10.9 Formula List

$$S{S}_{xx}=\mathrm{\Sigma }{x}^2-\frac{1}{n}{\left(\mathrm{\Sigma }x\right)}^{2}S{S}_{xy}=\mathrm{\Sigma }xy-\frac{1}{n}\left(\mathrm{\Sigma }x\right)\left(\mathrm{\Sigma }y\right)S{S}_{yy}=\mathrm{\Sigma }{y}^2-\frac{1}{n}{\left(\mathrm{\Sigma }y\right)}^2$$

Correlation coefficient:

$$r=\frac{S{S}_{xy}}{\sqrt{S{S}_{xx}·S{S}_{yy}}}$$

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

$$\widehat{y}={\widehat{\mathit{\beta }}}_{1}x+{\widehat{\mathit{\beta }}}_0\text{\hspace{0.17em}where}{\widehat{\mathit{\beta }}}_1=\frac{S{S}_{xy}}{S{S}_{xx}}\text{\hspace{0.17em}and}{\widehat{\mathit{\beta }}}_0=\stackrel{-}{y}-{\widehat{\mathit{\beta }}}_1\stackrel{-}{x}$$

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

$$SSE=S{S}_{yy}-{\widehat{\mathit{\beta }}}_{1}S{S}_{xy}.$$

Sample standard deviation of errors:

$$s_{\mathit{\epsilon }}=\sqrt{\frac{SSE}{n\text{−}2}}$$

$100\left(1-\alpha \right)\%$ confidence interval for ${\mathit{\beta }}_1$:

$${\widehat{\mathit{\beta }}}_1\pm t_{\alpha ∕2}\frac{s_{\mathit{\epsilon }}}{\sqrt{S{S}_{xx}}}\left(df=n\text{−}2\right)$$

Standardized test statistic for hypothesis tests concerning ${\mathit{\beta }}_1$:

$$T=\frac{{\widehat{\mathit{\beta }}}_1-B_0}{s_{\mathit{\epsilon }}∕\sqrt{S{S}_{xx}}}\left(df=n\text{−}2\right)$$

Coefficient of determination:

$$r^2=\frac{S{S}_{yy}-SSE}{S{S}_{yy}}=\frac{S{S}_{xy}^2}{S{S}_{xx}S{S}_{yy}}={\widehat{\mathit{\beta }}}_1\frac{S{S}_{xy}}{S{S}_{yy}}$$

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

$${\widehat{y}}_p\pm t_{\alpha ∕2}{s}_{\mathit{\epsilon }}\sqrt{\frac{1}{n}+\frac{{\left(x_p-\stackrel{-}{x}\right)}^2}{S{S}_{xx}}}\left(df=n\text{−}2\right)$$

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

$${\widehat{y}}_p\pm t_{\alpha ∕2}{s}_{\mathit{\epsilon }}\sqrt{1+\frac{1}{n}+\frac{{\left(x_p-\stackrel{-}{x}\right)}^2}{S{S}_{xx}}}\left(df=n\text{−}2\right)$$