Skip to main content
Statistics LibreTexts

10.9: Formula List

  • Page ID
    549
    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Learning Objectives

    • Listing of all formulas used throughout the chapter.

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

    Correlation coefficient:

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

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

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

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

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

    Sample standard deviation of errors:

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

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

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

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

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

    Coefficient of determination:

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

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

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

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

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

    ˆ=7771.44


    10.9: Formula List is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.