12.4: Chapter 12 Formulas
- Page ID
- 24078
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| \(SS_{xx} = (n-1) s_{rx}^{2}\) \(SS_{yy} = (n-1) s_{y}^{2}\) \(SS_{xy} = \sum (xy) - n \cdot \bar{x} \cdot \bar{y}\) |
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 \(R_{adj}^{2} = 1 - \left( \frac{\left(1 - R^{2}\right) (n-1)}{(n - p - 1)} \right)\) |
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