Formula List

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10.9 Formula List

S S_{x x} = Σ x^{2} - \frac{1}{n} {(Σ x)}^{2} S S_{x y} = Σ x y - \frac{1}{n} (Σ x) (Σ y) S S_{y y} = Σ y^{2} - \frac{1}{n} {(Σ y)}^{2}

Correlation coefficient:

r = \frac{S S_{x y}}{\sqrt{S S_{x x} \cdot S S_{y y}}}

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{S S_{x y}}{S S_{x x}} and {\hat{β}}_{0} = \bar{y} - {\hat{β}}_{1} \bar{x}

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

S S E = S S_{y y} - {\hat{β}}_{1} S S_{x y} .

Sample standard deviation of errors:

s_{ε} = \sqrt{\frac{S S E}{n − 2}}

$100 (1 - α) %$ confidence interval for $β_{1}$ :

{\hat{β}}_{1} \pm t_{α ∕ 2} \frac{s_{ε}}{\sqrt{S S_{x x}}} (d f = n − 2)

Standardized test statistic for hypothesis tests concerning $β_{1}$ :

T = \frac{{\hat{β}}_{1} - B_{0}}{s_{ε} ∕ \sqrt{S S_{x x}}} (d f = n − 2)

Coefficient of determination:

r^{2} = \frac{S S_{y y} - S S E}{S S_{y y}} = \frac{S S_{x y}^{2}}{S S_{x x} S S_{y y}} = {\hat{β}}_{1} \frac{S S_{x y}}{S S_{y y}}

$100 (1 - α) %$ confidence interval for the mean value of y at $x = x_{p}$ :

{\hat{y}}_{p} \pm t_{α ∕ 2} s_{ε} \sqrt{\frac{1}{n} + \frac{{(x_{p} - \bar{x})}^{2}}{S S_{x x}}} (d f = n − 2)

$100 (1 - α) %$ prediction interval for an individual new value of y at $x = x_{p}$ :

{\hat{y}}_{p} \pm t_{α ∕ 2} s_{ε} \sqrt{1 + \frac{1}{n} + \frac{{(x_{p} - \bar{x})}^{2}}{S S_{x x}}} (d f = n − 2)