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10.3: Chapter 10 Formulas

EC225: Research Methods and Statistics for Economics · LibreTexts

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    From LibreTexts, under CC BY-SA 4.0, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    \(SS_{xx} = (n-1) s_{rx}^{2}\)
    \(SS_{yy} = (n-1) s_{y}^{2}\)
    \(SS_{xy} = \sum (xy) - n \cdot \bar{x} \cdot \bar{y}\)    		 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
    \(R_{adj}^{2} = 1 - \left( \frac{\left(1 - R^{2}\right) (n-1)}{(n - p - 1)} \right)\)    		  

    Regression ANOVA table with equations.

    This page titled 10.3: Chapter 10 Formulas was last modified on Thu, 12 Jun 2025 19:18:17 GMT and is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Rachel Webb via source content that was edited to the style and standards of the LibreTexts platform.