The two regression lines are given by $3x + 2y = 26$ and $6x + y = 31$. Calculate the correlation coefficient $r$ between $x$ and $y$.
Answer Key:
-0.5
Step-by-Step Solution & Explanation
Let the regression line of $y$ on $x$ be $3x + 2y = 26 \implies 2y = -3x + 26 \implies y = -\frac{3}{2} x + 13$, so $b_{yx} = -1.5$.
Then the regression line of $x$ on $y$ is $6x + y = 31 \implies 6x = -y + 31 \implies x = -\frac{1}{6} y + \frac{31}{6}$, so $b_{xy} = -\frac{1}{6}$.
Checking product:
$b_{yx} \times b_{xy} = (-1.5) \times \left(-\frac{1}{6}\right) = \frac{1.5}{6} = 0.25 \le 1$
This is a valid assignment. The correlation coefficient is:
$r = -\sqrt{b_{yx} \times b_{xy}} = -\sqrt{0.25} = -0.5$