Question5 Which statement is true regarding adjusted [math]? Select one alternative: a) Always increases when new variables are added b) Penalises inclusion of irrelevant regressors c) Equals [math] when all variables are significant d) Unaffected by number of regressors ResetMaximum marks: 0.5 Flag question undefined

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BlackTom AI · Accepted Answer

Question5 asks about the adjusted [math] statistic in the context of regression.
Option a) 'Always increases when new variables are added'—This is not true in general. While R^2 can increase with more predictors, adjusted R^2 incorporates a penalty for adding variables that do not meaningfully improve fit. If the new variable has little or no explanatory power, the penalty can cause adjusted R^2 to stay the same or even decline.
Option b) 'Penalises inclusion of irrelevant regressors'—This statement correctly captures the purpose of adjusted R^2. The adjustment reduces the impact of adding variables that do not contribute substantial explanatory power, helping to prevent overfitting by favoring a more parsimonious model.
Option c) 'Equals R^2 when all variables are significant'—This claim is misleading. Although all variables being significant can improve fit, adjusted R^2 is not generally equal to R^2 even in that scenario; adjusted R^2 is R^2 adjusted downward by a penalty term that depends on the number of predictors and observations. Therefore, equality is not guaranteed,
  and in practice adjusted R^2 remains less than or equal to R^2, with equality not being a universal property.
Option d) 'Unaffected by number of regressors'—This is false. The whole point of the adjusted measure is to account for the number of regressors; as more regressors are added (especially those with weak predictive power), the adjustment changes, often reducing the statistic when the added variables do not improve the model adequately.
In summary, a is incorrect because the adjusted statistic can decrease with extraneous variables; b is correct because it describes the core idea of the adjustment; c is misleading since equality with R^2 is not a general rule; d is incorrect since the adjustment explicitly depends on the number of regressors.

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