1. **Problem Statement:** We have a linear regression model for Trips: $$Trips = a + b \times Households + c \times Employment + \varepsilon$$ Given data for 10 zones with Households, Employment, and Observed Trips, we want to understand how to calculate the coefficient of determination, $R^2$, which measures the goodness of fit of the model. 2. **Formula for $R^2$:** $$R^2 = 1 - \frac{SS_{res}}{SS_{tot}}$$ where: - $SS_{res} = \sum (y_i - \hat{y}_i)^2$ is the residual sum of squares, - $SS_{tot} = \sum (y_i - \bar{y})^2$ is the total sum of squares, - $y_i$ are observed values, - $\hat{y}_i$ are predicted values from the model, - $\bar{y}$ is the mean of observed values. 3. **Steps to calculate $R^2$:** - Calculate the mean of observed trips: $$\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i$$ - Fit the regression model to find coefficients $a$, $b$, and $c$ using least squares. - Use the model to predict trips $\hat{y}_i$ for each zone. - Compute $SS_{res} = \sum (y_i - \hat{y}_i)^2$. - Compute $SS_{tot} = \sum (y_i - \bar{y})^2$. - Calculate $R^2 = 1 - \frac{SS_{res}}{SS_{tot}}$. 4. **Explanation:** $R^2$ tells us the proportion of variance in the observed trips explained by the model. An $R^2$ close to 1 means a good fit; close to 0 means poor fit. 5. **Note:** To find $a$, $b$, and $c$, you would typically use matrix algebra or statistical software to perform multiple linear regression on the data provided. Since the problem does not provide coefficients or ask for numerical $R^2$, this explanation guides how to compute it from the data and model.