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2.6 Linear Regression Models

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The Least-Squares Regression Line

The least-squares regression line (LSRL)

\hat{y} = a + bx

Where:

${\hat{y}}$ (read "y-hat") = predicted value of the response variable
${a}$ = y-intercept (predicted ${y}$ when ${x = 0}$ )
${b}$ = slope (change in predicted ${y}$ for each 1-unit increase in ${x}$ )
${x}$ = value of the explanatory variable

Formulas for slope and intercept:

b = r \cdot \frac{s_y}{s_x}

a = \bar{y} - b\bar{x}

Where ${r}$ is the correlation, ${s_y}$ and ${s_x}$ are the standard deviations, and ${\bar{x}}$ , ${\bar{y}}$ are the means.

Key fact: The LSRL always passes through the point ${(\bar{x}, \bar{y})}$ .

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