Linear Model vs Least-Squares Regression Line

A linear model summarizes a scatterplot with one straight line, so any $x$ you choose turns into a prediction. It is written $\hat{y} = a + bx$, where $\hat{y}$ (read y-hat) is the predicted response, $a$ is the y-intercept, and $b$ is the slope. For example, if $\hat{y} = 12 + 3x$ predicts a plant's height in centimeters from weeks of growth, then at $x = 5$ weeks the model gives $12 + 3(5) = 27$ centimeters. Fit a linear model only when the scatterplot and the residual plot show a linear form, since the equation will happily produce numbers for curved data too.

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 $(\bar{x}, \bar{y})$.