
\hline
\em Stratum Statistics & $\bar y_1 = 33.9000$ & $\bar y_2 = 25.125$ &
$\bar y_3 = 19.000$ \\
                   & $s^2_1 = 35.358$ & $s^2_2 = 232.411$ & $s^2_3 = 87.636$\\
\hline
\end{tabular}
\end{center}



\begin{enumerate}


\item\pts{6}  Compute $\bar y_{st}$ and $\bar y_{srs}$.  If they
  should be the same, explain why.  If they should be different,
  explain why {\em [use the back of this page to show work, if you
      need to]}.

\vspace*{\fill}

\clearpage


\item\pts{6} Compute SE$(\bar y_{st})$ and use this to create an
  approximate 95\% confidence interval for the mean household TV
  viewing time (hrs/wk) in Jefferson County.

\vspace*{\fill}

\item\pts{8} If we were to treat this as an SRS rather than a
  stratified sample, the sample variance would be $s^2=83.658$. 
\begin{enumerate}
\item Estimate the DEFF (design effect) for this design.
\item Was it a good idea to do the stratified sample rather than an
  SRS?  Why or why not?
\end{enumerate}
\vspace*{\fill}

\end{enumerate}

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\clearpage

\item\pts{18} {\em Approval Rating of Sen. Diane Feinstein, Part I (3 parts).}

In late February 2009, the Field Research Corporation
(http://www.field.com/fieldpollonline) conducted a poll of Democrats
and Republicans in California to assess voters' approval of US
Sentator Diane Feinstein, and judge whether any potential candidates
might be able to unseat her in the midterm elections in November 2010.

Approximately equal numbers of Democrats (342) and Republicans (298)
were contacted by phone; 267 of the Democrats and 98 of the
Republicans approved of the job Sen. Feinstein was doing.

According to the Ballot Access News
(http://www.ballot-access.org/2008/120108.html), there were 7,683,495
registered Democrats and 5,428,052 registered Republicans in
California at approximately the same time as the Field Poll was being
taken.
	



\begin{enumerate}

\item\pts{6} Treating the sample of Democrats as an SRS w/o replacement,
  compute $\hat p_{dem}$, the proportion of Democrats who approve of
  Sen. Feinstein, and its standard error, ${\rm SE}(\hat p_{dem})$.

\vspace*{\fill}

\item\pts{6} Do the same for Republicans: compute $\hat p_{rep}$ and
  ${\rm SE}(\hat p_{rep})$.

\vspace*{\fill}

\clearpage

\item\pts{6} A confidence interval for the difference between the true
  proportions of Democrats and Republicans who approve of Senator
  Feinstein can be computed with the help of the pooled SE, $\sqrt{{\rm
      SE}(\hat p_{dem})^2 + {\rm SE}(\hat p_{rep})^2}$.  Compute this
  pooled SE and give an approximate 95\% confidence interval for this
  difference in proportions of voters who approve of Sen. Feinstein.

\vspace*{\fill}

\end{enumerate}

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

\clearpage

\item\pts{24} {\em Approval Rating of Sen. Diane Feinstein, Part II (4
  parts).}

\begin{enumerate}

\item\pts{6} Combining Democrats and Republicans, and treating the
  survey as an SRS w/o replacement, estimate the overall proportion of
  voters in California who approve of the job Sen. Feinstein has been
  doing. (You do not have to compute the SE).

\vspace*{\fill}

\item\pts{6} We really do not know if the Field Poll was an SRS,
  what the response rate was, etc., but it seems clear that the
  proportions of Democrats and Republicans in the poll were different
  from the population proportions in California.  Using political
  party registration to define two post-strata, compute
  post stratification weights to apply to this survey.

\vspace*{\fill}

\clearpage

\item\pts{6} Using the weights you computed in part (b), compute the
  weighted proportion of voters in California who approve of
  Sen. Feinstein's job (you do not have to compute the SE).

\vspace*{\fill}

\item\pts{6} Use the values of the weights to explain\footnote{If you
  were unable to complete part (c), use the weights to make a
  prediction about whether your answer to part (c) would be higher or
  lower than your answer to part (a), and explain why.} the difference
  between your answers in parts (a) and (c).

\vspace*{\fill}


\end{enumerate}



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

\clearpage

% Examples from pp 167-168 of blue Groves book...

% Examples from lecture 09...

\item\pts{18} {\em Data Collection Methods (3 parts.)} In each
  situation below, choose the best mode of data collection, and give a
  reason why.  For the ``why'' question, write complete, clear
  sentences.  Your response does not need to fill all of the available
  space\footnote{On the other hand, if you need more space, use the
    back of the page.}.  Strong reasoning and clear writing count much
  more than length.

  Think about tradeoffs between feasibility, cost, response rate \&
  nonresponse bias, availability of information on which respondents
  can base their answers, etc.  You may assume there is a good
  sampling frame, and that a strong sampling method like SRS,
  stratified or clustered sampling, will be used.\\[-0.75em]


\begin{enumerate}

\item\pts{6} Your market research firm has been asked to conduct a
  survey in Pittsburgh to see who would purchase a new, sweet, fruity
  carbonated soft drink.
\begin{itemize}
\item Which mode of data collection (circle the best answer)?
\begin{enumerate}
\item Telephone survey.
\item Face to Face interviews.
\item Internet (web-based) survey.
\end{enumerate}
\item Why?

\vspace*{\fill}

\clearpage

\end{itemize}

\item\pts{6} The Governor of your state has asked you to design a
  survey to assess the views of residents of the state toward public
  transportation and to evaluate how likely they would be to use
  alternative modes of transportation.
\begin{itemize}
\item Which mode of data collection (circle the best answer)?
\begin{enumerate}
\item Telephone survey.
\item Face to Face interviews.
\item Internet (web-based) survey.
\end{enumerate}
\item Why?

\vspace*{\fill}

\end{itemize}


\item\pts{6} The American Statsitical Association wishes to do a
  survey of its members across the United States on their views toward
  certification of statisticians (statisticians would need pass a test
  in order to get a license to practice statistics, like doctors, some
  engineering professions, etc.).  
\begin{itemize}
\item Which mode of data collection (circle the best answer)?
\begin{enumerate}
\item Telephone survey.
\item Face to Face interviews.
\item Internet (web-based) survey.
\end{enumerate}
\item Why?

\vspace*{\fill}

\end{itemize}

\end{enumerate}


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

\comment{

\clearpage

% Examples from lecture 09...

\item\pts{??} Another 36-303 group wants to do a survey of PAT bus
  riders' attitudes toward a proposed schedule of bus fare increases.
  In their survey proposal they specify the {\em target population} to
  be all adult residents of Pittsburgh's East End (Oakland, Shadyside,
  Squirrell Hill \& Point Breeze).  They plan to collect data by
  approaching people waiting at six specific bus stops near where
  these students live, between 8am and 10am for one whole work-week
  (Mon-Fri), for face-to-face interviews.
\begin{enumerate}

\item\pts{??} Did this group specify a {\em sampling frame} for their
  survey?  If so, say what it is.  If not, specify a sampling frame
  that the group could work with.

\vspace*{\fill}

\item\pts{??} Write down two very likely sources of {\em coverage
  error} for this survey as currently proposed.

\vspace*{\fill}

\item\pts{??} Make one big suggestion that this group could implement
  to improve their sampling plan.

\vspace*{\fill}

\end{enumerate}

}

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

\end{enumerate}

\end{document}

