midterm_2012_solution

# Multiple linear regressions with detailed steps got

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Multiple Linear Regressions with detailed steps got partial credit. 5. (20) The add states “Improve your GRE score. Guaranteed. On average our students improve their scores by 8 points on the new GRE scale (150 points on the old scale.) “ GRE = β0 + β1*GRE0 + β2*PREP+β3*GRE0*PREP + e. GRE = final GRE score (old scale) GRE0 = practice GRE score one month prior to the test (old scale) PREP = 1 for students who took the preparatory class 0 for students who did not take the preparatory class. (a) Explain how one should use the above regression to test the claim in the ad. (Outline all the steps: null hypothesis, alternative hypothesis, test, confidence interval). Answer: The improvement for the students in the program is GRE – GRE0 = β0 + β1*GRE0 + β2 + β3*GRE0 – GRE0 + e. Its average value is β0 + β2 + (β1 + β3 -1)*Average (GRE0). H0: β0 + β2 + (β1 + β3 -1)*Average(GRE0) = 150 Ha: β0 + β2 + (β1 + β3 -1)*Average(GRE0) ≠ 150 Significance level: 5% t-stat = (b0 + b2 + (b1 + b3 -1)*Average(GRE0) -150)/s.e.(b0 + b2+(b1 + b3-1)*Average(GRE0))

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Confidence interval [-1.96, 1.96]. If the t-stat is in the confidence interval, then the ad’s claim is correct. Possible versions: square2 H0: β2+β3*Average (GRE0) = 150 for testing if the students improve compared to their peers who do not enroll in the preparatory class. square2 T-test can be replaced by the F-test. (b) Suppose the estimate of β3 is equal to b3 = -0.1. Interpret the coefficient. Answer (any of the below two versions is fine): d GRE/d PREP = β2+β3*GRE0. The higher the initial GRE score of a student, the lesser is the effect of the preparatory class. Exactly by how much? – By 0.1 point of the final GRE score per each point of the initial GRE score. Alternatively, d GRE/d GRE0 = β1 +β3*PREP. That is, the preparatory class reduces the gap between the students with high and low initial scores. Exactly by how much? – By 1/10th of the original gap. (c) Suppose β2 = 220 and β3 = -0.3. Do all the students benefit from the preparatory class? (The maximum GRE0 score in the sample is 700). Who benefits most from the preparatory class: students with a higher GRE or lower GRE? Explain. Answer: The benefit = β2+β3* GRE0 = 220 – 0.3*GRE0 >= 220 – 0.3*700 =10. Therefore, all the students in the sample would benefit from the preparatory class. However, since β3 < 0, students with a higher GRE0 (and therefore higher GRE) will benefit less.

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