Not only is it true that the first slope doesn't have to be less than one, it could plausibly be greater than one.
Suppose the course material is really cumulative, so that some students "get it" and take off, while other students fall by the wayside. Then scoring well on the first test predicts scoring well on the second midterm, while scoring poorly on the first predicts scoring really badly on the second. Then the slope of your least-squares-fit line could easily be greater than 1.
In other words, the mean could stay the same, due to above-average students (on the first test) getting better, and below-average students getting worse. There's no reason to suppose that below-average students will magically get better.
Comments
Not only is it true that the first slope doesn't have to be less than one, it could plausibly be greater than one.
Suppose the course material is really cumulative, so that some students "get it" and take off, while other students fall by the wayside. Then scoring well on the first test predicts scoring well on the second midterm, while scoring poorly on the first predicts scoring really badly on the second. Then the slope of your least-squares-fit line could easily be greater than 1.
In other words, the mean could stay the same, due to above-average students (on the first test) getting better, and below-average students getting worse. There's no reason to suppose that below-average students will magically get better.