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\title{36-303 Sampling, Surveys \& Society \\ Homework 01 Solutions}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  PROB1  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Question 1}
See Table~\ref{tab:question1}.\\
\begin{table}[tbp]
\begin{center}
\caption{Survey: The State of Our Nation's Youth}
\begin{tabular}{lp{10cm}}

\hline
Sponsor & The Horatio Alger Association of Distinguished American\\
Collector & collected by Peter D. Hart Research Associates \\
Purpose & {to give voice to the issues, opinions and thoughts of our nation's young people.} \\
Year started & 2005\\
Target Population & students age 13 to 19 who identified themselves as ninth-through-twelfth graders \\
Sampling Frame & compiled list provided by american student list, the well-respected national list management firm, which
specializes in maintaining lists of k-12 students\\
Sample Design & sample was created by selecting 505 geographic points randomly and proportionate to the population of each region and, within
each region, by size of place.  All telephone numbers have an equal chance to be included.  One student
from each selected household was included, selected by a systematic procedure to provide a balance of respondents by sex\\
Use of Interviewer & interviewer administered\\
Mode of Administration & telephone interview\\
Computer Assistance & not clear\\
Reporting Unit & student\\
Time dimension & conducted from May 9 to 12, 2005\\
Frequency & every year\\
Interviews per Round of Survey & one\\
Levels of observation & student\\
Web Link & \url{ http://www.horatioalger.com/pdfs/state05.pdf/}\\
\hline
\end{tabular}
\end{center}
\label{tab:question1}
\end{table}

(b)
I couldn't find any information about computer assistance, although I think they probably used computer assistance
in randomizing geographic points and telephone numbers.  Also, it is simply stated that one student from each household was selected by a systematic procedure
to provide an approximate balance of respondents by sex, readers would like to know more details about the systematic procedure to decide on the validity of the survey.
It is generally common to get nonresponse in telephone survey, I couldn't find information about how was this problem
dealt with in the survey so there might be some bias in the findings.  I am not sure if the stratification method used in this survey
was enough to draw conclusions about complex issues such as diversity of America's youth.             
   
(c)

\[ \frac{2*\sqrt{0.5*0.5}}{\sqrt{1005}}=3.15 \% \]



\section{Question 2}
See Table~\ref{tab:question2}. \\

\begin{table}[tbp]
\begin{center}
\caption{Survey: Raising the Bar: Employers' views on college learning in the wake of the economic downturn}
\begin{tabular}{lp{10cm}}
\hline
Sponsor &  The Association of American Colleges and Universities \\
Collector & hart Research Associates\\
Purpose & $(1)$find out if the majority of employers are expect to increase their hiring in the next year.  If so, what kind of education
background are most favorable to them for hiring
			 $(2)$ find out what kind of skills for new graduates do employers favor for hiring
			 $(3)$find out if the employers are satisfied with college today in educating their employees to face challenges.   If not, what needs to be improved in their
			 opinion\\
Year started & 2009 \\
Target Population & employers whose organizations have at least 25 employees and report that $25\%$ or more of their
new hires hold either an associates degree from a two-year college or a bachelor's degree from a four-year college\\
Sampling Frame & not clear\\
Sample Design & not clear\\
Sample Size & 302 employers\\
Use of Interviewer & yes, by associates at Hart research\\
Mode of Administration & not clear\\
Computer Assistance & not clear\\
Reporting Unit & selected executives at private sector and non-profit organizations, including owners, CEOs, presidents, C-suite level executives,
and vice presidents\\
Time dimension & conducted from October 27 to November 17,2009\\
Frequency & single application\\
Interviews per Round of Survey & one\\
Levels of observation & employer, company\\
Web Link & \url{ http://www.aacu.org/leap/documents/2009_EmployerSurvey.pdf}\\
\hline
\end{tabular}
\end{center}
\label{tab:question2}
\end{table}

(b) The information we actually have about the design and application of this survey is minimal. We do not know basic information such as the sampling frame 
and sampling design.  

(c)  \[ \frac{2*\sqrt{0.5*0.5}}{\sqrt{302}}=5.75 \% \]
\section{Question 3}
\subsection{(a)}
\begin{eqnarray*}
E[X_i] &=& E[X_1] \mbox{~~~(because the $X_i$'s are iid)}\\
&=& 1 \cdot p + 0 \cdot (1-p)\\
&=& p
\end{eqnarray*}
\begin{eqnarray*}
V[X_i] &=& E[X_i^2] - E[X_i]^2\\
&=& E[X_1^2] - E[X_1]^2\\
&=& 1^2\cdot p + 0^2 \cdot p - p^2\\
&=& p-p^2=p(1-p)
\end{eqnarray*}
\subsection{(b)}
\begin{eqnarray*}
E\left[\sum_{i=1}^{n}X_i\right] &=& \sum_{i=1}^{n}E[X_i]\\
&=& \sum_{i=1}^{n}E[X_1] \\
&=& np
\end{eqnarray*}





\begin{eqnarray*}
V\left[\sum_{i=1}^{n}X_i\right] &=& \sum_{i=1}^{n}V[X_i]\\
&=& \sum_{i=1}^{n}V[X_1] \\
&=& np(1-p)
\end{eqnarray*}
Note that the identity  $V\left[\sum_{i=1}^{n}X_i\right] = \sum_{i=1}^{n}V[X_i]$ is true because we know that the random variables $X_1, X_2, ..., X_n$ are independent. This is {\bf not} true in general.
\subsection{(c)}
Using the results from the previous sections,
\begin{eqnarray*}
E\left[\hat{p}\right] &=& E\left[{Y \over n}\right]\\
&=& {1 \over n}E\left[Y\right]\\
&=& {1 \over n} \cdot np \\
&=& p
\end{eqnarray*}
The above property is called "unbiasedness". In this case we say that $\hat{p}$ is an {\it unbiased} estimator of $p$.\\

Again using the results from the previous sections,
\begin{eqnarray*}
V\left[\hat{p}\right] &=& V\left[{Y \over n}\right]\\
&=& {1 \over n^2}V\left[Y\right]\\
&=& {1 \over n^2} \cdot np(1-p) \\
&=& {p(1-p) \over n}
\end{eqnarray*}

\section{Question 4}
\subsection{(a)}
\begin{eqnarray*}
P[X=1]= P[X=1, Y=4] + P[X=1, Y=3]= \frac{1}{2}\\
P[X=2]=  1- P[X=1]=\frac{1}{2}\\
P[Y=3]= P[X=1, Y=3] + P[X=2, Y=3]= \frac{1}{2}\\ 
P[Y=4] = 1- P[Y=3]= \frac{1}{2}\\
\end{eqnarray*}

Now, to prove that X and Y are not independent, we just need to show a counter example:
\begin{eqnarray*}
(P[X=1,Y=4]=\frac{1}{8}) \neq (P[X=1] *P[Y=4]=\frac{1}{4})\\
\end{eqnarray*}

\subsection{(b)}
assuming X and Y are independent, then
\begin{eqnarray*}
P[X=x | Y=y] = \frac{P[X=x, Y=y] }{P[Y=y]} =\frac{P[X=x] * P[Y=y]}{P[Y=y]} = P[X=x]\\
\end{eqnarray*}



\end{document}
