% 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}
\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}{9.9in} \setlength{\textwidth}{6.9in}
%\setlength{\columnsep}{0.3125in} \setlength{\topmargin}{0in}
\setlength{\headheight}{0in} \setlength{\headsep}{-1in}
\setlength{\parindent}{1pc}
\setlength{\oddsidemargin}{-.2in}  % Centers text.
\setlength{\evensidemargin}{-.2in}

% 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 19\\
Inference on Slope}

% For single author (just remove % characters)
\author{Moo K. Chung\\
mchung@stat.wisc.edu}
%\date{Feb 20, 2003}
% For two authors (default example)
\maketitle \thispagestyle{empty}
\subsection*{Concepts}
\begin{enumerate}
\item Given a model $$Y_j= \beta_0 + \beta_1
x_j + \epsilon_j,$$ we wish to test if $H_0: \beta_1 = 0$. Assume
that $\epsilon \sim N(0,\sigma^2)$. The least squares estimator of
$\beta_1$ is $\hat \beta_1 = \frac{S_{xy}}{S_{xx}}=\sum_{j=1}^n
c_jY_j,$ where $c_j = (x_j -\bar x)/S_{xx}.$
\item From $\sum_{j=1}^n c_j = 0$, $\sum_{j=1}^n c_jx_j=1$ and $\sum_{j=1}^n c_j^2 = S_{xx}^{-1}$ we can show
that $\hat \beta_1 \sim N(\beta_1, \sigma^2/S_{xx})$.
\item Inference on the slope parameter $\beta_1$ is based on $$T = \frac{\hat \beta_1 - \beta_1}{S_{\hat \beta_1}} \sim
t_{n-2},$$ where $S_{\hat \beta_1} = \hat \sigma/\sqrt{S_{xx}}
=\sqrt{\frac{SSE}{(n-2)S_{xx}}}$. From Lecture 18 formula, $SSE =
S_{yy}-S_{xy}^2/S_{xx}$, we get $S_{\hat \beta_1} =
\frac{1}{\sqrt{n-2}}\sqrt{\frac{S_{yy}}{S_{xx}}-(\frac{S_{xy}}{S_{xx}})^2}$.
There is a reason $S_{\hat \beta_1}$ is written in this way (see
example 1).

We reject $H_0$ if $|t|>t_{\alpha/2,n-2}$ at $100(1-\alpha)$
significance.

\end{enumerate}
\subsection*{In-class problems}
{\em Example 1.} 10 students took tow midterm exams.
\begin{verbatim}
> x<-c(80, 75, 60, 90, 99, 60, 55,
85, 65, 70)
> y<-c(70, 60, 70, 72, 95, 66, 60,
80, 70, 60)
> summary(lm(y~x))
Call: lm(formula = y ~ x)
 Residuals:
    Min      1Q  Median   3Q     Max
-10.908  -6.312   1.758 4.354 10.836
 Coefficients:
 Estimate Std. Error t value Pr(>|t|)
(Intercept) 29.48  13.23  2.23  0.056
 x           0.55   0.18  3.14  0.014
\end{verbatim}
The $P$-value here is based on two-sided test. Let's see if we
are getting the same $P$-value using Concept 3. Under $H_0$,
$\beta_1 =0$. So $t=\hat \beta_1/S_{\hat \beta_1}$. $\hat \beta_1
=0.55$. In R, we first compute
$\frac{S_{yy}}{S_{xx}}-(\frac{S_{xy}}{S_{xx}})^2$:
\begin{verbatim}
>cov(y,y)/cov(x,x)-(cov(x,y)/cov(x,x))^2
[1] 0.2476849
> Sbeta<-sqrt(0.2477)/sqrt(8)
> Sbeta
[1] 0.1759616
> 0.5523/Sbeta
[1] 3.138752
> 2*pt(-3.138,8)
[1] 0.01384698
\end{verbatim}
$P$-value$=P(|T|>3.138)=0.014$.\\

{\em Example 2.} There was a mistake in the solution to example 3,
lecture 18. Let $S_x^2$ be the sample variance of $x_j$. Since
$S_{xx} = \sum_{j=1}^n (x_j -\bar x)^2$, $S_{xx} = (n-1)S_x^2$,
$S_{yy} = (n-1)S_y^2$. So $S_{xy}/(n-1)$ is the sample covariance
between $x_j$ and $y_j$. In R, ${\tt cov}$ and ${\tt var}$ are
sample covariance and sample variance respectively. The correct
way to estimate $\sigma^2 = {\bf Var} Y_j$ is to use the
following R computation.
\begin{verbatim}
> 9*(cov(y,y)-beta1*cov(x,y))/8
[1] 58.48568
\end{verbatim}
Alternately, we can use
\begin{verbatim}
> 9*(cov(y,y)-cov(x,y)^2/cov(x,y))/8
[1] 58.4815
\end{verbatim}
Due to the rounding-off error, you are getting the different
values for the third decimal place.
\subsection*{Self-study problems}
Example 12.10., 12.11.

\end{document}