Which of the following is true concerning linear regression using the least squares method?

• A. The least squares method finds the line that passes through every data point exactly.
• B. The least squares method finds the line that minimizes the sum of the squared vertical distances between the observed values and the predicted values, that is, it minimizes $\sum {(y_i-{\hat{y}}_i)}^2$.
• C. The least squares method maximizes the sum of the squared residuals between the observed and predicted values.
• D. The least squares method minimizes the sum of the absolute differences between the observed and predicted values, that is, it minimizes $\sum |y_i-{\hat{y}}_i|$.

Answer: (B)

Verified step by step guidance

1. Understand that linear regression using the least squares method aims to find the best-fitting line through a set of data points by minimizing the discrepancies between observed and predicted values.
2. Recall that the 'residual' for each data point is the vertical distance between the observed value and the predicted value on the regression line, calculated as \(e_i = y_i - \hat{y}_i\).
3. The least squares method specifically minimizes the sum of the squared residuals, which is expressed as \(\sum_{i=1}^n (y_i - \hat{y}_i)^2\), where \(y_i\) are observed values and \(\hat{y}_i\) are predicted values from the regression line.
4. Note that the least squares line does not necessarily pass through every data point; instead, it balances the residuals to minimize their squared sum, which reduces the overall error in prediction.
5. Recognize that other methods, such as minimizing the sum of absolute differences or maximizing residuals, are not characteristics of the least squares method.