Step 1: Concept Overview:
The correlation coefficient, \(r\), is derived from two regression lines: Y regressed on X, and X regressed on Y. Their slopes, \(b_{YX}\) and \(b_{XY}\), are the regression coefficients. The square of \(r\) equals the product of these coefficients.
Step 2: Core Formula:
The correlation coefficient \(r\) is the geometric mean of the regression coefficients:\[ r = \pm \sqrt{b_{YX} . b_{XY}} \]The sign of \(r\) matches the sign of the regression coefficients.
Step 3: Detailed Solution:
Assume the first equation represents Y regressed on X, and the second represents X regressed on Y.Line 1 (Y on X): \(8X - 10Y + 66 = 0\)Rewrite as \(Y = b_{YX}X + a\):\[ 10Y = 8X + 66 \]\[ Y = \frac{8}{10}X + \frac{66}{10} \]Thus, \(b_{YX} = \frac{8}{10} = 0.8\).Line 2 (X on Y): \(40X - 18Y = 264\)Rewrite as \(X = b_{XY}Y + c\):\[ 40X = 18Y + 264 \]\[ X = \frac{18}{40}Y + \frac{264}{40} \]Thus, \(b_{XY} = \frac{18}{40} = \frac{9}{20} = 0.45\).Calculate \(r^2\):\[ r^2 = b_{YX} . b_{XY} = 0.8 \times 0.45 = \frac{8}{10} \times \frac{45}{100} = \frac{360}{1000} = 0.36 \]Then, \(r\) is the square root. Since \(b_{YX}\) and \(b_{XY}\) are positive, \(r\) is positive.\[ r = \sqrt{0.36} = 0.6 \]Note: Our assumption is valid because \(|r| = 0.6 \le 1\). Reversing the assumption leads to \(b_{YX} = 40/18\) and \(b_{XY} = 10/8\), with a product greater than 1, which is impossible.
Step 4: Final Result:
The correlation coefficient between X and Y is 0.6.