Which of the following is not one of the purposes of a line of regression in linear regression using the least squares method ($\text{least squares}$)?

• A. To determine the strength and direction of the relationship between two variables
• B. To establish a causal relationship between the independent and dependent variables
• C. To minimize the sum of the squared differences between observed and predicted values ($\sum {(y-ŷ)}^2$)
• D. To predict the value of the dependent variable based on the independent variable ($y=\beta 0+\beta 1x$)

Answer: (B)

Verified step by step guidance

1. Understand that the line of regression in linear regression is primarily used to model the relationship between two variables by fitting a line that best represents the data points.
2. Recall that the least squares method finds the line that minimizes the sum of the squared differences between the observed values and the values predicted by the line, ensuring the best fit.
3. Recognize that the regression line helps in predicting the value of the dependent variable given a value of the independent variable, which is a key purpose of regression analysis.
4. Know that the regression line can be used to assess the strength and direction (positive or negative) of the linear relationship between the two variables, often summarized by the correlation coefficient.
5. Identify that establishing a causal relationship is not a purpose of the regression line itself; regression shows association but does not prove causation without further experimental or contextual evidence.