Professor A. Voidim looked through the midterm exam marks of his 100 students and wondered whether there was a way to motivate them to do better on the final exam. He decided to try out two ideas. First, he sent a harsh email to the 10 students whose midterm marks were the lowest. He criticized their performance and said they should quit the video games and parties and hit the books. Second, he wrote to the top 10 students, praised their achievement, and wished them even more success on the final exam. When he had marked the final exam, he checked out the marks of the students who were in the top 10 on the midterm and was dismayed to find that their final exam marks were on average 5.5% lower than on the midterm. “So much for encouraging them!” he thought. But the lowest 10 students on the midterm had increased their marks on the final by an average of 4.1%. “Harsh, critical emails work better than congratulatory ones,” thought Professor A. Voidim. “I’ll send out more harsh emails in future.”

Using the data in the file ch07_MCSP_Motivating_Students, assess whether the emails affected student performance. Would we expect similar emails to be effective in other classes where the correlation between midterm and final exam marks is lower?

Let us call the students with the 10 lowest marks on the midterm Group A, and the students with the 10 highest marks on the midterm Group B.

(a) Perform a linear regression of the final exam marks using the midterm marks as the explanatory variable (checking the four conditions).

(b) If the emails affected student performance, the residuals for the students who received emails would be expected to be higher/lower than the residuals for the rest of the class. Check to see whether this is true (i) for Group A, and (ii) for Group B.

(c) Based on the regression equation, compare the expected marks on the final exam with the marks students actually got on the midterm exam averaged for Groups A and B. Compare your results for (b), (c), and Professor A. Voidim’s results.

(d) Calculate the means and standard deviations of the midterm marks and of the expected final exam marks from the regression. How many standard deviations below/above the mean is the average of the midterm exam marks of the students in Groups A and B? How many standard deviations below/above the mean is the average of the expected final exam marks (from the regression) of the students in Groups A and B?

(e) Calculate the correlation coefficient between the midterm and the final exam marks for all students. What is the connection between this correlation coefficient and your answers to (d)?

(f) Suppose we had a different set of data, with the same means and standard deviations, but with a different correlation coefficient. If the correlation coefficient is lower in the new data, would the effect that Professor A Voidim observed (the students in Group A improving and the students in Group B getting lower marks) be greater or less? Include the extreme case with correlation coefficient equal to zero.

(g) Did Professor A. Voidim’s emails affect student performance? If not, why are the final exam marks higher for the students in Group A and lower for those in Group B?