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Let's go through each part of the question with the given data and calculations:

### Given Information
- **Least-squares regression line equation**: \( \hat{y} = 43.31 - 0.63x \)
- **Data table** with observed values of \( x \) and \( y \) along with various squared error terms.

### Task Breakdown

1. **(a)** The question asks why the least-squares regression line "best fits" the data. The line minimizes the **sum of squared residuals** between the observed \( y \)-values and the predicted \( \hat{y} \)-values. Therefore, the correct answer is likely **sum of squared residuals**, and this is given as the **column sum of 3.2691**.

2. **(b)** This part involves determining the variation in \( y \) explained by the regression model. This is captured by the **regression sum of squares (SSR)**, calculated by the squared differences between the predicted \( \hat{y} \) values and the mean \( \bar{y} \). Here, the **column sum is 15.0480**.

3. **(c)** The proportion of total variation explained by the model, or the **coefficient of determination \( R^2 \)**, is calculated as:
   \[
   R^2 = \frac{\text{SSR}}{\text{SST}}
   \]
   where:
   - **SSR (sum of squares due to regression)** = 15.0480
   - **SST (total sum of squares)** = 18.3880 (the sum of \( (y - \bar{y})^2 \))

   Plugging in the values:
   \[
   R^2 = \frac{15.0480}{18.3880} \approx 0.82
   \]
   So, the answer for (c) is **0.82**.

### Answers
- **(a)** Sum of squared residuals: **3.2691**
- **(b)** Sum of squares due to regression: **15.0480**
- **(c)** \( R^2 \) value: **0.82**

Would you like a deeper explanation of any part of these calculations or concepts?

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### Related Questions

1. What are residuals in the context of regression analysis, and why do we minimize their squares?
2. How is the sum of squares due to regression (SSR) different from the sum of squared residuals?
3. What does the coefficient of determination \( R^2 \) represent in a regression model?
4. Can \( R^2 \) ever be greater than 1, and why or why not?
5. How would the regression line equation change if we observed more data points?

### Tip
Remember, the higher the \( R^2 \) value, the better the model explains the variability in the data. An \( R^2 \) close to 1 indicates a good fit!

Bivariate data obtained for paired variables x and y are shown in the table 'Sample data.' The data are plotted in the scatter plot in Figure 1, which displays the least-squares regression line. The equation for this line is given as ŷ = 43.31 - 0.63x. The problem involves interpreting this regression model, calculating sums of squares, and determining the proportion of variation explained by the regression.

Learn to interpret and calculate the least-squares regression line for paired bivariate data. This problem includes calculating sums of squares, identifying the variation explained by the regression, and understanding the coefficient of determination (R²) with a sample dataset.

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