Which of the following statements is/are TRUE for the regression lines of $Y$ on $X$ and $X$ on $Y$?
Options:
(A) Both regression lines intersect at the point of means $(\bar{x}, \bar{y})$
(B) The regression coefficients $b_{yx}$ and $b_{xy}$ must have the same algebraic sign as the correlation coefficient $r$
(C) If $|r| = 1$, the two regression lines are identical (coincident)
(D) If $r = 0$, the two regression lines are mutually perpendicular
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
By classical regression analysis:
1. The lines are $y - \bar{y} = b_{yx}(x - \bar{x})$ and $x - \bar{x} = b_{xy}(y - \bar{y})$, both passing through $(\bar{x}, \bar{y})$ (A is true).
2. Since $b_{yx} = r\frac{\sigma_y}{\sigma_x}$ and $b_{xy} = r\frac{\sigma_x}{\sigma_y}$ and standard deviations are positive, $b_{yx}$, $b_{xy}$, and $r$ always share the same sign (B is true).
3. When $|r| = 1$, all data points lie on a single line, so both regression lines coincide (C is true).
4. The angle between the two lines satisfies $\tan\theta = \frac{1 - r^2}{|r|}\left(\frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}\right)$. As $r \to 0$, $\theta \to 90^\circ$, so the lines become perpendicular ($y = \bar{y}$ and $x = \bar{x}$) (D is true).