State True or False: If `b_(xy) lt 0` and `b_(yx) lt 0` then ‘r’ is `gt0`

Answer

To determine whether the statement "If \( b_{xy} < 0 \) and \( b_{yx} < 0 \) then \( r > 0 \)" is true or false, we can analyze the relationship between the regression coefficients and the correlation coefficient.

### Step-by-Step Solution:

1. **Understanding the Regression Coefficients**:
- The regression coefficients \( b_{xy} \) and \( b_{yx} \) represent the relationship between two variables \( x \) and \( y \).
- \( b_{xy} \) indicates how \( y \) changes with respect to \( x \), and \( b_{yx} \) indicates how \( x \) changes with respect to \( y \).

2. **Given Conditions**:
- We are given that \( b_{xy} < 0 \) and \( b_{yx} < 0 \). This means both regression coefficients are negative.

3. **Relationship Between Regression Coefficients and Correlation Coefficient**:
- There is a known relationship between the regression coefficients and the correlation coefficient \( r \):
\[ r^2 = b_{xy} \cdot b_{yx} \]
- This implies that the square of the correlation coefficient is equal to the product of the two regression coefficients.

4. **Analyzing the Signs**:
- Since both \( b_{xy} < 0 \) and \( b_{yx} < 0 \), their product \( b_{xy} \cdot b_{yx} \) will be positive (as the product of two negative numbers is positive).
- Therefore, \( r^2 > 0 \), which implies that \( r \) can be either positive or negative.

5. **Determining the Sign of \( r \)**:
- However, since both regression coefficients are negative, the correlation coefficient \( r \) will also be negative.
- This is because the correlation coefficient reflects the direction of the relationship: both variables decrease together, leading to a negative correlation.

6. **Conclusion**:
- Since we have established that \( r < 0 \) given the conditions \( b_{xy} < 0 \) and \( b_{yx} < 0 \), the statement "If \( b_{xy} < 0 \) and \( b_{yx} < 0 \) then \( r > 0 \)" is FALSE.

### Final Answer:
**False**