4.5 A researcher runs an experiment to measure the impact a short nap has on memory. 200 participants in the sample are allowed to take a short nap of either 60 minutes or 75 minutes. After waking up, each of the participants takes a short test on short-term recall. Each participant is randomly assigned one of the examination times, based on the flip of a coin. Let Yi denote the number of points scored on the test by the ith participant

(0cYic100), let Xi denote the amount of time for which the participant slept prior to taking the test (Xi = 60 or 75), and consider the regression model Yi = b0 + b1Xi + ui.

a. Explain what the term ui represents. Why will different students have different values of ui?

b. What is E(ui Xi)? Are the estimated coefficients unbiased?

c. What concerns might you have about ensuring compliance among participants?

d. The estimated regression is Yi = 55 + 0.17 Xi.

i. Compute the estimated regression’s prediction for the average score of participants who slept for 60 minutes before taking the test. Repeat for 75 minutes and 90 minutes.

ii. Compute the estimated gain in score for a participant who is given an additional 5 minutes on the exam.