In simple linear regression analysis, which of the following best describes the method of least squares?

• A. It finds the line that maximizes the correlation coefficient between the independent and dependent variables.
• B. It finds the line that passes through the origin and maximizes the sum of the observed values.
• C. It finds the line that minimizes the sum of the absolute differences between the observed and predicted values.
• D. It finds the line that minimizes the sum of the squared vertical distances between the observed values and the predicted values, that is, it minimizes ${\displaystyle \sum i_{=1}{(}^{\mathrm{n}}{)}^2(y_i-{\hat{y}}_i{)}^2}$.

Answer: (D)

Verified step by step guidance

1. Understand that in simple linear regression, we aim to find the best-fitting line through the data points that relate an independent variable \(x\) to a dependent variable \(y\).
2. Recall that the method of least squares specifically looks for the line that minimizes the sum of the squared vertical distances (residuals) between the observed values \(y_i\) and the predicted values \(\hat{y}_i\) on the line.
3. Express the residual for each data point as \(e_i = y_i - \hat{y}_i\), where \(\hat{y}_i = b_0 + b_1 x_i\) is the predicted value from the regression line with intercept \(b_0\) and slope \(b_1\).
4. Formulate the objective function to minimize as the sum of squared residuals: \(S = \sum_{i=1}^n (y_i - (b_0 + b_1 x_i))^2\).
5. Recognize that the least squares method finds the values of \(b_0\) and \(b_1\) that minimize \(S\), ensuring the best fit by minimizing the squared vertical distances, not by maximizing correlation or minimizing absolute differences.