Question

a certain standardized test measures students knowledge in math and english. the scatterplot displays the scores for 10 randomly selected students. the equation $hat{y}=54.16 + 0.87x$ is called the least - squares regression line because it is least able to make accurate predictions for the data. makes the sum of the squared residuals as large as possible. makes the sum of the squared residuals as small as possible. causes the relationship between english scores and math scores to be strongest.

Explanation:

The least - squares regression line is defined as the line that minimizes the sum of the squared residuals. Residuals are the differences between the observed \(y\) - values (in this case, math scores) and the predicted \(\hat{y}\) - values (from the regression equation). A smaller sum of squared residuals means a better fit of the line to the data points.

• The first option is incorrect because the least - squares regression line is designed to make the best (not the least) predictions in the sense of minimizing the sum of squared errors.
• The second option is incorrect because we want to minimize (not maximize) the sum of squared residuals for a good fit.
• The fourth option is incorrect. The strength of the relationship between two variables (in this case, English and math scores) is measured by the correlation coefficient \(r\), not by the least - squares regression line formula itself. The least - squares regression line is about minimizing the sum of squared residuals for prediction purposes.

Answer:

makes the sum of the squared residuals as small as possible.