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\begin{document}

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%\date{January 23, 2003}

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\title{Stat 312: Lecture 9\\
Chi-squared distributions}

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\author{Moo K. Chung\\
mchung@stat.wisc.edu}
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\section*{Concepts}
\begin{enumerate}

\item Let $X_1,\cdots,X_n$ be a random sample from
$N(\mu,\sigma^2)$.
$$ \frac{(n-1)S^2}{\sigma^2} = \frac{1}{\sigma^2}\sum_{j=1}^n(X_j
- \bar X)^2 \sim \chi_{n-1}^2$$
\item Critical values for $\chi_n^2$
$$P(\chi_{1-\alpha/2,n}^2 < \chi_n^2 < \chi_{\alpha/2,n}^2)= 1
-\alpha$$

\item $100(1-\alpha)\%$ CI for $\sigma^2$:
$$\frac{(n-1)}{\chi_{\alpha/2,n-1}}s^2 < \sigma^2 <
\frac{(n-1)}{\chi_{1-\alpha/2,n-1}}s^2$$
\end{enumerate}

\section*{In-class problems}
Continuing 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 variance of 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}

Example 7.15.
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%\centering
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%\caption{$t_{1000}}
%\end{figure}




\section*{Self-study problems}
Excercise 7.43.

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