If the slope beta1 is zero in simple linear regression, what does this imply about the relationship between X and Y?

If the slope beta1 is zero in simple linear regression, what does this imply about the relationship between X and Y?

• A. There is a strong positive linear relationship.
• B. There is a strong negative linear relationship.
• C. There is no linear relationship between X and Y.
• D. The regression line perfectly predicts Y from X.

Answer: (C)

The slope of the regression line measures how Y changes on average as X increases. If that slope is zero, changing X does not change the average value of Y in this linear model—the best-fitting line is horizontal. In other words, there is no linear relationship between X and Y because Y does not vary with X in the linear sense. You can still have Y and X related in other, nonlinear ways, but within the straight-line relationship, the effect of X is null. A nonzero positive slope would show a positive linear relationship, a nonzero negative slope would show a negative linear relationship, and a perfect prediction would require the residuals to be zero, which isn’t implied by a zero slope.

The slope of the regression line measures how Y changes on average as X increases. If that slope is zero, changing X does not change the average value of Y in this linear model—the best-fitting line is horizontal. In other words, there is no linear relationship between X and Y because Y does not vary with X in the linear sense. You can still have Y and X related in other, nonlinear ways, but within the straight-line relationship, the effect of X is null. A nonzero positive slope would show a positive linear relationship, a nonzero negative slope would show a negative linear relationship, and a perfect prediction would require the residuals to be zero, which isn’t implied by a zero slope.