Suppose a college offers weekly one-on-one tutoring sessions for students who sign up for it. In an economics course, approximately half the students sign up for the free tutoring. In general, students who are harder working are more likely to sign up for tutoring and they study more. As a result, the average student who signed up for tutoring studies economics for 20 hours per week, while the average student who does not, studies for 10 hours per week. Figure 1 represents the relationship between hard-workings, studying, tutoring and test scores, and Equation 1 shows the relationship between studying, tutoring and test scores.

(a) The college decides to evaluate the impact of the tutoring program on test scores (β1) by comparing the average test score for people who used the tutoring ser- vices and those who did not. Do you expect that this will produce an unbiased estimate of the impact of tutoring on test scores? If not, how much do you expect the estimate to be off by?

(b) One year the tutoring center is understaffed. They are only able to offer tutoring services to 30 of the students of the 60 who sign up. They randomly choose 30 of them to receive the tutoring they requested and deny it to the remaining 30. How could you use this unfortunate situation to construct an unbiased estimate of β1? Explain what it is about this situation that allows you to estimate the impact of tutoring differently from in part (a).

Test Score = 50 + β1Tutoring + 2 × Hours Studying/week

 

Figure 1: Relationship of Hard-work, Tutoring, and Test Score Hard-workingness Tutoring Studying Impact of 1 hour of Study on E(Score] 31 Impact of Tutoring on E[Score] Test Score