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10.9 Formula List

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

$$r=\frac{S{S}_{xy}}{\sqrt{S{S}_{xx}\xb7S{S}_{yy}}}$$
Least squares regression equation (equation of the least squares regression line):

$$\widehat{y}={\widehat{\mathit{\beta}}}_{1}x+{\widehat{\mathit{\beta}}}_{0}\text{\hspace{1em}}\text{\hspace{0.17em}where}\text{\hspace{1em}}{\widehat{\mathit{\beta}}}_{1}=\frac{S{S}_{xy}}{S{S}_{xx}}\text{\hspace{1em}}\text{\hspace{0.17em}and}\text{\hspace{1em}}{\widehat{\mathit{\beta}}}_{0}=\stackrel{-}{y}-{\widehat{\mathit{\beta}}}_{1}\stackrel{-}{x}$$
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:

$${s}_{\mathit{\epsilon}}=\sqrt{\frac{SSE}{n\text{\u2212}2}}$$
$100\left(1-\alpha \right)\%$ confidence interval for ${\mathit{\beta}}_{1}$:

$${\widehat{\mathit{\beta}}}_{1}\pm {t}_{\alpha \u22152}\text{\hspace{0.17em}}\frac{{s}_{\mathit{\epsilon}}}{\sqrt{S{S}_{xx}}}\text{\hspace{1em}}\left(df=n\text{\u2212}2\right)$$
Standardized test statistic for hypothesis tests concerning ${\mathit{\beta}}_{1}$:

$$T=\frac{{\widehat{\mathit{\beta}}}_{1}-{B}_{0}}{{s}_{\mathit{\epsilon}}\u2215\sqrt{S{S}_{xx}}}\text{\hspace{1em}}\left(df=n\text{\u2212}2\right)$$
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 $x={x}_{p}$:

$${\widehat{y}}_{p}\pm {t}_{\alpha \u22152}\text{\hspace{0.17em}}{s}_{\mathit{\epsilon}}\text{\hspace{0.17em}}\sqrt{\frac{1}{n}+\frac{{\left({x}_{p}-\stackrel{-}{x}\right)}^{2}}{S{S}_{xx}}}\text{\hspace{1em}}\left(df=n\text{\u2212}2\right)$$
$100\left(1-\alpha \right)\%$ prediction interval for an individual new value of *y* at $x={x}_{p}$:

$${\widehat{y}}_{p}\pm {t}_{\alpha \u22152}\text{\hspace{0.17em}}{s}_{\mathit{\epsilon}}\text{\hspace{0.17em}}\sqrt{1+\frac{1}{n}+\frac{{\left({x}_{p}-\stackrel{-}{x}\right)}^{2}}{S{S}_{xx}}}\text{\hspace{1em}}\left(df=n\text{\u2212}2\right)$$