Question

If the two regression lines are given by \(8X-10Y+66=0\) and \(40X-18Y=264\), then the correlation coefficient between X and Y is:

• A. 0.424
• B. 0.524
• C. -0.492
• D. 0.6

Hint:

To identify which equation is Y on X and which is X on Y, calculate both possible pairs of regression coefficients. The correct pair is the one for which their product is less than or equal to 1.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The correlation coefficient, \(r\), can be determined from the two regression lines. One line represents the regression of Y on X, and the other represents the regression of X on Y. The slopes of these lines are the regression coefficients, \(b_{YX}\) and \(b_{XY}\). The square of the correlation coefficient is the product of these two regression coefficients.

Step 2: Key Formula or Approach:
The correlation coefficient \(r\) is given by the geometric mean of the regression coefficients: \[ r = \pm \sqrt{b_{YX} . b_{XY}} \] The sign of \(r\) is the same as the sign of the regression coefficients.

Step 3: Detailed Explanation:
Let's assume the first equation is the regression of Y on X and the second is the regression of X on Y. Line 1 (Y on X): \(8X - 10Y + 66 = 0\) Rearrange to the form \(Y = b_{YX}X + a\): \[ 10Y = 8X + 66 \] \[ Y = \frac{8}{10}X + \frac{66}{10} \] So, the regression coefficient of Y on X is \(b_{YX} = \frac{8}{10} = 0.8\). Line 2 (X on Y): \(40X - 18Y = 264\) Rearrange to the form \(X = b_{XY}Y + c\): \[ 40X = 18Y + 264 \] \[ X = \frac{18}{40}Y + \frac{264}{40} \] So, the regression coefficient of X on Y is \(b_{XY} = \frac{18}{40} = \frac{9}{20} = 0.45\). Now, calculate the square of the correlation coefficient: \[ r^2 = b_{YX} . b_{XY} = 0.8 \times 0.45 = \frac{8}{10} \times \frac{45}{100} = \frac{360}{1000} = 0.36 \] The correlation coefficient is the square root of this value. Since both \(b_{YX}\) and \(b_{XY}\) are positive, \(r\) must also be positive. \[ r = \sqrt{0.36} = 0.6 \] Note: Our assumption was correct because \(|r| = 0.6 \le 1\). If we had assumed the opposite, we would get \(b_{YX} = 40/18\) and \(b_{XY} = 10/8\), whose product is greater than 1, which is impossible.
Step 4: Final Answer:
The correlation coefficient between X and Y is 0.6.