10-9. Formula List

10.9 Formula List

$SS_{xx}=Σx^2− \frac{1}{n}(Σx)^2$ ,  $SS_{xy}=Σxy− \frac{1}{n}(Σx)(Σy)$ , $SS_{yy}=Σy^2− \frac{1}{n}(Σy)^2$

Correlation coefficient: $r= \frac{SS_{xy}} {\sqrt{SS_{xx}⋅SS_{yy}}}$

Least squares regression equation (equation of the least squares regression line): $\hat{y}=\hat{β_1}x+\hat{β_0}$  where $\hat{β_1}= \frac{SS_{xy}} {SS_{xx}}$  and $\hat{β_0} =\bar{y}−\hat{β_1} \bar{x}$

Sum of the squared errors for the least squares regression line: $SSE=SS_{yy}−\hat{β_1}SS_{xy}$.

Sample standard deviation of errors: $s_ε=\sqrt{ \frac {SSE}{n−2}}$

$100(1−α)\%$ confidence interval for $β_1$ : $\hat{β_1}±t_{α∕2} \frac {s_ε} {\sqrt{SS_{xx}}}$

Standardized test statistic for hypothesis tests concerning $β_1$: $T=(\hat{β_1}−B_0) / \frac {s_ε} {\sqrt{SS_{xx}}}$ ( $df=n−2$ )

Coefficient of determination: $r^2 = \frac{SS_{yy}−SSE} {SS_{yy}} = \frac{SS^2_{xy}}{ SS_{xx}SS_{yy}} = \hat{β_1} \frac{SS_{xy}}{SS_{yy}}$

$100(1−α)\%$ confidence interval for the mean value of $y$ at $x=x_p$ : $\hat{y_p} ±t_{α∕2} s_ε \sqrt{\frac{1}{n} + \frac{(x_p−\bar{x})^2} {SS_{xx}}}$ ( $df=n-2$ )

$100(1−α)\%$ prediction interval for an individual new value of $y$ at $x=x_p$: $\hat{y_p} ±t_{α∕2} s_ε \sqrt{1 + \frac{1}{n} + \frac{(x_p−\bar{x})^2} {SS_{xx}}}$ ( $df=n-2$ )