\setcounter{ExampleCounter}{1}
It may be helpful to think back to the normal distribution and the t distribution as we meet the chi-square distribution (typically written $\chi^2$-distribution, as $\chi$ is the Greek letter chi, pronounced `kai').
\begin{itemize}
\item The area under a portion of the graph is equal to the proportion of the distribution in that range (the probability of being in that range)
\item Like the t distribution, the graph of the $\chi^2$-distribution depends on the \emph{degrees of freedom} (more on that later)
\item The graph below shows $\chi^2$-distributions with degrees of freedom ranging from 1 to 8
\begin{center}
\begin{tikzpicture}[scale=0.5,domain=.001:16,samples=200,thick]
\clip (-1,-1) rectangle (17,10);
\foreach[count=\k,evaluate={\z=\k>2?"(0,0)--":"";\c=10*\k}]
\g in {sqrt(pi),1,sqrt(pi)/2,1,3/4*sqrt(pi),2,15/8*sqrt(pi),6}
\draw[color=blue!\c!red,yscale=30] \z
plot (\x,{exp(ln(\x/2)*\k/2-ln(\x)-\x/2-ln(\g))});
\end{tikzpicture}
\end{center}
\item For the sake of simplicity, from here on we'll use a graph like the one below (which happens to have 3 degrees of freedom), but remember that the shape of the graph varies in reality
\begin{center}
\begin{tikzpicture}
\begin{axis}[no markers, axis lines*=none, xlabel=$x$,
hide y axis,hide x axis,width=12cm, height=6cm]
%\addplot[domain=\chiright:15]
%{chisquare(x,\grauliber)} \closedcycle;
%\addplot[domain=0.01:\chiright]
%{chisquare(x,\grauliber)} \closedcycle;
\addplot[samples at={0.2,0.19,...,0}, thick]
{chisquare(x,\grauliber)} -- (0,0);
\addplot[samples=50, thick, smooth, domain=0.2:15]
{chisquare(x,\grauliber)};
\draw (0,0) -- (150,0);
\end{axis}
\end{tikzpicture}
\end{center}
\item Just as with the normal and t distributions, we'll have a test statistic (before we had $z$ and $t$ statistics; now we'll have a $\chi^2$ statistic
\end{itemize}
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\pagebreak
\subsection{Calculation}
To use the calculator to find areas under the graph of the $\chi^2$-distribution,
\begin{itemize}
\item Open the distributions menu (\texttt{2ND} $\to$ \texttt{DISTR})
\item Look for the $\chi^2$\texttt{cdf} option
\item Enter the lower and upper bounds and the degrees of freedom, in that order
\item Example for TI-83: to calculate the area above 2.5 when $df=5$, you would type in $\chi^2$\texttt{cdf(2.5,1000000,5)}.
\end{itemize}
With that, we're ready to start testing goodness-of-fit.
\vfill
\pagebreak