\setcounter{ExampleCounter}{1}

\begin{formula}{Rule of Thumb for Outliers}
If a point is more than two standard deviations away from the regression line, it is considered an outlier.

The standard deviation used is the standard deviation of the residuals $y-\hat{y}$.

\paragraph{Finding $s$} The standard deviation of the residuals is listed in the results of the \texttt{LinRegTTest}.
\end{formula}

\begin{example}{Regression Line}
A random sample of 11 statistics students produced the following data, where $x$ is the third exam score out of 80, and $y$ is the final exam score out of 200.
\begin{center}
\begin{tabular}{c c}
$x$ & $y$\\
\hline
\\
65 & 175\\
67 & 133\\
71 & 185\\
71 & 163\\
66 & 126\\
75 & 198\\
67 & 153\\
70 & 163\\
71 & 159\\
69 & 151\\
69 & 159
\end{tabular}
\end{center}

Below is the scatter plot with the regression line included.

\begin{center}
\begin{tikzpicture}
\begin{axis}[
    xmin=60, xmax=80,
    ymin=110, ymax=210,
    axis lines=center,
    axis on top=false,
    domain=0:1,
    x=0.4cm,
    y=0.04cm,
    xtick={60,65,...,80},
    xticklabels={60,65,...,80},
    ytick={110,120,130,...,200,210},
    yticklabels={110,120,130,...,200,210},
    axis lines=middle,
    axis line style={->},
    x label style={at={(axis description cs:0.5,-0.1)},anchor=north},
    y label style={at={(axis description cs:-0.11,.5)},rotate=90,anchor=south},
    xlabel={Third Test Score},
    ylabel={Final Exam Score},
    grid=none
    ]
	\addplot [blue,only marks,mark size=2] table {
	65 175
	67 133
	71 185
	71 163
	66 126
	75 198
	67 153
	70 163
	71 159
	69 151
	69 159
	};
	\addplot [blue, thick, domain=60:80, samples=10] {4.827*x-173.513};
\end{axis}
\end{tikzpicture}
\end{center}

Find any outliers.\\

\begin{center}
\includegraphics[width=3in]{CalcLinRegTTest4}
\end{center}
\[s=16.41\]

Draw bounds two standard deviations above and below the regression line.

\begin{center}
\begin{tikzpicture}
\begin{axis}[
    xmin=60, xmax=80,
    ymin=110, ymax=210,
    axis lines=center,
    axis on top=false,
    domain=0:1,
    x=0.4cm,
    y=0.04cm,
    xtick={60,65,...,80},
    xticklabels={60,65,...,80},
    ytick={110,120,130,...,200,210},
    yticklabels={110,120,130,...,200,210},
    axis lines=middle,
    axis line style={->},
    x label style={at={(axis description cs:0.5,-0.1)},anchor=north},
    y label style={at={(axis description cs:-0.11,.5)},rotate=90,anchor=south},
    xlabel={Third Test Score},
    ylabel={Final Exam Score},
    grid=none
    ]
	\addplot [blue,only marks,mark size=2] table {
	65 175
	67 133
	71 185
	71 163
	66 126
	75 198
	67 153
	70 163
	71 159
	69 151
	69 159
	};
	\addplot [blue, thick, domain=60:80, samples=10] {4.827*x-173.513};
	\addplot [red, dotted, thick, domain=60:80, samples=10] {4.827*x-140.693};
	\addplot [red, dotted, thick, domain=60:80, samples=10] {4.827*x-206.333};
\end{axis}
\end{tikzpicture}
\end{center}

The one outlier is the point \[(65,175).\]
\end{example}









