Question

the weight (in pounds) and height (in inches) for a child were measured every few months over a two - year period. the results are displayed in the scatterplot. the equation $hat{y}=17.4 + 0.5x$ is called the least - squares regression line because it
is least able to make accurate predictions for the data.
makes the strongest association between weight and height.
minimizes the sum of the squared distances from the actual $y$-value to the predicted $y$-value.
maximizes the sum of the squared distances from the actual $y$-value to the predicted $y$-value.

Explanation:

The least - squares regression line is defined as the line that minimizes the sum of the squared residuals (the squared distances from the actual \(y\) - value to the predicted \(\hat{y}\) - value). The formula for the sum of squared residuals is \(SSE=\sum_{i = 1}^{n}(y_{i}-\hat{y}_{i})^{2}\), and the least - squares regression line is the one that makes this sum as small as possible.

• 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. The strength of the association between two variables is measured by the correlation coefficient \(r\), not by the least - squares regression line formula itself. The regression line is based on the existing association (measured by \(r\)) to make predictions.
• The fourth option is incorrect. Maximizing the sum of squared distances would be the opposite of what the least - squares regression line is intended to do.

Answer:

minimizes the sum of the squared distances from the actual \(y\) - value to the predicted \(y\) - value.