In linear regression, the intercept is the predicted response when every predictor equals zero. For one predictor, the fitted equation is ŷ = β₀ + β₁x: β₀ is the intercept, while β₁ is the change in the prediction for a one-unit change in x. The intercept and slope position the regression line to minimize differences between observed and predicted values under a linear model. section 4.1.1 section 4.2.2
Predicting a running split
A fitted model predicts Stage 2 running time from Stage 1 time as ŷ = −3.0778 + 1.3083x, in minutes. At x = 193, the predicted Stage 2 time is about 249 minutes. The intercept, −3.0778, is the model’s prediction at a Stage 1 time of zero—not a typical running time in the observed data. Interpret it as part of the fitted equation rather than as a practical forecast at zero. section 4.2.3 section 4.1.1
With statsmodels, add the constant to the predictor before fitting ordinary least squares; the constant receives the intercept estimate. The regression table reports it under `coef` alongside the slope. Use those coefficients to calculate predictions, then assess fit rather than treating the coefficients alone as proof of predictive quality. section 4.2.2 section 4.2.3
>>> x = sm.add_constant(x)
The `sm.add_constant(x)` call adds the constant used to estimate the intercept. section 4.2.2
