The two line of regression are perpendicular if

Options

A. `b_(yx).b_(xy)=1`
B. `b_(yx).b_(xy)=-1`
C. `b_(xy).b_(yx)=0`
D. `b_(xy)+b_(yx)=0`

Answer

To determine when the two lines of regression are perpendicular, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Regression Lines**:
- The two lines of regression are typically represented as:
- Regression line of Y on X: \( Y = b_{yx}X + a_{y} \)
- Regression line of X on Y: \( X = b_{xy}Y + a_{x} \)
- Here, \( b_{yx} \) is the slope of the regression line of Y on X, and \( b_{xy} \) is the slope of the regression line of X on Y.

2. **Condition for Perpendicular Lines**:
- For two lines to be perpendicular, the product of their slopes must equal -1. This is a fundamental property of perpendicular lines in geometry.
- Therefore, we need to check the condition:
\[ b_{yx} \cdot b_{xy} = -1 \]

3. **Correlation Coefficient**:
- The correlation coefficient \( r \) relates the slopes of the regression lines:
\[ b_{yx} = r \cdot \frac{s_y}{s_x} \quad \text{and} \quad b_{xy} = r \cdot \frac{s_x}{s_y} \]
- Where \( s_y \) and \( s_x \) are the standard deviations of Y and X, respectively.

4. **Finding the Product of Slopes**:
- Now, substituting the expressions for \( b_{yx} \) and \( b_{xy} \):
\[ b_{yx} \cdot b_{xy} = \left(r \cdot \frac{s_y}{s_x}\right) \cdot \left(r \cdot \frac{s_x}{s_y}\right) \]
- Simplifying this gives:
\[ b_{yx} \cdot b_{xy} = r^2 \]

5. **Condition for Perpendicularity**:
- For the lines to be perpendicular, we set:
\[ r^2 = -1 \]
- However, since \( r^2 \) cannot be negative, we consider the case when \( r = 0 \), which indicates no correlation between the variables.

6. **Conclusion**:
- Thus, the two lines of regression are perpendicular if:
\[ b_{yx} \cdot b_{xy} = 0 \]
- This corresponds to the scenario where the correlation coefficient \( r = 0 \), indicating that the variables are independent.

### Final Answer:
The two lines of regression are perpendicular if \( b_{yx} \cdot b_{xy} = 0 \).