Learn to interpret regression equations in context — slopes, intercepts, predictions, and more.
A linear regression equation predicts the value of a response variable (y) from an explanatory variable (x).
where b₀ is the y-intercept (predicted y when x = 0) and b₁ is the slope (change in ŷ for each 1-unit increase in x).
The slope tells you the rate of change. Always say: "For each additional [1 unit of x], ŷ [increases / decreases] by [|slope|] [units of y], on average."
A positive slope means as x increases, y increases. A negative slope means as x increases, y decreases.
The y-intercept is the predicted value of y when x = 0. It is only meaningful in context if x = 0 is a realistic and sensible value. If x = 0 makes no practical sense, the y-intercept is not meaningful.
To predict y, substitute a specific x value into the equation and compute ŷ. Be cautious about extrapolation — predicting far outside the range of observed x values is unreliable.
Equation: ŷ = −0.12x + 9.5 where x = weeks of training, y = mile time (minutes).
Slope: For each additional week of training, predicted mile time decreases by 0.12 minutes, on average. (Negative slope → getting faster.)
y-intercept (9.5): A person with 0 weeks of training is predicted to run a mile in 9.5 minutes. This is plausible, so it is meaningful here.
Prediction at x = 20: ŷ = −0.12(20) + 9.5 = −2.4 + 9.5 = 7.1 minutes.