In simple linear regression Y = α + βX + ε, what does β represent?

In simple linear regression Y = α + βX + ε, what does β represent?

• A. β is the average change in Y for a one-unit increase in X, holding other factors constant.
• B. β is the intercept of the regression line.
• C. β is the correlation between X and Y.
• D. β is the predicted value of Y when X = 0.

Answer: (A)

In simple linear regression, the slope β is the change in the expected value of Y for a one-unit increase in X. Since the model implies E[Y|X] = α + βX (assuming the errors average to zero), increasing X by 1 shifts the conditional mean of Y by β. A positive β means Y tends to rise with X, a negative β means Y tends to fall. This is not the intercept (which is α), not the correlation between X and Y, and not a specific predicted value of Y at a given X (that would be α + βX).

In simple linear regression, the slope β is the change in the expected value of Y for a one-unit increase in X. Since the model implies E[Y|X] = α + βX (assuming the errors average to zero), increasing X by 1 shifts the conditional mean of Y by β. A positive β means Y tends to rise with X, a negative β means Y tends to fall. This is not the intercept (which is α), not the correlation between X and Y, and not a specific predicted value of Y at a given X (that would be α + βX).