\documentclass[12pt]{article} 
%\usepackage{amsbsy} % for \boldsymbol and \pmb 
%\usepackage{graphicx} % To include pdf files!
\usepackage{amsmath}
\usepackage{amsbsy}
\usepackage{amsfonts}
%\usepackage[colorlinks=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref} % For links
\usepackage{fullpage}           % Good for US Letter paper

\topmargin=-0.75in
\textheight=9.5in
\usepackage{fancyhdr} 
\renewcommand{\headrulewidth}{0pt}  % Otherwise there's a rule under the header
\setlength{\headheight}{15.2pt}
\fancyhf{}
\pagestyle{fancy} 
\cfoot{Page \thepage {} of 2}
%

%\pagestyle{empty} % No page numbers


\begin{document}
%\enlargethispage*{1000 pt} 

\begin{flushright}
Name \underline{\hspace{60mm}} \\ 
$\,$ \\
Student Number \underline{\hspace{60mm}}
\end{flushright}   
\vspace{5mm}


\begin{center}   
{\Large \textbf{STA 442/2101 F 2012 Quiz 3}}\\
\vspace{1 mm}
\end{center}



\begin{enumerate} 

    \item An advertising agency obtains a list of several million ``live" email addresses, meaning that the owner of the email address has responded to a commercial message some time within the past 6 months. Before sending out junk email to the whole list, the advertising agency decides to test three versions of the junk email. Three independent random samples of size 500 are selected from the list, and one version of the email is sent to each sample. The response variable is whether the recipient replies within seven days: Yes or No.

        \begin{enumerate}
            \item (3 points) State a reasonable model for this problem.
            
            
\vspace{110mm}            
            
            
            
            
            \item (1 point) What is the parameter space $\Theta$?

\vspace{20mm}

            \item (1 point) State the null hypothesis in symbols.
        \end{enumerate}



\newpage
    \item (5 points) One version of the delta method says that if $X_1, \ldots, X_n$ are a random sample from a distribution with mean $\mu$ and variance $\sigma^2$, and $g(x)$ is a function  whose  derivative is continuous in a neighbourhood of  $x=\mu$, then $\sqrt{n}\left( g(\overline{X}_n)- g(\mu) \right) \stackrel{d}{\rightarrow} T \sim N(0,g^\prime(\mu)^2\sigma^2)$. In many applications, both $\mu$ and $\sigma^2$ are functions of some parameter $\theta$.  
    
Let $X_1, \ldots, X_n$ be a random sample from a chi-square distribution with parameter $\nu$, so that $E(X_i)=\nu$ and $Var(X_i)=2\nu$. Find a function $g(x)$ such that the limiting distribution of 
$Z_n = \sqrt{n}\left(g(\overline{X}_n)-g(\nu)\right)$ is \emph{standard} normal --- that is $Z_n \stackrel{d}{\rightarrow} Z \sim N(0,1)$. Show your work. 

\end{enumerate} 

\end{document}

% State a model next time.
    \item (2 points) A polling firm asks a random sample of $n$ registered voters in Quebec whether Quebec should separate from Canada and become an independent nation:  Yes or No. State a reasonable model for the data.
    
\vspace{25 mm}