The two lines of regression for a bivariate dataset are given by:
$3x + 2y = 26 \quad \text{and} \quad 6x + y = 31$
The Karl Pearson correlation coefficient $r$ between $X$ and $Y$ is ________ (round off to two decimal places).
Answer Key:
-0.5
Step-by-Step Solution & Explanation
Let us identify which equation corresponds to regression of $y$ on $x$ versus $x$ on $y$:
**Hypothesis 1:**
- Let $3x + 2y = 26$ be the regression line of $y$ on $x$:
$2y = -3x + 26 \implies y = -\frac{3}{2}x + 13 \implies b_{yx} = -1.5$
- Let $6x + y = 31$ be the regression line of $x$ on $y$:
$6x = -y + 31 \implies x = -\frac{1}{6}y + \frac{31}{6} \implies b_{xy} = -\frac{1}{6}$
Compute $r^2$:
$r^2 = b_{yx} \cdot b_{xy} = \left(-\frac{3}{2}\right) \left(-\frac{1}{6}\right) = \frac{3}{12} = \frac{1}{4} = 0.25 \le 1$
Since $r^2 \le 1$, this assignment is statistically valid.
Because both regression coefficients are negative, $r$ must be negative:
$r = -\sqrt{0.25} = -0.50$
(Note: Swapping assignments yields $r^2 = 4 > 1$, which violates $|r| \le 1$).