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\begin{center}
\large\bf
36-303: Sampling, Surveys and Society\\
Exam 1\\
Thu Feb 17, 2011
\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 & 18 & \\ 
  &    & \\ \hline
4 & 20 & \\ 
  &    & \\ \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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\let\oitemize\itemize
\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 Following are some of the most important decisions that can
  be made as part of designing a survey.  Which one \ul{\bf doesn't
    belong}?  % groves, p 33
\begin{enumerate}
\item How will the potential sample members be identified and
  selected?
\item What approach will be used to contact those sampled, and how
  much effort will be devoted to trying to collect data from those who
  are hard to reach or reluctant to respond?
\item How much effort will be devoted to evaluating and testing
  the questions that are asked?
\item What mode will be used to pose questions and collect answers
  from respondents?
\item None of the above (I.e., all are important decisions in survey
  design).
\end{enumerate}

\item\pts5 It is possible to construct a representative sample without
  random sampling, but it is more difficult to argue that it is really
  representative.
\begin{enumerate}
\item True
\item False
\end{enumerate}

\item\pts5 {\em Beneficence} is a basic ethical principle for research
  with human subjects.  Which of the following is \ul{\bf not} an aspect
  of beneficence? % groves, p 378
\begin{enumerate}
% \item Avoid harming the subjects if possible.
\item Maximize possible benefits and minimize possible harms, to
  subjects.
% \item Do not do the research if the benefit does not outweigh the
%   risks.
\item Decide when to do a study because the benefits outweigh the
  risks to subjects; and when to forego a study because the risks
  outweigh the benefits.
\item Make sure the subjects get the benefit of some compensation
  for their participation.
\item None of the above (I.e., all are aspects of beneficence).
\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 without
  replacement from the target population?

\begin{enumerate}

\item You can force the standard error of sample estimates to be zero
 by making the sampling fraction large enough, as long as the response
 rate is equal to 1.

\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 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} According to a 2005 survey of American workers reported
  in the San Francisco Chronicle
  (http://www.sfgate.com/cgi-bin/article.cgi?f=/g/a/2005/07/11/wastingtime.TMP),
  the average worker admits to frittering away 2.09 hours per 8-hour
  workday, not including lunch and scheduled break-time.  44.7\% of
  the respondents polled cited web surfing as their \#1 distraction at
  work. Socializing with co-workers came in second at 23.4\%.\\

  Answer the following questions (3 parts):
\begin{enumerate}
\item\pts{6} What is the {\bf Target Population} for this survey?

\vspace*{\fill}

\item\pts{6} The ``Survey Methodology'' section of the article states,
\begin{quote}
America Online and Salary.com conducted in-depth research relating to
time wasted at work among 10,044 respondents during May and June,
2005. Populations surveyed included AOL users, Salary.com Salary
Wizard users and corporate human resource professionals. Respondents
were asked to indicate how much time they wasted in an average
workday, assuming a workweek consisting of five 8-hour
days. Demographic information, including respondent geography, job
category and gender was also collected in the study. Data was analyzed
by Salary.com's team of Certified Compensation Professionals.
\end{quote}
Do you think the {\bf coverage error} for this sampling method, for
the target population in part (a), is high or low?  Give at least one
reason for your answer.

\vspace*{\fill}

\vspace*{\fill}

\hfil{\sl[Continued on next page]}
\clearpage

\item\pts{6} Give two other potential problems with the sampling
  method chosen for this survey.  Use words from class to name the
  problem, and then use specific ideas from class, and/or evidence
  from the survey methodology paragraph, to explain what you mean.\\

\ul{Problem 1}:

\vspace*{\fill}

\hrule

\vspace*{0.25in}

\ul{Problem 2}:

\vspace*{\fill}

\end{enumerate}

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\clearpage

\item\pts{18} {\sl[continuation]} Suppose you wish to replicate this
  survey, for the Target Population consisting of employees of
  Carnegie Mellon on the Pittsburgh campus.  According to the
  2009--2010 CMU FactBook, there are 4732 employees (faculty and
  staff) on the Pittsburgh campus.
\begin{enumerate}

\item\pts{6} Suppose you are able to get a simple random sample of 100
  CMU Pittsburgh employees, and (remarkably) all of them respond to
  your survey.  You find that 40\% of the respondents list web surfing
  as their \#1 distraction.  Construct a confidence interval for $p$,
  the true proportion of Pittsburgh Campus CMU employees for whom web
  surfing is the \#1 distraction.  {\em (Hint: The SD is
    $\sqrt{p(1-p)}$.)} \\

(You may not need all the space provided.)

\vspace*{\fill}

\hfil{\sl[continued on next page]}

\clearpage

\item\pts{6} Suppose you wanted to get a confidence interval in part
  (a) with a margin of error of 0.05 (5\%).  What would the sample
  size need to be, if you were doing simple random sampling {\em with
    replacement}?

\vspace*{\fill}

\item\pts{6} Suppose you wanted to get a confidence interval in part
  (a) with a margin of error of 0.05 (5\%).  What would the sample
  size need to be, if you were doing simple random sampling {\em
    \ul{without} replacement}?

\vspace*{\fill}

\end{enumerate}

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\clearpage

% Groves, p. 401, #2, modified.

\item\pts{20} Each situation below raises a potential issue about {\bf
  informed consent}.  If you were a member of the IRB, would you
  approve the research activity, or would you require a change?
  Justify your ``approve'' or ``disapprove'' answer in each case.

\def\ans{
\begin{enumerate}
\item[]\rule{1em}{1pt} Approve.
\item[]\rule{1em}{1pt} Disapprove.  {\em Please Change the Following:}\\
\vspace*{\fill}
\item[] Reason:\\
\vspace*{\fill}
\end{enumerate}
}

\begin{enumerate}
\item\pts{5} As a part of a face-to-face survey, the interviewer is
  asked to observe how many books are visible in the respondent's
  living room.
\ans

\item\pts{5} As part of a telephone survey, the first few moments of
  the interaction between the interviewer and the potential respondent
  are recorded, for use as examples in teaching other interviewers.
\ans

\clearpage

\item\pts{5} Before approaching a parent-child pair for a face-to-face
  interview to study parent-child interations, the interviewer makes
  structured observations of how the parent and child interact in a
  public park.  The observations are used to select parent-child pairs
  for the sample, and become part of the data collected for each
  parent-child pair in the sample.  
\ans


\item\pts{5} As part of a study of whether voter intentions are
  affected by the sponsor of the survey, some respondents are told
  that the sponsor is an organization different from the one actually
  conducting the survey.
\ans


\end{enumerate}

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\clearpage

% Groves, p. 288, \#2, modified.

\item\pts{xx} Below are several survey questions.  For each question:
  (i) indicate a potential problem with the question {\em using
  specific ideas from the lecture notes 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 was your income in the past year?\\
  \hspace*{1.8in} Answer here:
  \rule{3in}{1pt}''}   
\ans

\item {\em ``Do you favor or oppose universal health care in the
  United States?''}  
\ans

\item {\em ``I am happier than usual these days. (Circle the best
  answer below)\\
\begin{tabular}{rcccl}
                1 & 2 & 3 & 4 & 5  \\
Strongly agree    &   &   &   &    Strongly disagree.''
\end{tabular}
% \begin{description}
% \item[]\rule{1em}{1pt} Strongly agree.
% \item[]\rule{1em}{1pt} Generally agree.
% \item[]\rule{1em}{1pt} Neither agree nor disagree.
% \item[]\rule{1em}{1pt} Generally disagree.
% \item[]\rule{1em}{1pt} Strongly disagree.''
% \end{description}
}%
\ans

\item {\em ``What kind of music do you listen to when you study?\\
\begin{tabular}{p{2in}p{2in}}
\begin{itemize}
\item[]\rule{1em}{1pt} Rap.
\item[]\rule{1em}{1pt} Classical.
\end{itemize}
&
\begin{itemize}
\item[]\rule{1em}{1pt} Classic Rock.
\item[]\rule{1em}{1pt} Industrial Music.
\end{itemize}
\end{tabular}
}%
\ans

\end{enumerate}

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\end{enumerate}

\end{document}

