Predicted value (y-hat)

The predicted value $\hat{y}$ (read y-hat) is the height of the fitted line at a chosen $x$. It is manufactured by the model, not measured on anybody. That is the entire job of the hat: $y$ is the number in the data set and $\hat{y}$ is the number the line returns for the same individual, and the two are equal only by coincidence.

Take six students who study 1, 2, 3, 4, 5 and 6 hours and score 60, 72, 65, 78, 71 and 86. Their least-squares line is $\hat{y} = 58 + 4x$. At $x = 4$ the predicted value is $\hat{y} = 58 + 4(4) = 74$. The student who actually studied 4 hours scored 78, so observed equals predicted plus residual: $78 = 74 + 4$. Run every $x$ through the line and the six predicted values are 62, 66, 70, 74, 78 and 82, not one of which matches the score that student earned.

The sentence to avoid is "the model says a student who studies 4 hours will score 74." The line returns the same 74 for anyone who studied 4 hours, and real students scatter around that value: in these data the misses run as large as 7 points. Written properly it is "the model predicts about 74 points," and the word predicted is what an AP interpretation is graded on.

A predicted value inherits every limit of the line that produced it. Substituting an $x$ the study never reached is extrapolation, and the arithmetic gives no warning: $x = 12$ hours returns $58 + 4(12) = 106$ on a test scored out of 100. Predicted values also depend on the direction of the fit, so this line predicts scores from hours and cannot be rearranged to predict hours from scores.