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10.9: Formula List

  • Page ID
    549
    • Anonymous
    • LibreTexts

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    Learning Objectives
    • Listing of all formulas used throughout the chapter.

    \[SS_{xx}=\sum x^2-\frac{1}{n}\left ( \sum x \right )^2\; \; SS_{xy}=\sum xy-\frac{1}{n}\left ( \sum x \right )\left ( \sum y \right )\; \; SS_{yy}=\sum y^2-\frac{1}{n}\left ( \sum y \right )^2 \nonumber \]

    Correlation coefficient:

    \[r=\frac{SS_{xy}}{\sqrt{SS_{xx}SS_{yy}}} \nonumber \]

    Least squares regression equation (equation of the least squares regression line):

    \[\hat{y}=\hat{\beta _1}x+\hat{\beta _0}\; \; \text{where}\; \; \hat{\beta _1}=\frac{SS_{xy}}{SS_{xx}}\; \; \text{and}\; \; \hat{\beta _0}=\bar{y}-\hat{\beta _1}\bar{x} \nonumber \]

    Sum of the squared errors for the least squares regression line:

    \[SSE=SS_{yy}-\hat{\beta _1}SS_{xy} \nonumber \]

    Sample standard deviation of errors:

    \[S_\varepsilon =\sqrt{\frac{SSE}{n-2}} \nonumber \]

    \(100(1-\alpha )\%\) confidence interval for \(\beta _1\):

    \[\hat{\beta _1}\pm t_{\alpha /2}\frac{S_\varepsilon }{\sqrt{SS_{xx}}}\; \; \; (df=n-2) \nonumber \]

    Standardized test statistic for hypothesis tests concerning \(\beta _1\):

    \[T=\frac{\hat{\beta _1}-B_0}{S_\varepsilon /\sqrt{SS_{xx}}}\; \; \; (df=n-2) \nonumber \]

    Coefficient of determination:

    \[r^2=\frac{SS_{yy}-SSE}{SS_{yy}}=\frac{SS_{xy}^{2}}{SS_{xx}SS_{yy}}=\hat{\beta _1}\frac{SS_{xy}}{SS_{yy}} \nonumber \]

    \(100(1-\alpha )\%\) confidence interval for the mean value of \(y\) at \(x=x_p\):

    \[\hat{y_p}\pm t_{\alpha /2}S_\varepsilon \sqrt{\frac{1}{n}+\frac{(x_p-\bar{x})^2}{SS_{xx}}} \; \; \; (df=n-2) \nonumber \]

    \(100(1-\alpha )\%\) prediction interval for an individual new value of \(y\) at \(x=x_p\):

    \[\hat{y_p}\pm t_{\alpha /2}S_\varepsilon \sqrt{1+\frac{1}{n}+\frac{(x_p-\bar{x})^2}{SS_{xx}}} \; \; \; (df=n-2) \nonumber \]

    ˆ=7771.44


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