Problem Statement
The two regression equations for variables $x$ and $y$ are $2x + 3y = 6$ and $5x + 7y = 12$.
(i) Determine which line is the regression line of $y$ on $x$ and which is of $x$ on $y$.
(ii) Find the mean values $\bar{x}$ and $\bar{y}$.
(iii) Calculate the correlation coefficient $r$.
Verified Solution & Marking Scheme
Assume and Test Regression Coefficients
Assumption 1: Let $2x + 3y = 6$ be $y$ on $x$ $\implies 3y = -2x + 6 \implies y = -\frac{2}{3}x + 2 \implies b_{yx} = -\frac{2}{3}$.
Let $5x + 7y = 12$ be $x$ on $y$ $\implies 5x = -7y + 12 \implies x = -\frac{7}{5}y + \frac{12}{5} \implies b_{xy} = -\frac{7}{5}$.
Check product $b_{yx} \times b_{xy} = \left(-\frac{2}{3}\right)\left(-\frac{7}{5}\right) = \frac{14}{15} \approx 0.9333 < 1$.
Since $0 \le b_{yx} \cdot b_{xy} < 1$, the assumption is correct!
Find Means by Solving System Simultaneously
The regression lines pass through $(\bar{x}, \bar{y})$:
$2\bar{x} + 3\bar{y} = 6 \implies 10\bar{x} + 15\bar{y} = 30$
$5\bar{x} + 7\bar{y} = 12 \implies 10\bar{x} + 14\bar{y} = 24$
Subtracting gives $\bar{y} = 6$.
Substitute $\bar{y} = 6$: $2\bar{x} + 3(6) = 6 \implies 2\bar{x} + 18 = 6 \implies 2\bar{x} = -12 \implies \bar{x} = -6$.
Calculate Correlation Coefficient r
Both regression coefficients are negative, so $r$ must be negative:
$r = -\sqrt{b_{yx} \cdot b_{xy}} = -\sqrt{\frac{14}{15}} \approx -0.966$