First, clarify the assumptions of linear regression: residuals are zero-mean, approximately normal, and homoscedastic. Then, analyze each event's probability using these properties, noting that event (1) is about the sign of the error (50% chance), event (2) depends on the distribution of y (which may be centered around 20 or not), and event (3) is about the tail probability of residuals exceeding 1.5 standard deviations (about 13.4% if normal). Finally, rank them based on these probabilities, justifying with the properties.
Pro tip: Emphasize that the ranking depends on the distribution of y and the model assumptions; without knowing the mean of y, event (2) could be more or less probable. A strong candidate will state assumptions and consider edge cases.
Recall that in linear regression, residuals have mean zero, are approximately normally distributed, and have constant variance (homoscedasticity).
The regression overestimates y for a new x if the residual is negative. Since residuals are zero-mean and symmetric (under normality), the probability is approximately 0.5.
The absolute residual exceeds 1.5 times the standard deviation of training residuals. Under normality, P(|Z| > 1.5) ≈ 0.1336, so about 13.4%.
The probability that a new y > 20 depends on the distribution of y. Without additional information, we cannot determine it precisely, but we can compare it to the others by considering typical scenarios.
Compare the probabilities: event (1) is ~50%, event (3) is ~13.4%, and event (2) is unknown but likely less than 50% unless the mean of y is well above 20. Typically, event (2) might be the least probable if y is centered near 20, but it could be more probable if the mean is high. Justify based on assumptions.
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