In linear regression using the least squares method, the line of best fit for a scatterplot always passes through which pair of points?

• A. The point with the largest $x$-value and the point with the smallest $y$-value
• B. The point with the smallest $x$-value and the point with the largest $y$-value
• C. The first and last data points in the dataset
• D. The mean of $x$ and the mean of $y$, that is, the point $(\overline{x},\overline{y})$

Answer: (D)

Verified step by step guidance

1. Recall that in linear regression using the least squares method, we fit a line to minimize the sum of squared vertical distances between the observed data points and the line.
2. Understand that the equation of the regression line is typically written as \(y = b_0 + b_1 x\), where \(b_0\) is the intercept and \(b_1\) is the slope.
3. Recognize that the regression line always passes through the point of averages, which is the point with coordinates \((\overline{x}, \overline{y})\), where \(\overline{x}\) is the mean of all x-values and \(\overline{y}\) is the mean of all y-values.
4. This happens because the least squares estimates satisfy the condition that the residuals sum to zero and the line balances the data around the means.
5. Therefore, the line of best fit always passes through the point \((\overline{x}, \overline{y})\), not necessarily through any specific data points like the smallest or largest x or y values.