We expect that students who do well on the midterm exam in a course will usually also do well on the final exam. Gary Smith of Pomona College looked at the exam scores of all 346 students who took his statistics class over a 10-year period. The least-squares line for predicting final-exam score from mid-term exam score was γ=46.6 + 0.41x.
Octavio scores 10 points above the clas mean on the midterm. How many points above the class mean do you predict that he will score on the final. (Hint: what is the predicted final-exam score for the class mean midterm score of x-bar?) This is an example of regression to the mean, the phenomenon that gave "regression" its name: students who do well on the midterm will on average do less well on the final, but still above the class mean.)