\documentclass[12pt]{article}
\usepackage{verbatim, amssymb, amsmath, float, fullpage}
\begin{document}
\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: Health Behavior in School-Age Children Study}
\begin{tabular}{lp{10cm}}

\hline
Sponsor & Cross national effort with the collaboration of the World Health Organization\\
Collector & Collected by each participant country by an HBSC team and all countries data gathered and stored at the Norwegian Social Science Data Services. \\
Purpose & {Gain new insight into, and increase understanding of young people's health and well-being, health behaviours and their social context. The results must be allow inter-country comparisons.} \\
Year started & 1982\\
Target Population & Young people attending school, aged 11, 13 and 15 years old. \\
Sampling Frame & Varies per country. Usually a list of classes. A list of  schools if a list of classes is not available.\\
Sample Design & Clustered probability design. For each country classes (or schools) are the PSU and the students are randomly selected within them.\\
Sample Size & Varies per country. A total of 162,306 samples were collected counting all countries.\\
Use of Interviewer & None. Self administered. Supervised by teachers. In some countries, a team of trained interviewers was employed.\\
Mode of Administration & Self administered.\\
Computer Assistance & None.\\
Reporting Unit & Student\\
Time dimension & Administered over a 7-8 month period from October to May.\\
Frequency & Every four years.\\
Interviews per Round of Survey & One\\
Levels of observation & Countries, classes (or schools where not available) and students.\\
Web Link & { http://www.hbsc.org/}\\
\hline
\end{tabular}
\end{center}
\label{tab:question1}
\end{table}

A potential problem with this survey are the cross country comparisons. Although there exists a common protocol document to ensure a consistent application, the actual implementation of the survey within each country depends on the local team. This may raise suspicions about the quality and comparability of the data.\\

The technical description given in annex 1 of the main document is a little ``light". The document points to the main survey protocol document \begin{verbatim}
http://www.hbsc.org/downloads/Protocol_Section1.pdf
\end{verbatim} 
for further details, but this document is not accessible.

\section{Question 2}
See Table~\ref{tab:question2}. Credit to Grace DeForest.\\

\begin{table}[tbp]
\begin{center}
\caption{Survey: Hiremenow.com}
\begin{tabular}{lp{10cm}}
\hline
Sponsor & Hireme.com \\
Collector & Hireme.com\\
Purpose & - To determine if employers are willing to watch video resumes.\newline
			 - To determine the appropriate length of a video resume.\\
Year started & 2007 \\
Target Population & Employers seeking temporary and contract-work employees.\\
Sampling Frame & 300 human resource and business managers\\
Sample Design & Not clear\\
Sample Size & 300 subjects\\
Use of Interviewer & Not clear.\\
Mode of Administration & Not clear.\\
Computer Assistance & Not clear\\
Reporting Unit & Selected employers\\
Time dimension & administered over a one-month period\\
Frequency & Single application\\
Interviews per Round of Survey & not clear\\
Levels of observation & Human resources and business managers. Not clear about other levels.\\
Web Link & www.hiremenow.com\\
\hline
\end{tabular}
\end{center}
\label{tab:question2}
\end{table}

The information we actually have about the design and application of this survey is minimal. We do not know basic things such as how the respondents were selected or how the instrument was administered or which the sampling frame was. All the information available comes from press releases and there are not any accessible technical documents. Without this information the results of this survey are hard to believe.

\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*}

\end{document}
