The image contains a problem involving regression lines and correlation. Let me extract and analyze it.

Problem Description:

In a partially blurry laboratory record, only the regression lines and some statistics are provided:

1. Regression lines: $5y = 4x + 33$ $20x = 9y + 107$

2. Tasks:

   • Find the mean values of $x$ and $y$.
   • Find the correlation coefficient ($r$) between $x$ and $y$.
   • Find the standard deviations ($S_x$ and $S_y$).

Explanation of Solution:

1. Regression Lines Rearranged:

Rewriting both equations in the standard regression form:

• From $5y = 4x + 33$, divide through by 5: $y = 0.8x + 6.6 \quad \text{(Line of y on x)}$
• From $20x = 9y + 107$, divide through by 20: $x = 0.45y - 5.35 \quad \text{(Line of x on y)}$

2. Finding the Means:

The regression lines always pass through the mean values ($\bar{x}, \bar{y}$).

• Using $y = 0.8x + 6.6$, substitute $x = \bar{x}$, $y = \bar{y}$: $\bar{y} = 0.8\bar{x} + 6.6$

• Using $x = 0.45y - 5.35$, substitute $x = \bar{x}$, $y = \bar{y}$: $\bar{x} = 0.45\bar{y} - 5.35$

Solve these two simultaneous equations for $\bar{x}$ and $\bar{y}$.

3. Correlation Coefficient ($r$):

The slopes of the regression lines ($b_{yx}$ and $b_{xy}$) are given by:

• $b_{yx} = 0.8$
• $b_{xy} = 0.45$

The correlation coefficient is: $r = \sqrt{b_{yx} \cdot b_{xy}}$ Substitute the values to find $r$.

4. Standard Deviations ($S_x$ and $S_y$):

The relationship between regression coefficients and standard deviations is: $b_{yx} = \frac{S_y}{S_x} \quad \text{and} \quad b_{xy} = \frac{S_x}{S_y}$

Using these, find $S_x$ and $S_y$.

Would you like me to compute the values in detail? Let me know!

Related Questions:

1. What are the properties of regression coefficients?
2. How do we derive the formula for the correlation coefficient ($r$) from regression lines?
3. How are regression equations used in predicting values?
4. What assumptions underlie regression analysis?
5. How is the mean of $x$ and $y$ used in least squares regression?

Tip: The correlation coefficient $r$ is always between -1 and 1. The closer $|r|$ is to 1, the stronger the linear relationship between $x$ and $y$.