Suppose we regress student’s ATAR scores on parent’s education and parent’s income. If we run the regression again but also include the student’s height as an additional regressor:[Fill in the blank]

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In this question, we’re adding height as an additional regressor to a regression of ATAR on parent education and parent income, so we need to think about R^2 vs adjusted R^2 behavior when a regressor is added.
Option a: 'The R^2 for the regression will either increase or decrease.' In ordinary least squares, the regular R^2 cannot decrease when you add a regressor; it either stays the same or increases because it’s the fraction of variance explained by the model and more predictors can only explain more or the same amount of variance. So this statement is not accurate as written, since a decrease is not possible for R^2 itself.
Option b: 'The adjusted R^2 for the regression will either stay the same or increase.' This is the standard property of adjusted R^2: it includes a penalty for adding additional regressors that do not improve the fit sufficiently. If the height variable provides useful information beyond what is already captured by parental education and income, adjusted R^2 will rise; if not, it may stay the same. Therefore this statement correctly captures the usual behavior.
Option c: 'Both (a) and (b) are true.' Since (a) is incorrect about R^2 potentially decreasing, this combination cannot be correct.
Option d: 'The adjusted R^2 for the regression will decrease.' As noted, adjusted R^2 can decrease if the new regressor adds little explanatory power relative to its penalty, but it is not guaranteed to decrease; it can stay the same or increase. So this statement is not universally true.

Suppose we regress student’s ATAR scores on parent’s education and parent’s income. If we run the regression again but also include the student’s height as an additional regressor:[Fill in the blank]

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