Sum of Squares Criterion In addition to the value of another measurement used to assess the quality of a model is the sum of squares of the residuals. Recall from Section 10-2 that a residual is (the difference between an observed y value and the value predicted from the model). Better models have smaller sums of squares. Refer to the U.S. population data in Table 10-7.
c. Verify that according to the sum of squares criterion, the quadratic model is better than the linear model.

Verified step by step guidance

1. Identify the residuals for both the linear and quadratic models. A residual is calculated as the difference between the observed value \(y_i\) and the predicted value \(\hat{y}_i\) from the model, i.e., \(r_i = y_i - \hat{y}_i\).
2. Square each residual to get \(r_i^2 = (y_i - \hat{y}_i)^2\). This ensures all differences are positive and emphasizes larger errors.
3. Sum all the squared residuals for each model separately to get the sum of squares of residuals (SSR): \(SSR = \sum_{i=1}^n (y_i - \hat{y}_i)^2\).
4. Compare the SSR values for the linear and quadratic models. The model with the smaller SSR is considered better according to the sum of squares criterion because it fits the data more closely.
5. Conclude that if the quadratic model has a smaller SSR than the linear model, then the quadratic model is better based on this criterion.

Key Concepts

Residuals

Residuals are the differences between observed values and the values predicted by a model. They measure how far off the model's predictions are from actual data points. Smaller residuals indicate a model that fits the data more closely.

Sum of Squares of Residuals (SSR)

The sum of squares of residuals is the total of each residual squared, quantifying the overall discrepancy between observed and predicted values. It is used to assess model accuracy; a lower SSR means the model predictions are closer to the actual data.

Model Comparison Using SSR

Comparing models involves evaluating their SSR values; the model with the smaller SSR is considered better because it fits the data more accurately. In this question, verifying that the quadratic model has a smaller SSR than the linear model shows it is a better fit.