Which of the following is NOT true about simple linear regression using the least squares method?

• A. The errors ($e$, residuals) are assumed to have constant variance.
• B. The least squares method minimizes the sum of the squared vertical distances between observed and predicted values.
• C. The slope of the regression line is always $+$.
• D. The relationship between the independent and dependent variable is assumed to be linear.

Answer: (C)

Verified step by step guidance

1. Understand the assumptions and properties of simple linear regression using the least squares method. These include assumptions about the errors (residuals), the method of fitting the line, and the nature of the relationship between variables.
2. Recall that the errors (residuals) in simple linear regression are assumed to have constant variance, which is known as homoscedasticity. This means the spread of residuals should be roughly the same across all levels of the independent variable.
3. Remember that the least squares method works by minimizing the sum of the squared vertical distances (residuals) between the observed data points and the predicted values on the regression line. This is the fundamental principle behind fitting the regression line.
4. Recognize that the relationship between the independent variable (predictor) and the dependent variable (response) is assumed to be linear, meaning the regression line is a straight line that best describes this relationship.
5. Identify that the statement 'The slope of the regression line is always positive' is NOT true because the slope can be positive, negative, or zero depending on the data. The slope indicates the direction and strength of the relationship, and it is not constrained to be positive.