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\begin{document}

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%\date{January 23, 2003}

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\title{Stat 312: Lecture 8\\
 Student's $t$-distribution}
% For single author (just remove % characters)
\author{Instructor's valentine gift to students :)\\
Moo K. Chung\\
mchung@stat.wisc.edu}
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\maketitle
\thispagestyle{empty}
\section*{Concepts}
\begin{enumerate}

\item When $n$ large,
$$Z \sim \frac{\bar X -\mu}{S/\sqrt{n}} \sim N(0,1).$$

\item What if $n$ is small?
$$T \sim \frac{\bar X -\mu}{S/\sqrt{n}} \sim t_{n-1},$$
a t distribution with $n-1$ degrees of freedom (Review STAT311).

\item As $n \to \infty$, $t_{n} \to N(0,1)$.

\item Critical values:
$$P(-t_{\alpha/2,n-1} < T < t_{\alpha/2,n-1})=1-\alpha.$$

\item Suppose $X_1,\cdots,X_n \sim N(\mu,\sigma^2)$, where $\mu$
and $\sigma$ are unknown. $100(1-\alpha)$ CI for $\mu$ is
$$\bar x \pm t_{\alpha/2}s/\sqrt{n}.$$
\end{enumerate}

\section*{In-class problems}
\begin{verbatim}
> x<- -50:50/10
> plot(x,dt(x,1000),type='l')

> qt(.975, df = c(1:10,20,100,1000,10000))
 [1] 12.706205  4.302653  3.182449
 [4]  2.776445  2.570582  2.446912
 [7]  2.364624  2.306004  2.262157
[10]  2.228139  2.085963  1.983972
[13]  1.962339  1.960201
> qnorm(0.975)
[1] 1.959964
> pt(2.228,10)
[1] 0.9749941
\end{verbatim}
%\begin{figure}[t]
%\centering
%\renewcommand{\baselinestretch}{1}
%\includegraphics[scale=0.5]{tdensity.ps}
%\caption{$t_{1000}}
%\end{figure}

Example 7.12. Fat content of 10 randomly selected hot dogs.
Assuming that observations come from normal distribution, find a
95$\%$ CI for the population mean fat content.
\begin{verbatim}
> x<-c(25.2,21.3,22.8,17.0,29.8,21.0,
       25.5,16.0,20.9,19.5)
> sd(x)
[1] 4.13414
> mean(x)+qt(0.975,9)*sd(x)/sqrt(10)
[1] 24.85739
> mean(x)-qt(0.975,9)*sd(x)/sqrt(10)
[1] 18.94261
\end{verbatim}
Assuming $X_i \sim N(\mu,\sigma^2 = 4.13^2)$, find a $95\%$ CI for
$\mu$.
\begin{verbatim}
> mean(x)+qnorm(0.975)*4.13/sqrt(10)
[1] 24.45975
> mean(x)-qnorm(0.975)*4.13/sqrt(10)
[1] 19.34025
\end{verbatim}
\section*{Self-study problems}
Exercise 7.29., 7.33.
\section*{Assignment III.}
Due Feb 27, 11:00am.\\
Exercise 7.32, 7.34., 7.42., 7.44., 8.10.\\
The first midterm will cover chapters 6,7 and 8.1. Sample midterm
problems will be posted on Feb. 25.
\end{document}