Solution
Which Regression Line Is Which
Show the problem again
The two least-squares regression lines of a dataset (one of $y$ on $x$, the other of $x$ on $y$) are $3x + 2y = 26$ and $6x + y = 31$, but you are not told which is which. What is the correlation coefficient $r$ between $x$ and $y$? Give a reduced fraction.
Worked solution
Write the slopes of each line in both orientations. Line 1 as $y$ on $x$ has slope $-3/2$; as $x$ on $y$ has slope $-2/3$. Line 2 as $y$ on $x$ has slope $-6$; as $x$ on $y$ has slope $-1/6$. We have $b_{yx} \cdot b_{xy} = r^2 \le 1$. If line 2 were $y$ on $x$, line 1 would be $x$ on $y$ and the product $(-6)(-2/3) = 4 > 1$, impossible. So line 1 is $y$ on $x$ and line 2 is $x$ on $y$: $r^2 = (-3/2)(-1/6) = 1/4$. Both slopes are negative, so $r = -1/2$.
Source: Classic textbook exercise on the two regression lines; numbers original. Statement written for AxiomIQ.