In linear regression using the least squares method, what is the primary purpose of the regression line?

• A. To guarantee that all observed data points lie exactly on a straight line
• B. To prove that the explanatory variable causes changes in the response variable
• C. To maximize the sum of absolute residuals to increase prediction accuracy
• D. To provide the best-fitting linear relationship between the explanatory and response variables by minimizing the sum of squared residuals

Answer: (D)

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 explanatory (independent) variable and the response (dependent) variable.
2. Recognize that the 'least squares method' is a technique used to determine this best-fitting line by minimizing the sum of the squared differences (residuals) between the observed values and the values predicted by the line.
3. Recall that residuals are the vertical distances between the observed data points and the regression line, calculated as \(\text{residual} = y_i - \hat{y}_i\), where \(y_i\) is the observed value and \(\hat{y}_i\) is the predicted value from the regression line.
4. Note that the least squares method minimizes the sum of squared residuals, expressed mathematically as \(\sum (y_i - \hat{y}_i)^2\), to ensure the best possible linear fit to the data.
5. Conclude that the primary purpose of the regression line is to provide the best-fitting linear relationship by minimizing these squared residuals, rather than forcing all points to lie exactly on the line or proving causation.