Correlation and Regression

$r$ ranges from $-1$ to $1$.

• $r = 1$: perfect positive linear correlation.
• $r = -1$: perfect negative linear correlation.
• $r = 0$: no linear correlation.

$r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$

where $S_{xy} = \sum xy - \frac{\sum x \sum y}{n}$, $S_{xx} = \sum x^2 - \frac{(\sum x)^2}{n}$, $S_{yy} = \sum y^2 - \frac{(\sum y)^2}{n}$.

$y = a + bx$ where $b = \frac{S_{xy}}{S_{xx}}$ and $a = \bar{y} - b\bar{x}$.

The line passes through $(\bar{x}, \bar{y})$.

$H_0: \rho = 0$ (no correlation). Compare $r$ with critical values.