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MONT 106N, Identifying Patterns \\
Discussion on Computing Correlation Coefficients\\
October 26, 2009
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\noindent
{\it Background}
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The process for computing the correlation coefficient of
two data sets $x_i$ and $y_i$, $i = 1, ..., n$ is as follows:
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\begin{itemize}
\item First, compute $ave_x, ave_y$, and $SD_x, SD_y$.
\item Then convert the $x_i$ and $y_i$ to standard units:
$$z_i = \frac{x_i - ave_x}{SD_x}, \qquad w_i = \frac{y_i - ave_y}{SD_y}.$$
\item Finally,
\begin{align*}
r &= {\rm ave\ of\ products\ } z_iw_i\\
  &= \frac{1}{n}(z_1w_1 + z_2w_2 + \cdots + z_nw_n).
\end{align*}
\end{itemize}
We also saw that the computations could be arranged in a 
convenient tabular format to facilitate the various steps.
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\noindent
{\it Practice Problem}
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The heights  (in inches) and weights (in pounds) of $n = 10$ offensive backfield 
high school football players were measured as follows:
$$
\begin{matrix} 
{\rm Player} & {\rm Height} & {\rm Weight} \\
1 & 73 & 185\\
2 & 71 & 175\\
3 & 75 & 200\\
4 & 72 & 210\\
5 & 72 & 190\\
6 & 75 & 195\\
7 & 67 & 150\\
8 & 69 & 170\\
9 & 71 & 180\\
10& 69 & 175
\end{matrix}
$$
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\begin{enumerate}
\item[A)] Construct a scatter plot for this data (use the graph paper provided).  What
can you say from the general shape of your scatter plot?
\item[B)] Compute $ave_x$, $ave_y$, $SD_x$, and $SD_y$.
\item[C)] Convert all the $x_i$ and $y_i$ to the appropriate standard
units.
\item[D)] Compute the correlation coefficient $r$.  What does the value say about
the association between height and weight for these football players.  (Of course,
by extension, one would like to say the same thing about \emph{all} high school
football offensive backfield players!)
\item[E)] Compute the equation of the SD-line and add it to your scatter plot.
\end{enumerate}
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