State True or False: The following data is not consistent: `byx+ bxy =1.3` and `r = 0.75`

Answer

To determine whether the given data is consistent or inconsistent, we need to analyze the relationship between the regression coefficients and the correlation coefficient.

### Step-by-Step Solution:

1. **Identify the Given Data**:
- We have two regression coefficients: \( b_{yx} + b_{xy} = 1.3 \)
- The correlation coefficient \( r = 0.75 \)

2. **Understand the Relationship**:
- The relationship between the regression coefficients and the correlation coefficient is given by:
\[ |r|^2 = b_{xy} \cdot b_{yx} \]
- This implies that the square of the correlation coefficient is equal to the product of the two regression coefficients.

3. **Calculate \( |r|^2 \)**:
- Given \( r = 0.75 \):
\[ |r|^2 = (0.75)^2 = 0.5625 \]

4. **Use the Arithmetic Mean-Geometric Mean Inequality**:
- According to the inequality:
\[ |b_{xy} + b_{yx}| \geq 2 \sqrt{b_{xy} \cdot b_{yx}} \]
- Substituting the values we have:
\[ |1.3| \geq 2 \sqrt{b_{xy} \cdot b_{yx}} \]

5. **Express the Product of Regression Coefficients**:
- We know from the previous step that:
\[ b_{xy} \cdot b_{yx} = |r|^2 = 0.5625 \]
- Therefore:
\[ 2 \sqrt{b_{xy} \cdot b_{yx}} = 2 \sqrt{0.5625} = 2 \cdot 0.75 = 1.5 \]

6. **Compare the Values**:
- Now we compare:
\[ 1.3 \geq 1.5 \]
- This statement is **false**.

7. **Conclusion**:
- Since \( 1.3 < 1.5 \), it violates the arithmetic mean-geometric mean inequality, indicating that the data is inconsistent.

### Final Answer:
- The statement is **True**: The data is inconsistent.