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\bf 36-303: Sampling, Surveys and Society\\
Exam 2 Solutions
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\begin{enumerate}

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

\begin{enumerate}

\item\pts{5} \ldots{}
% You have conducted a survey in which you have collected
%   data on respondents' ages and incomes; all respondents are between
%   20 and 60 years old, and earn between \$25,000 and \$250,000.  Some
%   cases (respondents) did not report their incomes:  
  older people were
  less likely to report their income, and among people who did report
  incomes, older people tended to earn more than younger people.\\[-0.75em]

  Which imputation method below is {\bf not} recommended (or, circle
  iv. if all are OK)?
\begin{enumerate}\small\em
% \item Delete cases from the data set for which there is no income
%   data.
\item[i.] For each missing income, fill in the mean of all of the incomes
  you do have.
\end{enumerate}

\item\pts{5} \ldots{}
%  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\em
\item[ii.] 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}.
\end{enumerate}

\item \pts{5} Which of the following is {\bf not} one of the
  recommended things to work on, to reduce the tendency of survey
  subjects to not respond?
\begin{enumerate}\small\em
\item[ii.] Choice of stratified or cluster sampling.
\end{enumerate}

\item\pts{5} In one-stage clustered sampling, the ICC $\rho$ measures
\begin{enumerate}\small\em
\item[iii.] The correlation between observations in the same cluster.
\end{enumerate}


\end{enumerate}

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% \clearpage

\item\pts{20} {\em Cluster Sampling (3 parts).}

% One of the joys of summer is driving by a roadside stand and buying
% bunches of fresh sweetcorn to take home and cook.  Of course, roadside
% stands have to satisfy health inspection requirements like all other
% food sellers.

\ldots{} joys of summer\ldots{} One roadside stand sells corn in
bunches of 3 ears each; they have 580 such bunches for sale.  A food
inspector samples 12 of these bunches and counts the number of worm
fragments he finds on each ear in all 12 bunches, to determine the
average number of worm fragments per ear of corn.  The inspector's
data are as follows:
%
\begin{center}
\begin{tabular}{c|*{12}{c}}
 \multicolumn{1}{c}{}& \multicolumn{12}{c}{Bunch} \\
&  01 &  02 &  03 &  04 &  05 &  06 &  07 &  08 &  09 &  10 &  11 &  12 \\
\hline
\hline
Ear 1 & 1 & 4 & 0 & 3 & 4 & 0 & 5 & 3 & 7 & 3 & 4 & 0 \\
Ear 2 & 5 & 2 & 1 & 6 & 9 & 7 & 5 & 0 & 3 & 1 & 7 & 0 \\
Ear 3 & 7 & 4 & 2 & 6 & 8 & 3 & 1 & 2 & 5 & 4 & 9 & 0 \\
\hline
mean  & 4.33 & 3.33 & 1.00 & 5.00 & 7.00 & 3.33 & 3.67 & 1.67 & 5.00 & 2.67 & 6.67 & 0.00 
\end{tabular}
\end{center}
\begin{eqnarray*}
   s^2_{\bar y_i} & = & \frac{1}{12-1}\sum_{i=1}^{12} (\bar y_i - \bar
   y_{cl})^2 ~ = ~ 4.53\\
%
  s^2_{y_{ij}} & = & \frac{1}{36-1} \sum_{i=1}^{12}\sum_{j=1}^3 (y_{ij} -
  \bar y_{srs})^2 ~ = ~ 7.38 
\end{eqnarray*}
%
\begin{enumerate}


\item\pts{6}  Compute $\bar y_{cl}$ and $\bar y_{srs}$; explain why
  they should be same/different.

{\em 
One-stage cluster sampling with equal cluster sizes is {\em
  self-weighting}, which means that $\bar y_{cl} = \bar y_{srs}$.  It
is easiest to calculate
\[
   \bar y_{cl} = \frac{1}{12}(4.33 +  3.33 +  1.00 +  5.00 +  7.00 +  3.33
   +  3.67 +  1.67 +  5.00 +  2.67 +  6.67 + 0.00 ) = 3.64
\]

}
% \clearpage


\item\pts{6}  Compute SE$(\bar y_{cl})$ and use this to create an
  approximate 95\% confidence interval for the mean number of worm
  fragments per ear in the entire population of $580\times3=1740$ ears
  of corn.
{\em 
\[
    \Var(\bar y_{cl}) = (1-n/N)\frac{1}{n}s^2_{\bar y_i} =
    (1-12/580)\frac{1}{12}4.53 = 0.3697 
\]
so
\[
    SE(\bar y_{cl}) = \sqrt{0.3697} = 0.61
\]
The confidence interval can be either of the following (full credit
for either one):
\[
   3.64 \pm (1.96)(0.61) = (2.44, 4.84)
\]
or
\[
   3.64 \pm (2)(0.61) = (2.42, 4.86)
\]
}

\item\pts{8} Estimate
\begin{itemize}
\item The DEFF (design effect) for this design; and
\item The ICC (intra-cluster correlation, $\rho$).
\end{itemize}
{\em

\[
   DEFF = \frac{Ms^2_{\bar y_i}}{s^2_{y_{ij}}} =
   \frac{(3)(4.53)}{7.38} = 1.84
\]
so
\[
   \rho = (DEFF - 1)/(M-1) = (1.84-1)/(3-1) = 0.42
\]
}

\vspace*{\fill}

% \item\pts{??} Explain what the ICC is, and why it is nonzero.
% 
% \vspace*{\fill}
%
% Putting this question in the M/C part...

\end{enumerate}

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% \clearpage

\item\pts{18} {\em Yale Work \& Life Survey, Part I (3 parts).}

\ldots there were 2,707 men and 2,609
women enrolled at Yale in 2005--2006, the academic year in which
Brescoll did her study.

In Brescoll's survey, 154 men responded, of which 134 planned to
become parents someday, and 315 women responded, of which 247 planned
to be parents someday\footnote{Other Likert-type questions more
  directly addressed the tradeoff between career and family, but we
  will just work with this yes/no question.}.

\begin{enumerate}

\item\pts{6} Treating the sample of men as an SRS w/o replacement,
  compute $\hat p_{men}$, the proportion of men at Yale who plan on
  becoming parents, and its standard error, ${\rm SE}(\hat p_{men})$.
{\em 

\[
    \hat p_{men} = 134/154 = 0.87
\]
so
\[
    \Var(\hat p_{men}) = (1-154/2707)\frac{1}{154}(0.87)(1-0.87) = 0.0006926
\]
and so
\[
     {\rm SE}(\hat p_{men}) = \sqrt{0.0006926} = 0.026 
\]

}


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

\[
    \hat p_{women} = 247/315 = 0.78
\]
so
\[
    \Var(\hat p_{women}) = (1-315/2609)\frac{1}{315}(0.78)(1-0.78) = 0.0004790
\]
and so
\[
     {\rm SE}(\hat p_{women}) = \sqrt{0.0004790} = 0.022 
\]

}

% \clearpage

\item\pts{6} Test whether there is a significant
  difference between men and women on this question\ldots
{\em 
\[
        z = \frac{\hat p_{men} - \hat p_{women}}
        {\sqrt{{\rm SE}(\hat p_{men})^2 + {\rm SE}(\hat p_{women})^2}}
        = \frac{0.87 - 0.78}{\sqrt{0.026^2 + 0.022^2}}
        = 2.63~;
\]
this value is well beyond $\pm1.96$ (limits for a Normal 95\%
confidence interval) and so we reject the null hypothesis that
$p_{men}-p_{women}=0$ at lower than the $0.05$ level, using a 2-sided
$z$-test comparing two sample proportions.

The conclusion we draw is that a significantly greater proportion of
men than women at Yale plan to be parents someday.  

(Depending on how much of the burden of parenthood men expect their
wives to carry, this may at least partly contradict the earlier
finding in the New York Times article.)

}

\end{enumerate}

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% \clearpage

\item\pts{24} {\em Yale Work \& Life Survey, Part II (4 parts).}

\begin{enumerate}

\item\pts{6} Combining men and women, and treating the survey as an
  SRS w/o replacement, estimate the overall proportion of students at
  Yale who plan on becoming parents.  (You do not have to compute the
  SE).
\[
    \hat p_{srs} = \frac{134 + 247}{154+315} = 0.8124 
\]

\item\pts{6} We really do not know if Brescoll's survey was an SRS,
  what the response rate was, etc., but it seems clear there is an
  imbalance among the respondents between men \& women, vs. the Yale
  student population.  Using sex to define two post-strata, compute
  post stratification weights to apply to this survey.

{\em 
\[
    w_{men} = (2707/(2707+2609))/(154/(154+315)) = 1.55
\]
and
\[
    w_{women} = (2609/(2707+2609))/(315/(154+315)) = 0.73
\]
}
% \clearpage

\item\pts{6} Using the weights you computed in part (b), compute the
  weighted proportion of students at Yale who plan on becoming parents
  (you do not have to compute the SE).
\[
     \hat p_w = \frac{w_{men}({\em yesses})_{men} + w_{women}({\em
         yesses})_{women}}{w_{men}({\em total})_{men} + w_{women}({\em
         total})_{women}} =
     \frac{(1.55)(134)+(0.73)(247)}{(1.55)(154)+(0.73)(315)} =  0.8279
\]

\item\pts{6} Use the values of the weights to explain the difference
  between your answers in parts (a) and (c).

{\em 
$\hat p_w$ is bigger than $\hat p_{srs}$ because the men, who had the
  higher proportion thinking they would be parents someday, are
  weighted more heavily (1.55 per person) than the women (0.73 per
  person). 
}
\end{enumerate}



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% \clearpage

% Examples from lecture 09...

\item\pts{18} A 36-303 group wants to do a survey of student attitudes
  toward Carnegie Mellon sports teams and sporting events.  In their
  survey proposal they specify the {\em target population} to be all
  currently enrolled undergraduates at Cargnegie Mellon, and they
  indicate that to collect data they will advertise for volunteers to
  take the survey on FaceBook.com, with a link to a formal survey
  instrument at QuestionPro.com.
\begin{enumerate}

\item\pts{6} 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.

{\em
Here are two, different, acceptable answers:
\begin{itemize}
\item They didn't explicitly state a sampling frame, but the sampling
  frame is effectively CMU students with facebook accounts.
\item They didn't specify a sampling frame.  An acceptable, usable
  sampling frame would be the list of undergraduates in C-Book.
\end{itemize}

}


\item\pts{6} Write down \ul{\bf two} very likely sources of {\em coverage
  error} for this survey as currently proposed.

{\em 

Here are two sources of coverage error; other answers may also be
acceptable.
\begin{itemize}
\item Not all CMU undergraduates have facebook accounts 
\item Since it is a volunteer survey, only volunteers will respond.
  As we know, volunteers are often different from the target
  population at large (for example, they care enough to volunteer!).
\end{itemize}
}

\item\pts{6} Make one big suggestion that this group could implement
  to improve their sampling plan.

{\em 
Here is one acceptable answer; other answers may also be acceptable:
\begin{quote}
Using C-Book as a sampling frame, take an SRS without replacement from
C-Book. Contact those students and invite them to take the survey on
QuestionPro.com (using email, FaceBook, or any other way that gives
you a high chance of actually getting to them).
\end{quote}


}

\end{enumerate}


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\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}

}

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\end{enumerate}

\end{document}

