\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 2}}\\
\vspace{1 mm}
\end{center}



\begin{enumerate} 

    \item (5 points) Let $X_1 , \ldots, X_n$ be a random sample from a Binomial distribution with parameters $3$ and $\theta$. That is, $P(X_i = x_i) = \binom{3}{x_i} \theta^{x_i} (1-\theta)^{3-x_i}$ for $x_i=0,1,2,3$, where $\theta$ is \emph{strictly} between zero and one. Let 
\begin{displaymath}
    \widehat{\theta}_n = \frac{1}{3}\sqrt{\frac{1}{n}\sum_{i=1}^n X_i^2}.
\end{displaymath}
Is $\widehat{\theta}_n$ a consistent estimator of $\theta$? Answer Yes or No. \textbf{Circle the word Yes or the word No}. Show your work. You may use without proof the fact that if $Y$ is a Binomial random variable with parameters $k$ and $\theta$, $E(Y)=k\theta$ and $Var(Y)= E(Y^2)-[E(Y)]^2 = k\theta(1-\theta)$.


\newpage
    \item (5 points) This is straight from the homework. A polling firm plans to ask a random sample of registered voters in Quebec whether Quebec should separate from Canada and become an independent nation:  Yes or No. Suppose we intend to test whether the true percent favouring independence is different from 50\%, and in fact the true percent is 53\%. What is the minimum sample size required to reject the null hypothesis at $\alpha=0.05$ with probability at least 0.80, using a 2-sided $Z_2$ test? Recall that
    \begin{displaymath}
    Z_2 = \frac{\sqrt{n}(\overline{Y}-\theta_0)}{\sqrt{\overline{Y}(1-\overline{Y})}}
\end{displaymath}
\emph{Write the required sample space on this paper, in the space below.} Just write the number and nothing else.
\vspace{10mm}
    
\textbf{Please attach your R printout to this quiz paper.} 
\begin{itemize}
    \item Please turn in \emph{only} the calculation requested.
    \item If the calculation is not on a separate sheet, please cross out irrelevant material.
    \item \textbf{Circle the required sample size on your printout.}
    \item It would be safest to write your name and student number on the printout you turn in, even though it is attached to the quiz paper.
  \end{itemize}

    
\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}