

\subsection{\bf Student's $t$ Distribution}
\subsection{\bf for Hypothesis Testing---II}

\begin{slide}
\begin{itemize}
\item The probability distribution is
\[
p(t) = \frac{\Gamma \left( \frac{f+1}{2} \right)}{\sqrt{n\pi}\ \Gamma
\left( \frac{f}{2} \right)}\ \ \frac{1}{\left( 1 + \frac{t^2}{f}
\right)^{(f+1)/2}}
\]

\item $f=1$ is Cauchy distribution.  

\item As $f\rightarrow\infty$, the $t$ distribution becomes Gaussian
with mean 0 and Standard Deviation 1.

\item This can be expected for as $f\rightarrow\infty$, using data to
find $\overline\sigma$, should be accurate.

\vspace*{10pt}
\centerline{\psfig{figure=fig24b.phys.ps,clip=true,width=3.75in}}
\vspace*{10pt}

\item If $\mu_1 > \mu_0$, choose $0< t_a < \left( \mu_1 - \mu_0
\right)$, and

\item one could say that 0 is true if $t<t_a$ and 1 is true if $t>t_a$.
\end{itemize}
\end{slide}

