In the context of linear regression using the least squares method, why is the graph shown considered a line of best fit?

• A. Because it minimizes the sum of the squared vertical distances between the observed data points and the line, that is, it minimizes $\sum {(y_i-{\hat{y}}_i)}^2$.
• B. Because it connects the first and last data points.
• C. Because it maximizes the sum of the residuals.
• D. Because it passes through every data point exactly.

Answer: (A)

Verified step by step guidance

1. Understand that in linear regression, the goal is to find a line that best represents the relationship between the independent variable (x) and the dependent variable (y).
2. Recognize that the 'line of best fit' is defined as the line that minimizes the sum of the squared vertical distances (residuals) between the observed data points and the line itself.
3. Recall that the vertical distance for each data point is the difference between the observed value \(y_i\) and the predicted value \(\hat{y}_i\) on the line, i.e., the residual \(e_i = y_i - \hat{y}_i\).
4. The least squares method finds the line by minimizing the sum of squared residuals, expressed mathematically as minimizing \(\sum_{i=1}^n (y_i - \hat{y}_i)^2\).
5. This minimization ensures that the overall error between the data points and the line is as small as possible, which is why the graph is considered the line of best fit.