To solve the problem:

1. Regression Equations:
   The given regression equations are: $8x - 10y + 66 = 0 \quad \text{and} \quad 40x - 18y = 217$

2. Finding the Means ($\bar{x}$ and $\bar{y}$):
   The means of $x$ and $y$ lie on both regression lines. Thus, we solve the equations simultaneously to find $\bar{x}$ and $\bar{y}$.

   Rewrite the equations in standard form: $y = \frac{8}{10}x + \frac{66}{10} \quad \text{(from the first equation)}$ $y = \frac{40}{18}x - \frac{217}{18} \quad \text{(from the second equation)}$

   Equating the two expressions for $y$: $\frac{8}{10}x + \frac{66}{10} = \frac{40}{18}x - \frac{217}{18}$ Simplify and solve for $x$, then substitute back to find $\bar{y}$.

3. Finding the Correlation Coefficient $r$:
   The correlation coefficient is related to the slopes of the regression lines. Denote: $b_{yx} = \text{slope of the regression line of } y \text{ on } x = -\frac{8}{10} = -0.8$ $b_{xy} = \text{slope of the regression line of } x \text{ on } y = -\frac{18}{40} = -0.45$

   The formula for $r$ is: $r = \pm\sqrt{b_{yx} \cdot b_{xy}}$ Use the given information to compute $r$.

I will now solve these equations step by step. sympy import symbols, Eq, solve, sqrt

Define variables

x, y = symbols('x y')

Regression equations

eq1 = Eq(8 * x - 10 * y + 66, 0) eq2 = Eq(40 * x - 18 * y, 217)

Solve for means (intersection of the regression lines)

means = solve((eq1, eq2), (x, y))

Slopes of the regression lines

b_yx = -8 / 10 # slope of y on x b_xy = -18 / 40 # slope of x on y

Correlation coefficient (absolute value)

r = sqrt(b_yx * b_xy)

means, rThe results are as follows:

1. Means of $x$ and $y$: $\bar{x} = \frac{1679}{128} \approx 13.11, \quad \bar{y} = \frac{547}{32} \approx 17.09$

2. Correlation coefficient $r$: $r = 0.6$

Let me know if you would like detailed steps for any part of the solution or further clarification.

Here are some related questions to explore:

1. How do the slopes of regression lines relate to the correlation coefficient?
2. Can the regression lines be used to predict values for $x$ or $y$?
3. What happens to the correlation coefficient if one slope is reversed?
4. How would the results change if the constants in the equations were altered?
5. Is it possible to determine if the correlation is positive or negative without knowing the signs of the slopes?

Tip: The correlation coefficient $r$ indicates the strength and direction of a linear relationship; ensure that its sign matches the context of the slopes provided.