If the two regression lines are given by Y = 0.6X + 4 and X = 0.4Y + 2, what is the Karl Pearson correlation coefficient r_XY?

The line Y = 0.6X + 4 expresses Y in terms of X, so it is the regression of Y on X, and b_yx = 0.6. The line X = 0.4Y + 2 expresses X in terms of Y, so it is the regression of X on Y, and b_xy = 0.4. The correlation coefficient is the geometric mean of the two regression coefficients: r² = b_yx × b_xy = 0.6 × 0.4 = 0.24 r = √0.24 = 0.49 Both regression coefficients are positive, so r is positive: r = +0.49. We should check that the identification of the lines is correct, since r² cannot exceed 1. Had we taken the lines the other way round, we would get b_xy = 1/0.6 = 1.667 and b_yx = 1/0.4 = 2.5, whose product is 4.17. This is greater than 1 and not possible, so the identification above is right. Option (A), 0.240, is the value of r², not of r. Hence the correct option is (B).