Table 10.10 shows hypothetical abilities and test scores. These 20 students have abilities ranging from 75 to 95, which we assume to be constant between third and fourth grade. For the four students at each ability level, two score five points above their ability and two score five points below their ability. Whether they score above or below their ability on the fourth grade test is independent of whether they scored above or below their ability on the third grade test. The uniform distribution of abilities and error scores is unrealistic, as are the severely limited values. This stark simplicity is intended to clarify the argument.

a. Estimate the simple linear regression model where third grade score is the dependent variable and ability is the explanatory variable.

b. Estimate the simple linear regression model where ability is the dependent variable and third grade score is the explanatory variable.

c. Explain how the difference between your answers to questions a and b reflect regression to the mean.

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Table 10.10: Exercise 10.48 Ability Third Grade Scores Fourth Grade Scores 75 70 70 75 70 80 75 80 70 75 80 80 80 75 75 80 75 85 80 85 75 80 85 85 85 80 80 85 80 90 85 90 80 85 90 90 90 85 85 90 85 95 90 95 85 90 95 95 95 90 90 95 90 100 95 100 90 95 100 100