Correlation and regression by computer

You will write four functions: the correlation coefficient, the slope and the intercept of the line of $y$ on $x$, and a prediction. The checker tests them on data with known answers, and on data that lie exactly on a line.

The idea. Use the sums $S_{xx} = \sum (x - \bar{x})^2$, $S_{yy}$ and $S_{xy}$. Then $r = \dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$, $b = \dfrac{S_{xy}}{S_{xx}}$ and $a = \bar{y} - b\bar{x}$.

Your tasks.

1. Finish correlation(x, y).
2. Finish slope(x, y), the slope of the line of $y$ on $x$.
3. Finish intercept(x, y).
4. Finish predict(x, y, new_x), which uses the line of $y$ on $x$.