Evaluating Regression Model Fit and Interpreting Model Results

Correct Answer: Adjusted R-squared, as it accounts for the number of independent variables in the model and penalizes the addition of insignificant variables.

The adjusted R-squared is the most appropriate measure for comparing the relative quality of different regression models. It accounts for the number of independent variables in the model and penalizes the addition of insignificant variables. This helps prevent overfitting and provides a more reliable comparison between models with different numbers of variables. The regular R-squared, while representing the proportion of variance explained by the model, does not account for the number of variables and can increase simply by adding more variables, even if they are not significant. The standard error of the regression measures the average distance between observed and predicted values, but it does not provide a clear basis for comparing the overall quality of different models.

Correct Answer: The model's assumptions of linearity and homoscedasticity are violated, indicating the model may not be appropriate for the data despite the high R-squared value.

The correct answer is that the model's assumptions of linearity and homoscedasticity are violated, indicating the model may not be appropriate for the data despite the high R-squared value. A clear pattern of increasing residuals as the predicted housing prices increase suggests that the variance of the residuals is not constant across all levels of the predicted values. This violates the assumption of homoscedasticity, which states that the variance of the residuals should be constant for all predicted values. Additionally, the pattern in the residuals suggests that the relationship between the predictor variables and the response variable may not be linear, violating the linearity assumption of the regression model. While the high R-squared value and significant F-statistic suggest that the model fits the data well, these metrics alone do not guarantee that the model's assumptions are met. Violating the assumptions can lead to biased or unreliable estimates of the coefficients and predictions. The other answers are incorrect because: - Simply disregarding the pattern in the residuals due to the high R-squared and significant F-statistic would be a mistake, as it ignores the violation of the model's assumptions. - Adding more predictor variables may improve the model's predictive power, but it does not address the issues of non-linearity and heteroscedasticity revealed by the residual plot.

Correct Answer: The model may be misspecified and a non-linear relationship between the dependent and independent variables should be considered.

The best answer is that the model may be misspecified and a non-linear relationship between the dependent and independent variables should be considered. While the high R-squared value and significant F-statistic suggest the model fits the data well overall, the non-linear pattern in the residual plots is a red flag. This pattern indicates that the linear model is not capturing the true relationship between the variables. The other answers are incorrect because: - A high R-squared and significant F-statistic alone do not confirm a model is well-specified if there are issues like non-linearity in the residuals. Further investigation is warranted. - The non-linear pattern is not necessarily caused by outliers. Removing data points to force the residuals into a linear pattern is not appropriate without valid justification.