Regression
Warm-up
Statistical investigation
Evidence
Residuals: the vertical distance between an observation and a fitted prediction reveals what the line misses.
Live analysis
Investigate the evidence, then check your reasoning.
The maths
Least-squares line: predicted y = a + bx, where b = Sxy/Sxx and a = ȳ − bx̄.
Residual = observed y − predicted y. RSS is the sum of squared residuals, in squared response units.
For a straight-line model with varying y, r² = 1 − RSS/TSS. A residual pattern may reveal model misspecification.
Unknown population SD: two-sided mean interval x̄ ± t critical × s/√n, using n − 1 degrees of freedom.
For proportions: Wald interval p̂ ± z critical × √[p̂(1 − p̂)/n]; endpoints are not clipped.
Confidence is a long-run property of the interval procedure. An individual interval either covers the parameter or does not.
Interpolation uses observed x-range values; extrapolation lies outside that range.
Prediction uncertainty is not the same as confidence in the regression mean. Extrapolation adds model risk.