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\title{Stat 312: Lecture 24\\
Goodness-of-fit Test for Distributions}
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\author{Moo K. Chung\\
mchung@stat.wisc.edu}
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\subsection*{Concepts}
\begin{enumerate}
\item Knowing the exact distribution of a sample is important for
statistical analysis. We want to test if a sample $x_1,\cdots,x_n$
follows a certain probability distribution $P$. We perform a
$\chi^2$ goodness-of-fit test on the null hypothesis
$$H_0: P(X \in C_i) = p_i \mbox{ for all } i=1,\cdots,k$$
For a continuous distribution, $C_i$ are intervals.
\end{enumerate}

\subsection*{In-class problems} {\em Example 1.} Let us see if the
distribution of output tuft weight (Example 14.11.) follows an
exponential distribution with
parameter $\lambda$.\\
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
  \hline
Interval & 0-8 & 8-16 & 16-24& 24-32 & 32-40 & 40-48 & 48-56 & 56-64 & 64-70 \\
  \hline
Observed $\#$& 20 & 8 & 7 & 1 & 2 & 1 & 0 & 1 & 0 \\
 \hline
\end{tabular}\\
{\em Solution.} The density of the exponential random variable $X$
is given by $f(x)=\frac{1}{\lambda}e^{-x/\lambda}, x \geq 0$. Note
that $\int_0^8 \frac{1}{\lambda}e^{-x/\lambda} \doteq 0.5$.
Solving this, we estimate $\hat \lambda = 11.54$. Based on this,
we compute the expected number of elements in each category. For
instance, the expected number of element in the interval $[8,16]$
is $40 \int_0^8 \frac{1}{11.54}e^{-x/11.54} \; dx =
40(\tt{pexp(16,1/11.54)}-\tt{pexp(8,1/11.54)})$.
\begin{verbatim}
> interval<-c(8,16,24,32,40,48,56,64,70)
> p<- pexp(interval,1/r)
[1] 0.50 0.75 0.87 0.94 0.97 0.98 0.99 0.996 0.998
> p[2:9]-p[1:8]
[1] 0.25 0.12 0.06 0.03 0.016 0.008 0.004 0.002
> 40*(p[2:9]-p[1:8])
10  5  2.5  1.3  0.6 0.3 0.2 0.06
\end{verbatim}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
Expected $\#$ & 20 & 10 & 5 & 2.5 & 1.3 & 0.6 & 0.3 & 0.2 & 0.06 \\
  \hline
\end{tabular}\\
Then we compute $\chi^2 = 50$ and the $P$-value is
$\tt{pchisq(50,7)}=0$. So we reject $H_0$ and conclude that the
exponential distribution is not a good fit.
\subsection*{Self-study problems} Example 14.11. This is slightly
different from the in-class example setting. Exercise 12.14.


\subsection*{Assignment 8} Due May 8, 11:00am.
14.6., 14.16., 14.18., 14.26., 14.32. 
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