In linear regression using the least squares method, the least squares regression line minimizes the sum of the $e^2$, where $e$ represents the residuals (the differences between observed and predicted values).

• A. products of the observed and predicted values
• B. squared vertical distances between the observed values and the predicted values, i.e., minimizes $\sum {(y_i-{\hat{y}}_i)}^2$
• C. absolute values of the residuals, i.e., minimizes $\sum \left|e_i\right|$
• D. squared horizontal distances between the observed values and the predicted values

Answer: (B)

Verified step by step guidance

1. Understand the goal of the least squares method in linear regression: it aims to find the line that best fits the data by minimizing the differences between observed values and predicted values.
2. Recall that the residual for each data point is the vertical distance between the observed value (actual data point) and the predicted value (point on the regression line).
3. Recognize that the least squares method specifically minimizes the sum of the squares of these residuals, which means it minimizes the sum of the squared vertical distances between observed and predicted values.
4. Note that minimizing squared vertical distances is preferred over minimizing absolute values of residuals or horizontal distances because squaring emphasizes larger errors and leads to a unique solution with nice mathematical properties.
5. Therefore, the least squares regression line is the one that minimizes the sum of the squared vertical distances between the observed values and the predicted values.