Question

If \(Y = mX + 4\) and \(X = 4Y + 5\) are the regression lines of Y on X and X on Y respectively, then m lies between the values:

• A. 0 and 1
• B. 0 and 0.5
• C. 0 and 0.25
• D. -1 and 1

Hint:

Use \(r^2 = b_{yx}\cdot b_{xy}\) and the fact that \(r^2\) cannot exceed 1.

Answer 1

The Correct Option is C

Step 1: Write $b_{yx}=m$ (slope of Y on X line) and $b_{xy}=4$ (slope of X on Y line).

Step 2: The correlation coefficient equals the signed geometric mean of the two regression coefficients: $r = \pm\sqrt{b_{yx}\,b_{xy}} = \pm\sqrt{4m}$.

Step 3: For $r$ to be a real number lying in $[-1,1]$, we need $4m \ge 0$ and $4m \le 1$, giving $0 \le m \le \frac14$.

Step 4: Since $b_{xy}=4$ is positive, $r$ must be positive too, and so must $b_{yx}=m$ (both regression coefficients share the sign of r). This rules out $m=0$ in practice.

Step 5: So m is restricted strictly between 0 and 0.25.

\[\boxed{0 < m < 0.25}\]