Solution
The Retest of the High Scorers
Show the problem again
Scores on two sittings of an exam have the same distribution in the population, with mean 500, standard deviation 100, and correlation 3/5; assume the relationship is linear. Among students who scored 680 on the second sitting, what is the expected score on the first sitting?
Worked solution
Regression is symmetric in the two variables: $E[X_1 \mid X_2 = s] = \mu + r\,(\sigma_1/\sigma_2)(s - \mu)$, and the standard deviations are equal. So the expectation is $500 + (3/5)(180) = 608$. The intuition that high second scores must have come from even higher first scores is wrong: selecting on the second score regresses the first just as selecting on the first regresses the second. The answer is **608**.
Source: Regression to the mean after Francis Galton. Statement written for AxiomIQ.