Predicted Value from a Regression Line

Once a line is fitted, prediction is arithmetic: multiply the predictor by the slope and add the intercept. A line of ŷ = 20 + 2.5x evaluated at x = 8 predicts 40. The hat on ŷ matters: it marks a fitted value, an average outcome for that x, not a promise about any individual. Real observations scatter around the line by the residual standard deviation, so a prediction interval for one new case is always much wider than the confidence interval for the line itself.

The classic trap is extrapolation. Within the range of the data a line can be an excellent summary; outside it, it is a guess dressed as arithmetic. Mark Twain skewered this perfectly in Life on the Mississippi: noting that the river had shortened by 242 miles over 176 years, he extrapolated back to conclude that the Lower Mississippi was once 1.3 million miles long and would eventually be a mile and three-quarters. Running the line backwards is legitimate and useful, solving 65 = 5 + 3x gives x = 20, but it is calibration, and it inherits all the uncertainty of the fit.