

\subsection{\bf 5.5\hspace{1em} Student's $t$ Distribution}
\subsection{\bf for Hypothesis Testing---I}

\begin{slide}
\begin{itemize}
\item Suppose that the $x_i$ are drawn from a normal (Gaussian)
distribution of unknown mean and Standard Deviation.
\begin{eqnarray*}
H^{(0)}\ \mbox{is hypothesis that mean} & = & \mu^{(0)} \\
H^{(1)}\ \mbox{is hypothesis that mean} & = & \mu^{(1)} 
\end{eqnarray*}

\item Define
\begin{eqnarray*}
t & = & \frac{\left( \overline{x} - \mu^{(0)}\right) 
\sqrt{n}}{\overline{\sigma}} \ \mbox{where} \\
\overline{x} & = & \frac{1}{n}\ {\displaystyle \sum^n_{i=1}} x_i \\
\overline{\sigma}^2 & = & \frac{1}{n-1}\ {\displaystyle \sum^n_{i=1}} 
\left( x_i - \overline{x} \right)^2
\end{eqnarray*}

\item Under hypothesis (0), $t$ follows a so-called $t$-distribution
with $n-1$ degrees of freedom.  $(f=n-1)$.  

\item $t$ would be Gaussian with unit standard deviation if
$\overline{\sigma}$ taken from theory---not from experiment.
\end{itemize}
\end{slide}

