91Ó°ÊÓ

What's my grade? In Professor Krugman's economics course, the correlation between the students' total scores prior to the final examination and their finalexamination scores is \(r=0.5\). The pre-exam totals for all students in the course have mean 280 and standard deviation 40 . The final-exam scores have mean 75 and standard deviation 8. Professor Krugman has lost Julie's final exam but knows that her total before the exam was 300 . He decides to predict her finalexam score from her pre-exam total. (a) What is the slope of the least-squares regression line of final-exam scores on pre-exam total scores in this course? What is the intercept? Interpret the slope in the context of the problem. (b) Use the regression line to predict Julie's final-exam score. (c) Julie doesn't think this method accurately predicts how well she did on the final exam. Use \(r^{2}\) to argue that her actual score could have been much higher (or much lower) than the predicted value.

Regression to the mean. 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. \({ }^{25}\) The least-squares line for predicting final exam score from midterm-exam score was \(y^{-} \bar{y}=46.6+0.41 x\). (Both exams have a 100-point scale.) Octavio scores 10 points above the class mean on the midterm. How many points above the class mean do you predict that he will score on the final? [Hint: Use the fact that the least-squares line passes through the point \(\left(x^{-}, y^{-}\right)(\bar{x}, \bar{y})\) and the fact that Octavio's midterm score is \(x^{-}+10 x+10\).] This is another example of regression to the mean: students who do well on the midterm will, in general, do less well, but still above average, on the final.