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: From \(Y = mX+4\), the regression coefficient of Y on X is \(b_{yx} = m\).

Step 2: From \(X = 4Y+5\), the regression coefficient of X on Y is \(b_{xy} = 4\).

Step 3: The correlation coefficient satisfies \(r^2 = b_{yx}\,b_{xy} = 4m\), and since \(0 \le r^2 \le 1\), we get \(0 \le 4m \le 1\), i.e. \(0 \le m \le \dfrac14\).

Step 4: Also, both regression coefficients must carry the same sign as \(r\). Since \(b_{xy}=4>0\), \(r\) is positive, so \(m=b_{yx}\) must also be positive.

Step 5: Hence m lies between 0 and 0.25.

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