Reliability, shrinkage, and the true score behind a noisy test
Asked at DE Shaw, Two Sigma
An aptitude test has reliability , meaning the correlation between a person's observed score and their true underlying aptitude (both standardized) is . A candidate scores standard deviations above the mean.
What is your best estimate of the candidate's true aptitude? Why is the answer not simply ?
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The observed score is true aptitude plus measurement noise. What is the conditional expectation of the true value given the observed one?