12.12 Chapter 12 Formulas

$SS_{xx} = (n-1) s_{rx}^{2}$ $SS_{yy} = (n-1) s_{y}^{2}$ $SS_{xy} = \sum (xy) - n \cdot \bar{x} \cdot \bar{y}$	Correlation Coefficient $r = \frac{SS_{xy}}{\sqrt{\left(SS_{xx} \cdot SS_{yy}\right)}}$
Correlation t-test $H_{0}: \rho = 0$ $H_{1}: \rho \neq 0$ $t = r \sqrt{\left( \frac{n-2}{1-r^{2}} \right)}$ $df = n-2$	Regression Equation (Line of Best Fit) $\hat{y} = b_{0} + b_{1}x$
Slope $b_{1} = \frac{SS_{xy}}{SS_{xx}}$	y-Intercept $b_{0} = \bar{y} - b_{1} \bar{x}$
Slope t-test $H_{0}: \beta_{1} = 0$ $H_{1}: \beta_{1} \neq 0$ $t = \frac{b_{1}}{\sqrt{ \left( \frac{MSE}{SS_{xx}} \right)}}$ $df = n - p - 1 = n - 2$	Slope/Model F-test $H_{0}: \beta_{1} = 0$ $H_{1}: \beta_{1} \neq 0$
Standard Error of Estimate $s_{est} = \sqrt{ \frac{\sum \left(y_{i} - \hat{y}_{i}\right)^{2}}{n-2}} = \sqrt{MSE}$	Residual $e_{i} = y_{i} - \hat{y}_{i}$
Prediction Interval $\hat{y} \pm t_{\alpha / 2} \cdot s_{est} \sqrt{\left( 1 + \frac{1}{n} + \frac{\left(x - \bar{x}\right)^{2}}{SS_{xx}} \right)}$	Coefficient of Determination $R^{2} = (r)^{2} = \frac{SSR}{SST}$
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 $H_{0}: \beta_{1} = \beta_{2} = \cdots = \beta_{p} = 0$ $H_{1}:$ At least one slope is not zero.
Adjusted Coefficient of Determination $R_{adj}^{2} = 1 - \left( \frac{\left(1 - R^{2}\right) (n-1)}{(n - p - 1)} \right)$

Adapted from Mostly Harmless Statistics by Rachel Webb (Portland State University, https://mostlyharmlessstat.wixsite.com/webpage), © Rachel Webb, licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.