To solve this problem, we need to handle two tasks:

1. Find the mean values of X and Y from the given regression equations.
2. Find the correlation coefficient based on the regression equations.

The given regression equations are:

1. $2Y - X - 50 = 0$
2. $3Y - 2X - 10 = 0$

Step 1: Solve the system of equations

We solve these equations simultaneously to find the values of $\bar{X}$ and $\bar{Y}$ (the mean values of $X$ and $Y$).

Equation 1:

$2Y - X - 50 = 0 \quad \Rightarrow \quad X = 2Y - 50$

Equation 2:

$3Y - 2X - 10 = 0$

Substitute $X = 2Y - 50$ into the second equation: $3Y - 2(2Y - 50) - 10 = 0$

Simplifying: $3Y - 4Y + 100 - 10 = 0 \quad \Rightarrow \quad -Y + 90 = 0$

$Y = 90$

Now that we have $Y = 90$, substitute this value into the first equation to find $X$: $X = 2(90) - 50 = 180 - 50 = 130$

Thus, the mean values are: $\bar{X} = 130, \quad \bar{Y} = 90$

Step 2: Find the Correlation Coefficient

The correlation coefficient $r$ is related to the regression coefficients. The general regression equations for two variables $X$ and $Y$ are: $X - \bar{X} = r \cdot \frac{\sigma_X}{\sigma_Y} \cdot (Y - \bar{Y})$ $Y - \bar{Y} = r \cdot \frac{\sigma_Y}{\sigma_X} \cdot (X - \bar{X})$ where $\sigma_X$ and $\sigma_Y$ are the standard deviations of $X$ and $Y$, respectively.

From the given regression equations, we can extract the regression coefficients:

• The regression equation of $X$ on $Y$ is $X = 2Y - 50$, so the regression coefficient $b_{XY} = 2$.
• The regression equation of $Y$ on $X$ is $Y = \frac{2}{3}X + \frac{10}{3}$, so the regression coefficient $b_{YX} = \frac{2}{3}$.

The correlation coefficient $r$ is the geometric mean of the regression coefficients: $r = \sqrt{b_{XY} \cdot b_{YX}} = \sqrt{2 \cdot \frac{2}{3}} = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}} \approx 1.155$

Final Answer:

• Mean of $X$: $130$
• Mean of $Y$: $90$
• Correlation coefficient $r$: $\frac{2}{\sqrt{3}} \approx 1.155$

Would you like any further details or clarifications?

Related questions:

1. What are the assumptions behind calculating the correlation coefficient?
2. How can we interpret the value of the correlation coefficient in this context?
3. How would the solution change if the regression equations were different?
4. What is the geometric interpretation of the correlation coefficient?
5. Can the correlation coefficient ever be greater than 1?

Tip: When solving regression-based problems, always check if the system of equations is consistent and leads to a meaningful solution.