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\begin{center}
\large\bf
36-303: Sampling, Surveys and Society\\
Exam 1\\
Thu March 1, 2012
\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.}\\
You may tear it out of the exam if you like.
\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 & 20 & \\ 
  &    & \\ \hline
4 & 25 & \\ 
  &    & \\ \hline
5 & 15 & \\ 
  &    & \\ \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 When making a public report on a survey, which of the following
  is \ul{\bf not} required?
\begin{enumerate}

\item Who sponsored it, who carried it out.

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

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

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

\end{enumerate}


\item\pts5 Which of the following statements is \ul{\bf most correct}:
\begin{enumerate}

\item If you do not have a random sample, it cannot be a
  representative sample.

\item The only valid random sampling for surveys is sampling without
  replacement.

\item It is possible to construct a representative sample without
  random sampling, but it is more difficult to argue that it is really
  representative.

\item If the sample is random, it is representative, regardless of the
  response rate.

\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

% Grave p 288, #5, slightly modified

\item\pts{20} You are going to test the following survey items in 
cognitive interviews with one person outside your project team.  
For each item below, indicate (by marking the appropriate blank) 
whether you think the item is OK as-is, and if not, what 
you would ask the person about the item, or about the things 
the person is thinking as he/she answers the item (to give 
you ideas to improve it).

\begin{enumerate}

\item\pts{5} During the past four weeks, beginning on [date 4 weeks ago] and 
ending today, have you done any exercise, including sports, physically 
active hobbies, and aerobic exercises, but not including any
activities carried out as part of your job or in the course of
ordinary housework?
\begin{description}

\item[\rule{1.2em}{1pt}] This item is OK as-is.
\item[\rule{1.2em}{1pt}] This item may not be OK.  I want to ask 
the person who is trying these items the following question, to 
get information to improve this item:

\end{description}

\vspace*{\fill}

\item\pts{5} How many times a week do you have milk, butter or other 
dairy products?
\begin{description}

\item[\rule{1.2em}{1pt}] This item is OK as-is.
\item[\rule{1.2em}{1pt}] This item may not be OK.  I want to ask 
the person who is trying these items the following question, to 
get information to improve this item:

\vspace*{\fill}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
\end{description}

% \item\pts{x} Living where you do now and meeting the expenses you 
% consider necessary, what would be the smallest income (before any deductions) 
% you and your family would need to make ends meet each month?
% \begin{description}
% 
% \item[\rule{1.2em}{1pt}] This item is OK as-is.
% \item[\rule{1.2em}{1pt}] This item may not be OK.  I want to ask 
% the person who is trying these items the following question, to 
% get information to improve this item:
% 
% \end{description}
% 
% \vspace*{\fill}


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


\item\pts{5} Some people feel the US federal government should take 
action to reduce the national debt, even if it means unemployment would 
go up a lot.  Others feel the government should take action to reduce
the rate of unemployment even if it means the national debt would go 
up a lot.  Where would you place yourself on the following 7-point scale?
\begin{center}
\begin{tabular}{*{7}{p{0.6in}}}
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
\multicolumn{3}{l}{Reduce} & & \multicolumn{3}{r}{Reduce} \\
\multicolumn{3}{l}{Debt} & & \multicolumn{3}{r}{Unemployment} \\
\end{tabular}
\end{center}
\begin{description}

\item[\rule{1.2em}{1pt}] This item is OK as-is.
\item[\rule{1.2em}{1pt}] This item may not be OK.  I want to ask 
the person who is trying these items the following question, to 
get information to improve this item:

\end{description}

\vspace*{\fill}

\item\pts{5} During the past 12 months, since [date], about how many
days did illness or injury keep you in bed more than half the day? 
Include days while you were an overnight patient in a hospital.
\begin{description}

\item[\rule{1.2em}{1pt}] This item is OK as-is.
\item[\rule{1.2em}{1pt}] This item may not be OK.  I want to ask 
the person who is trying these items the following question, to 
get information to improve this item:

\end{description}

\vspace*{\fill}


\end{enumerate}


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\clearpage

% Situational survey question...

\item\pts{20}\label{p:hospitals} 
The American Hospital Association (AHA) conducted a
survey on drug shortages in hospitals, in summer 2011
In a powerpoint press release 
(see http://www.aha.org/content/11/drug\-shortagesurvey.pdf), the 
survey methodology was described as follows:
\begin{itemize}\sf
\item Survey was sent to all community hospital CEOs on 
June 1, 2011 via fax and e-mail.
\item Data were collected through June 22, 2011.
\item Responses from 820 hospitals are included in 
analysis.
\item Respondents were broadly representative of the universe 
of community hospitals.
\item Survey questions were designed to assess the impact of 
drug shortages on patients and hospitals.
\item Nationally, there are about 2,800 urban hospitals,
1,300 critical access hospitals and 1,000 other rural hospitals.
\end{itemize}
Please answer the following questions about this survey.  If the 
information cannot be inferred from the bullets above, say so.
%
\begin{enumerate}

\item\pts{2} What is the {\em target population} for this survey?

\vspace*{\fill}

\item\pts{2} What are the {\em units} in the target population?

\vspace*{\fill}

\item\pts{2} What is the {\em sampling frame}?

\vspace*{\fill}

\item\pts{2} What {\em sampling method} was used?

\vspace*{\fill}

\item\pts{2} What {\em method of data collection} was used to contact
units in the sample and collect data from them?

\vspace*{\fill}

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

\item The press release reports that {\sf ``99.5\% of hospitals
reported experiencing one or more drug shortage in the last six
months and nearly half of the hospitals reported 21 or more drug
shortages.''}
\begin{enumerate}
\item\pts{5} Do you think this is useful information for the AHA?  
Why or why not?

\vspace*{\fill}

\item\pts{5} Do you think this is a scientifically valid inference
about the target population?  Explain carefully why or why not, 
using \ul{\em at least two} concepts from the lecture notes or textbook.

\vspace*{\fill}

\end{enumerate}

\end{enumerate}


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\clearpage
\
% continuation of the hospital survey

\item\pts{25} Now let's assume that the AHA survey is a {\em simple random 
sample (SRS) without replacement}, from a population of size 5,100 
(regardless of any answer you gave in problem \ref{p:hospitals}).  

\begin{enumerate}

\item\pts{10} 17\% of hospitals in the sample reported that patient 
treatment was delayed because of a drug shortage.  Calculate a 95\% 
confidence interval for the same percentage in the population, based 
on the 820 responses in the sample.

\vspace*{\fill}

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

\item\pts{5} What is the margin of error (ME) in part (a)?

\vspace*{\fill}


\item\pts{10} Suppose we only want a ME of 0.01.  What should the 
sample size be?

\vspace*{\fill}
\vspace*{\fill}
\vspace*{\fill}

% \item\pts{x} 

\end{enumerate}

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\clearpage

% Groves, p. 402, #8, modified.

\item\pts{15}\label{p:informed} \fig{f:consent} 
(page \pageref{f:consent} of this test) displays an 
informed-consent form for a web survey.  
\begin{enumerate}

\item\pts{12} Identify (using the paragraph numbers in \fig{f:consent}) any 
four of the six essential elements
for an informed consent form (by filling out the table below):\\[0.5em]
\begin{tabular}{|p{3in}|l|}
\hline
                      & {\bf Number of Paragraph} \\
{\bf Name of Element} & {\bf Where it Appears}     \\
\hline 
&            \\
&            \\ \hline
&            \\
&            \\ \hline 
&            \\
&            \\ \hline
&            \\
&            \\ \hline
\end{tabular}
~\\
\item\pts{3} Identify an element of informed consent forms that
is missing from \fig{f:consent}.

\end{enumerate}

\begin{figure}
\def\gap{0.5em}
\begin{center}
\begin{tabular}{|c|}
\hline
\begin{minipage}{5.75in}
~\\%[-\gap]
{\bf Welcome to the CAREGIVING: STRESS AND SUPPORT Survey (HUM00001234)}\\[\gap]
%
{\bf (1)} Dr. John Jones and Dr. Sarah Smith, of the University of Michigan 
Department of Psychology, invite you to be a part of a research study that
looks at the stresses that people experience when they are providing care
to a seriously ill family member.  The purpose of the study is to design 
better support programs for caregivers.  We are asking you to attend because you 
recently attended a meeting of a UM Caregivers Support Group.\\[\gap]
%
{\bf (2)} If you agree to be part of the research study, you will be asked to complete
an online survey about your experiences as a caregiver.  We expect this 
survey to take 30 to 45 minutes to complete.\\[\gap]
%
{\bf (3)} Although you may not receive any direct benefit from participating, we hope
that this study will contribute to the improvement of social support systems 
for those who provide care to others.\\[\gap]
%
{\bf (4)} Your responses to this survey are anonymous, meaning that the researchers 
will not be able to link your survey resposes to you.  The survey software
does not collect identifying information about you or your computer.  We plan 
to publish the results of this study, but we will not include any information
that would identify you. \\[\gap]
%
{\bf (5)} Participating in this survey is completely voluntary.  Even if you decide 
to paticipate now, you may change your mind and stop at any time.  You may
choose not to answer a question, or skip any part of the survey.  Simply 
click ``Next'' at the bottom of the survey page to move on to the next
question.  \\[\gap]
%
% {\bf (1)} If you have any questions about this survey, you can contact 
% John Jones, Ph.D., Department of Psychology, University of Michigan, 
% 123 East Hall, Ann Arbor MI, 48108, (734)  123-4567, jjones@umich.edu.\\[\gap]
%
{\bf (6)} The University of Michigan Behavioral Sciences Institutional Review Board has 
determined that this study is exempt from IRB oversight.\\[\gap]
%
{\bf (7)} By clicking on the link below, you are consenting to participate in this 
research study.
\begin{center}
\bf www.caregivingsurvey.net\\
~\\
If you do not wish to participate, click the ``x'' in the top right corner 
of your browser to exit.
%
\end{center}
\end{minipage}
~\\[\gap]
\hline
\end{tabular}
\end{center}
\caption{Informed consent form for problem \ref{p:informed}.}\label{f:consent}
\end{figure}

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\end{enumerate}

\end{document}

