% This is a simple LaTex sample document that gives a submission format
%   for IEEE PAMI-TC conference submissions.  Use at your own risk.

% Make two column format for LaTex 2e.
\documentclass[11pt,twocolumn]{article} %,twocolumn
\usepackage[dvips]{graphicx,graphics,color}
\usepackage{times,amsmath,amsfonts}

% Use following instead for LaTex 2.09 (may need some other mods as well).
% \documentstyle[times,twocolumn]{article}

% Set dimensions of columns, gap between columns, and paragraph indent
\setlength{\textheight}{9.2in} \setlength{\textwidth}{6.875in}
%\setlength{\columnsep}{0.3125in} \setlength{\topmargin}{0in}
\setlength{\headheight}{0in} \setlength{\headsep}{-1in}
\setlength{\parindent}{1pc}
\setlength{\oddsidemargin}{-.1875in}  % Centers text.
\setlength{\evensidemargin}{-.1875in}

% Add the period after section numbers.  Adjust spacing.
\newcommand{\Section}[1]{\vspace{-8pt}\section{\hskip -1em.~~#1}\vspace{-3pt}}
\newcommand{\SubSection}[1]{\vspace{-3pt}\subsection{\hskip -1em.~~#1}
        \vspace{-3pt}}
\newcommand{\bqn}{\begin{eqnarray}}
\newcommand{\eqn}{\end{eqnarray}}
\newcommand {\diff}[1] {\frac{\partial}{\partial #1}}
\newcommand{\jacob}[3]{\frac{\partial^2 #3}{\partial #1 \partial #2}}
\newcommand{\der}[2]{\frac{\partial #2}{\partial #1}}
\begin{document}

% Don't want date printed
%\date{January 23, 2003}

% Make title bold and 14 pt font (Latex default is non-bold, 16pt)
\title{Stat 312: Lecture 5\\
 Confidence Intervals}

% For single author (just remove % characters)
\author{Moo K. Chung\\
mchung@stat.wisc.edu}
% For two authors (default example)
\maketitle
\thispagestyle{empty}
\section*{Concepts}
\begin{enumerate}
\item For population parameter $\theta$, 95$\%$ confidence interval $(\hat \theta_L,\hat \theta_U)$ of $\theta$ is computed by solving $$P( \hat \theta_L \leq \theta \leq \hat \theta_U)=0.95$$
   
\item Let $X_i \sim N(\mu,\sigma^2)$ with known $\sigma^2$ and unknown $\mu$. 95$\%$ {\em confidence interval} for $\mu$ is $$\hat \mu_L =\bar x - 1.96 \cdot \sigma/\sqrt{n},\; \hat \mu_U=\bar x + 1.96 \cdot \sigma/\sqrt{n}.$$

\item  Let $X_i \sim N(\mu,\sigma^2)$ with known $\sigma^2$ and unknown $\mu$. $100(1-\alpha)\%$ confidence interval for $\mu$ is. 
$$\hat \mu_L =\bar x - z_{\alpha/2} \cdot \sigma/\sqrt{n},\; \hat \mu_U=\bar x + z_{\alpha/2} \cdot \sigma/\sqrt{n}.$$

\end{enumerate} 

\section*{In-class problems}
Continuing Exercise 6.25. Assuming $\sigma=18.86$, find $95\%$ confidence interval of $\mu$.
\begin{verbatim}
> b<-qnorm(0.975,0,sigma/sqrt(10))
> a<-qnorm(0.025,0,sigma/sqrt(10))
> b
[1] 11.68933
> a
[1] -11.68933
> mean(X)-b
[1] 372.7107
> mean(X)-a
[1] 396.0893

> qnorm(0.995)
[1] 2.575829
> qnorm(0.99)
[1] 2.326348
> qnorm(0.975)
[1] 1.959964
> qnorm(0.95)
[1] 1.644854
\end{verbatim}
\section*{Self-study problems}
Example 7.2.,7.3.,7.4.,7.5.

\section*{Assignment II.}
Due Feb 13. 11:00am.\\ 
Exercise 6.30., 7.4., 7.10., 7.14., 7.22., 7.26.\\

{\bf Bonus Problem} (Hard). Let $X_1,\cdots,X_n$ be a random sample with mean $\mu$ and variance $\sigma^2$. Among all estimators of the form $\hat \mu = \sum_{j=1}^n c_jX_J$, find MVUE and prove that it is in fact MVUE. Hand in this problem directly to the instructor {\em during the office hour}. No partial credit for this problem.

\section*{Midterms}
One page note and a calculator are allowed.\\
Midterm I. March 4, 11:00am-12:15pm (Tuesday).\\
Midterm II. April 15, 11:00am-12:15pm (Tuesday).\\
4 problems will be given: 1 easy, 2 medium difficulty and 1 challenging problems.
Mock midterm problems will be posted on the WEB one week before the midterms. 
\end{document}