Correlation Coefficient from Regression Lines

The correlation coefficient ($r$) between two variables $x$ and $y$ can be determined from the slopes of the two regression lines. If the regression line of $y$ on $x$ is $y = b_{yx}x + c_1$ and the regression line of $x$ on $y$ is $x = b_{xy}y + c_2$, then the square of the correlation coefficient is given by $r^2 = b_{yx} \cdot b_{xy}$. The sign of $r$ is the same as the sign of the slopes $b_{yx}$ and $b_{xy}$.

Identify Regression Line Slopes

We are given two lines:

• Line 1: $x + y + 11 = 0$
• Line 2: $2x + 3y + 4 = 0$

We need to determine which line corresponds to the regression of $y$ on $x$ ($b_{yx}$) and which corresponds to the regression of $x$ on $y$ ($b_{xy}$).

• Rewrite Line 1 as $x$ in terms of $y$: $x = -y - 11$. The slope $b_{xy}$ (coefficient of $y$) is $-1$.
• Rewrite Line 2 as $y$ in terms of $x$: $3y = -2x - 4 \implies y = -\frac{2}{3}x - \frac{4}{3}$. The slope $b_{yx}$ (coefficient of $x$) is $-\frac{2}{3}$.

Let's check the alternative assignment:

• Line 1 as $y$ on $x$: $y = -x - 11$, so $b_{yx} = -1$.
• Line 2 as $x$ on $y$: $2x = -3y - 4 \implies x = -\frac{3}{2}y - 2$, so $b_{xy} = -\frac{3}{2}$.

Calculate $r^2$ for this alternative assignment: $r^2 = b_{yx} \cdot b_{xy} = (-1) \cdot (-\frac{3}{2}) = \frac{3}{2}$. Since $r^2$ cannot be greater than 1, this assignment is incorrect.

Therefore, the correct slopes are $b_{yx} = -\frac{2}{3}$ and $b_{xy} = -1$.

Calculate Correlation Coefficient Squared ($r^2$)

Using the correct slopes:

$r^2 = b_{yx} \cdot b_{xy} = \left(-\frac{2}{3}\right) \times (-1) = \frac{2}{3}$

Determine the Sign of Correlation Coefficient ($r$)

The correlation coefficient $r$ must have the same sign as the slopes $b_{yx}$ and $b_{xy}$. In this case, both $b_{yx} = -\frac{2}{3}$ and $b_{xy} = -1$ are negative.

Therefore, the correlation coefficient $r$ must be negative.

$r = -\sqrt{\frac{2}{3}}$

The value of the correlation coefficient between $x$ and $y$ is $-\sqrt{2/3}$.