To evaluate the statements about the least-squares regression line, let's go through each one.

1. Statement A: The predicted value of $\hat{y}$ is an estimate of the mean value of the response variable for that particular value of the explanatory variable.

   • This statement is true. In regression, $\hat{y}$ represents the predicted value of the response variable (often the mean response) for a given value of the explanatory variable.

2. Statement B: The least-squares regression line minimizes the sum of squared residuals.

   • This statement is true. The least-squares regression method specifically finds the line that minimizes the sum of squared residuals (the squared vertical distances between observed values and predicted values).

3. Statement C: The least-squares regression line always contains the point $(0,0)$.

   • This statement is false. The regression line does not necessarily pass through the origin unless specifically constrained. Typically, it passes through the mean of the data points but not necessarily through $(0,0)$.

4. Statement D: The least-squares regression line always contains the point $(\bar{x}, \bar{y})$.

   • This statement is true. The least-squares regression line always passes through the mean of $x$ and $y$, which is the point $(\bar{x}, \bar{y})$.

5. Statement E: The least-squares regression line maximizes the sum of squared residuals.

   • This statement is false. The least-squares regression line minimizes the sum of squared residuals, not maximizes.

6. Statement F: The sign of the linear correlation coefficient, $r$, and the sign of the slope of the least-squares regression line, $b_1$, are the same.

   • This statement is true. The correlation coefficient $r$ has the same sign as the slope $b_1$, indicating the direction of the relationship (positive or negative).

7. Statement G: The predicted value of $\hat{y}$ is an estimate of the mean value of the explanatory variable for that particular value of the response variable.

   • This statement is false. The predicted value $\hat{y}$ estimates the response variable's mean value, not the mean of the explanatory variable.

Conclusion

The correct answers are A, B, D, and F.

Would you like further details on any of these points?

Here are 5 follow-up questions to expand your understanding:

1. How does changing the slope $b_1$ affect the fit of the least-squares regression line?
2. What happens if there is no correlation between $x$ and $y$ in a dataset? How would that affect $b_1$ and $r$?
3. How does outlier influence impact the least-squares regression line?
4. Can a least-squares regression line be used to make predictions outside the range of observed $x$-values?
5. Why is minimizing the sum of squared residuals particularly effective in regression?

Tip: The least-squares regression line is sensitive to outliers, which can significantly affect its slope and intercept. Always examine residuals to check for unusual data points.