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\begin{center}
\large\bf
36-303: Sampling, Surveys and Society\\
Exam 1\\
Thu Feb 21, 2008
\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 A formula sheet is provided on the next page 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 & 18 & \\ 
  &    & \\ \hline
3 & 20 & \\ 
  &    & \\ \hline
4 & 18 & \\ 
  &    & \\ \hline
5 & 24 & \\ 
  &    & \\ \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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\def\itemize{\oitemize\parsep=0pt\parskip=0pt\itemsep=0pt}

\begin{center}
\large\bf Some Useful Formulas From the Statistics of Survey Sampling
% \\
% \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{enumerate}

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\clearpage

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

\begin{enumerate}

\item\pts5 When making a public report on a survey, which of the following
  is \ul{\bf not} required?
\begin{enumerate}

\item Target population, sampling frame, sampling method, response
  rates.

\item The name of the statistical package used to do the analyses.

\item Sample size and precision (SE) of estimates.

\item Who sponsored it, who carried it out.

\end{enumerate}

\item\pts5 Recall the sample inclusion indicators $Z_i = 1$ if the $i^{th}$
  person in the population is in the sample, and $Z_i=0$ if not.  We
  showed in class that $Cov(Z_i,Z_j)<0$.  This means that
\begin{enumerate}

\item Seeing a high value (say, income) in the sample makes it more
  likely that the next value we see will be \ul{\bf low}.

\item Seeing a high value (say, income) in the sample makes it more
  likely that the next value we see will also be \ul{\bf high}.

\item If the $i^{th}$ person in the population is in the sample, then
  it's \ul{\bf more} likely that the $j^{th}$ person will also be in the
  sample.

\item  If the $i^{th}$ person in the population is in the sample, then
  it's \ul{\bf less}  likely that the $j^{th}$ person will also be in the
  sample.

\end{enumerate}

\item\pts5 {\em Beneficence} is one of the ethical obligations survey
  researchers have toward survey respondents.  The best definition of
  beneficience, according to our textbook, is
\begin{enumerate}

\item Ensuring that no harm comes to respondents as a result of the survey.

\item Ensuring that the survey benefits respondents in some way.

\item Acting to minimize possible harms and maximize possible benefits
  to respondents.

\item Making sure that the respondents benefit from the survey by
  getting a copy of the final report.

\end{enumerate}

\item\pts5 Two important fractions in sample surveys are the {\em sampling
  fraction} $n/N$ and the {\em response rate} $r/n$ (where $N$ is the
  population size, $n$ is the intended sample size, and $r$ is the
  number in the sample that actually responded).  Which of the
  following is \ul{\bf not} true, for a simple random sample with
  replacement from the target population?

\begin{enumerate}

\item To decrease variability in sample estimates, increase the
  sampling fraction.

\item To decrease possible bias in sample estimates, increase the
  response rate.

\item You can force the standard error of sample estimates to be zero
 by making the sampling fraction large enough.

\item You can get a more representative sample by increasing $n$,
  regardless of the response rate. 


\end{enumerate}

\end{enumerate}

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\clearpage

\item\pts{18} {\em Simple Random Sampling (3 parts).}

\begin{enumerate}

\item\pts6 Carefully define {\em Simple Random Sampling (SRS) \ul{\bf with}
  replacement}, and give an example.  Give enough detail that it is
  obvious that this is a good example.

\vspace*{\fill}


\item\pts6 Carefully define {\em Simple Random Sampling (SRS) \ul{\bf without}
  replacement}, and give an example.  Give enough detail that it is
  obvious that this is a good example.

\vspace*{\fill}


\clearpage

\item\pts6 Suppose we take a SRS without replacement of $n$ individuals
  from a population of $N$.  Let $y_i=1$ if the respondent feels
  people should be allowed to talk on their cell phone while driving,
  and $y_i=0$ if not.  Let
\[
	\hat p ~ = ~ \frac1n\sum_{i=1}^n Y_i ~ = ~ \frac1n\sum_{i=1}^N Z_iy_i
\]
be the estimated proportion of people who favor cell phone use by
drivers from the sample, and let
\[
        p ~ = ~ \frac1N\sum_{i=1}^N y_i
\]
be the population proportion.  Show that $\hat p$ is an unbiased
estimator for $p$, i.e., show that
\[
	E[\hat p] ~ = ~ p ~ .
\]

\end{enumerate}

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\clearpage

\item\pts{20} According to the Carnegie Mellon 2006--2007 FactBook
  (http://cmu.edu/ira/factbook/facts\-2007/facts2007\_main.htm),
  Carnegie Mellon employs 1421 full- and part-time faculty at the
  Pittsburgh campus (and 38 faculty at the CM-Qatar or CM-West~Coast
  campuses).  Your survey team is going to survey Pittsburgh campus
  faculty about their job satisfaction.  One of yout survey questions
  is
\begin{quote}
\em ``If you had it to do over again, would you apply for a faculty
  position at Carnegie Mellon? ~~~ Yes~/~No''
%   ``If you could have the same job you have now at a different
%   university, with the same salary and payroll benefits as you get at
%   Carnegie Mellon, would you take it?''
\end{quote}
Answer the following questions (4 parts).
\begin{enumerate}
\item\pts5 Let $p$ be the proportion of Pittsburgh-based CM faculty that
  would say yes.  Assuming you are doing SRS without replacement, how
  large must your sample size be, to estimate $p$ within a margin of
  error of $\pm0.10$, with a 95\% confidence interval?

\vspace*{\fill}

\clearpage

\item\pts5 It turns out you only have enough resources (time and money to
  process the responses, mainly) to do an email survey of 75 faculty,
  chosen at random from a complete email address list for the
  Pittsburgh-based CM faculty.  Amazingly, all 75 faculty reply, and of
  those, 60 respond ``yes'' to the question above.  Compute a 95\%
  confidence interval for $p$.

\vspace*{\fill}

\clearpage

\item\pts5 Now suppose that you sent out emails to a random sample of 200
  Pittsburgh-based CM faculty, but only 75 replied.  Of those 75 replies,
  60 respond ``yes''.  Compute the 95\% confidence interval for $p$.

\vspace*{\fill}

\item\pts5 Which of the confidence intervals, (b) or (c), is harder to
  believe as evidence about the whole population of Pittsburgh-based
  CM faculty?  Why?

\vspace*{\fill}

\end{enumerate}

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\clearpage

% Groves, problem 11.2, pp 374. - Ethics

\item\pts{18} There are many situations in which respondents may not report
  their true behaviors to survey researchers. This typically happens
  when the behaviors themselves are socially sensitive. 

  One researcher wanted to find out how accurately women report having
  had an abortion.  From medical records, she obtains the names and
  addresses of women wh have had an abortion at a particular clinic,
  and then sends each of them a letter inviting them to participate.
  Among other things, the letter states:
\begin{quote}
``The aim of the investigation is the collection of data on women's
  health and the factors influencing it, as well as how satisfied
  women are with the organization of medical care and their opinion
  about ways it might be reorganized.  The substantive part of the
  research is a survey in which data are collected directly from
  people. Respondents, such as yourself, were selected from the
  address register by a method of random selection\ldots''
\end{quote}
Answer the following questions (3 parts).
\begin{enumerate}

\item\pts6 What, if any, ethical principles do you think this study
  violates? (circle the roman numeral of the one best answer)

\begin{enumerate}

% \item Avoiding Fabrication, Falsification, or Plagiarism

\item Beneficence

\item Justice

\item Respect for Persons; Confidentiality 

\item Informed Consent

\item None of the above; it's just fine.

\item Not listed above; I think~ \rule{3.8in}{1pt}\\[2em]
\rule{5.4in}{1pt}~.

\end{enumerate}

% \item What consequences do you think this procedure will have, for the
%   findings the researcher comes up with?
%
% \vspace*{\fill}

% \item Suppose you were a respondent and discovered the real way that
%   your name came into the sample.  How would you feel?
% 
% BAG THIS Q
% 
% \vspace*{\fill}
% 
% \clearpage

\item\pts6 Suggest a different survey design and/or a different letter to
  respondents that could answer the initial research question and not
  raise any ethical flags.  (continue onto the next page if needed)

\vspace*{\fill}

\clearpage

\item[(b)] (More space to answer (b), if needed.)

\vspace*{\fill}

\item\pts6 Suppose this was a study of how accurately people report their
  voting behavior, and the names came from voter registration lists
  (which are publicly available), but again the respondents were not
  told the real way their names ended up in the sample.\\

 What, if any, ethical principles do you think this study
  violates?

\begin{enumerate}

% \item Avoiding Fabrication, Falsification, or Plagiarism

\item Beneficence

\item Justice

\item Respect for Persons; Confidentiality 

\item Informed Consent

\item None of the above; it's just fine.

\item Not listed above; I think~\rule{3.8in}{1pt}\\[2em]
\rule{5.4in}{1pt}~.

\end{enumerate}

\end{enumerate}


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\clearpage

\item\pts{24} Below are several survey questions.  For each question: (i)
  indicate a potential problem with the question {\em using terms we
    discussed in class on question writing}; (ii) suggest a way to
  rewrite it (as one or more questions, by providing more information,
  by improving grammar, etc.) that gets at the same thing while
  avoiding the problem you raised.

\def\ans{
\begin{enumerate}
\item\pts3 A Potential Problem:
\vspace{1.25in}
\item\pts3 Suggestion(s) For Rewrite:
\vspace{1.5in}
\end{enumerate}
}

\begin{enumerate}

\item {\em ``What brand of computer do you own?
\begin{itemize}
\item[]\rule{1em}{1pt} Dell
\item[]\rule{1em}{1pt} Apple''
\end{itemize}
}
\ans

\item {\em ``The United States should withdraw its armed forces from
  Iraq and Afghanistan within one year after the new US President
  takes office.  (Agree or disagree).''}  
\ans

\item {\em ``In the past week, what fraction of your time online have
  you spent on instant messaging? Answer here: \rule{3in}{1pt}''}
\ans

\item {\em ``Given the choice, would you prefer increasing employer
  contributions to 401(k) plans or increasing employee salaries and
  raising the investing limits on IRA's?''}  
\ans

\end{enumerate}

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\end{enumerate}

\end{document}

