Linear regression models, residuals and least-squares regression

Learning objectives

Use a linear regression model ŷ = a + bx to calculate a predicted value of the response variable, and explain why extrapolation beyond the range of the data is unreliable

Calculate a residual as the observed value minus the predicted value, and interpret its sign as showing that the model predicted too low or too high

Construct or interpret a residual plot, and use the presence or absence of a pattern in it to judge whether a linear model is appropriate

Explain that the least-squares regression line minimizes the sum of the squared residuals and passes through the point (x̄, ȳ), and determine its slope b = r × s_y ÷ s_x and intercept a = ȳ − b x̄ from summary statistics or read them from computer output

Interpret the slope and the y-intercept of a least-squares regression line in context with units, stating when the intercept has no sensible meaning

Interpret the coefficient of determination r² as the proportion of the variation in the response variable that is explained by the linear model with the explanatory variable