12.12 Chapter 12 Formulas

$S S_{x x} = (n - 1) s_{r x}^{2}$ $S S_{y y} = (n - 1) s_{y}^{2}$ $S S_{x y} = \sum (x y) - n \cdot \bar{x} \cdot \bar{y}$	Correlation Coefficient $r = \frac{S S_{x y}}{\sqrt{(S S_{x x} \cdot S S_{y y})}}$
Correlation t-test $H sub 0: ρ equals 0$ $H sub 1: ρ not equal to 0$ $t = r \sqrt{(\frac{n - 2}{1 - r^{2}})}$ $d f = n - 2$	Regression Equation (Line of Best Fit) $\hat{y} = b_{0} + b_{1} x$
Slope $b_{1} = \frac{S S_{x y}}{S S_{x x}}$	y-Intercept $b_{0} = \bar{y} - b_{1} \bar{x}$
Slope t-test $H sub 0: β sub 1 equals 0$ $H sub 1: β sub 1 not equal to 0$ $t = \frac{b_{1}}{\sqrt{(\frac{M S E}{S S_{x x}})}}$ $d f = n - p - 1 = n - 2$	Slope/Model F-test $H sub 0: β sub 1 equals 0$ $H sub 1: β sub 1 not equal to 0$
Standard Error of Estimate $s_{e s t} = \sqrt{\frac{\sum {(y_{i} - {\hat{y}}_{i})}^{2}}{n - 2}} = \sqrt{M S E}$	Residual $e_{i} = y_{i} - {\hat{y}}_{i}$
Prediction Interval $\hat{y} \pm t_{α / 2} \cdot s_{e s t} \sqrt{(1 + \frac{1}{n} + \frac{{(x - \bar{x})}^{2}}{S S_{x x}})}$	Coefficient of Determination $R^{2} = (r)^{2} = \frac{S S R}{S S T}$
Multiple Linear Regression Equation $\hat{y} = b_{0} + b_{1} x_{1} + b_{2} x_{2} + \cdot + b_{p} x_{p}$	Model F-Test for Multiple Regression $H_{0} : β_{1} = β_{2} = \dots = β_{p} = 0$ $H sub 1:$ At least one slope is not zero.
Adjusted Coefficient of Determination $R_{a d j}^{2} = 1 - (\frac{(1 - R^{2}) (n - 1)}{(n - p - 1)})$

Adapted from Mostly Harmless Statistics by Rachel Webb (Portland State University), hosted on LibreTexts (stats.libretexts.org) and licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.