# 10.9: Formula List

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