A researcher runs an experiment to measure the impact of a short nap on memory. There are 200 participants and they can take a short nap of either 60 minutes or 75 minutes. After waking up, each participant takes a short test for 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 (0 ≤ Yi ≤ 100), 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 = β0 + β1Xi + UI. (a) Explain what the term ui represents. Why will different participants have different values of ui? (b) What is E(ui|Xi)? Are the estimated coefficients unbiased? (c) What concerns might the researcher have about ensuring compliance among participants? (d) The estimated regression is Yi = 55 + 0.17Xi.
A researcher runs an experiment to measure the impact of a short nap on memory. There are 200 participants and they can take a short nap of either 60 minutes or 75 minutes. After waking up, each participant takes a short test for 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 (0 ≤ Yi ≤ 100), 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 = β0 + β1Xi + UI.
(a) Explain what the term ui represents. Why will different participants have different values of ui?
(b) What is E(ui|Xi)? Are the estimated coefficients unbiased?
(c) What concerns might the researcher have about ensuring compliance among participants?
(d) The estimated regression is Yi = 55 + 0.17Xi.
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 to nap.
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