Least-Squares Regression Line vs Slope of a Regression Line

The least-squares regression line is the best-fit line for predicting the response from the explanatory variable, chosen to minimize the total squared vertical distance from the points. Its equation is $\hat{y} = a + bx$, where $\hat{y}$ (read y-hat) is the predicted response, $a$ is the y-intercept, and $b$ is the slope. For example, $\hat{y} = 50 + 5x$ predicts a score of 75 when $x = 5$ hours of study. The line always passes through the point of averages $(\bar{x}, \bar{y})$.

The slope $b$ tells you how fast the predicted response rises or falls as the explanatory variable increases by one unit. A positive slope means the line goes up from left to right, and a negative slope means it goes down. For example, in $\hat{y} = 50 + 5x$ the slope 5 predicts 5 more points for each extra hour of study. It is computed as $b = r \cdot \frac{s_y}{s_x}$, where $r$ is the correlation and $s_y$ and $s_x$ are the standard deviations of $y$ and $x$.