Answer
$r=0.9958$,
best, not perfect.
Work Step by Step
Regression line: $y=mx+b,$
$m=\displaystyle \frac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^{2})-(\sum x)^{2}}\qquad b=\frac{\sum y-m(\sum x)}{n}$
$n=$ number of data points.
Correlation Coefficient: $r=\displaystyle \frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{n(\sum x^{2})-(\sum x)^{2}}\cdot\sqrt{n(\sum y^{2})-(\sum y)^{2}}}$
Interpretation
If $r$ is positive, the regression line has positive slope;
if $r$ is negative, the regression line has negative slope.
If $r=1$ or $- 1$, then all the data points lie exactly on the regression line;
if it is close to $\pm 1$, then all the data points are close to the regression line.
If $r$ is close to $0$, then $y$ does not depend linearly on $x.$
-----
(a)
Using Excel, we set up a table
\begin{array}{|c|c|c|c|c|c}
& x & y & xy & xx & yy \\
\hline & 1 & 3 & 3 & 1 & 9 \\
& 2 & 4 & 8 & 4 & 16 \\
& 5 & 6 & 30 & 25 & 36 \\\hline
\Sigma & {\bf 8} & {\bf 13} & {\bf 41} & {\bf 30} & {\bf 61} \\\hline
points & 3 & & 123 & 90 & 183 \\
& & & & & \\
m= & 0.730769 & & & & \\
b= & 2.384615 & & & & \\
& & & r= & 0.995871 & \\
\end{array}
m,b, and r are calculated according to the above formulas.
Regression line:$\qquad y=0.7308x+2.3846,$
Correlation Coefficient: $\qquad r=0.9958$,
which after calculating (b) and (c), turns out to be the best of the three
