Question

If \( b_{yx} + b_{xy} = 1.30 \) and \( r = 0.75 \) then the given data is inconsistent.

Hint:

For regression and correlation coefficients to be consistent, two key rules must hold: 1. Both regression coefficients must have the same sign as the correlation coefficient. 2. The correlation coefficient cannot be greater than the average of the regression coefficients.

Answer 1

One of the properties of regression coefficients (\(b_{yx}\) and \(b_{xy}\)) and the correlation coefficient (\(r\)) is that the arithmetic mean of the regression coefficients must be greater than or equal to the correlation coefficient. \[ \frac{b_{yx} + b_{xy}}{2} \geq |r| \] Let's check this condition with the given data:
\( b_{yx} + b_{xy} = 1.30 \)
\( r = 0.75 \)
The arithmetic mean of the regression coefficients is: \[ \frac{1.30}{2} = 0.65 \] Comparing this to the correlation coefficient: \[ 0.65 \geq 0.75 \] This statement is false. Since the condition is violated, the given data is inconsistent. Therefore, the original statement is True.