State True or False: If `b_(yx)=1.5` and `b_(xy)=1/3` then `r = 1/2`. The given data is consistent.

Answer

To determine whether the statement is true or false, we will analyze the given values and apply the correlation coefficient formula.

### Step-by-Step Solution:

1. **Identify Given Values:**
We are given:
- \( b_{yx} = 1.5 \)
- \( b_{xy} = \frac{1}{3} \)

2. **Use the Formula for Correlation Coefficient:**
The correlation coefficient \( r \) is related to the regression coefficients by the formula:
\[ r^2 = b_{xy} \cdot b_{yx} \]

3. **Substitute the Given Values into the Formula:**
Substitute \( b_{yx} \) and \( b_{xy} \) into the formula:
\[ r^2 = 1.5 \cdot \frac{1}{3} \]

4. **Calculate \( r^2 \):**
Performing the multiplication:
\[ r^2 = 1.5 \cdot \frac{1}{3} = \frac{1.5}{3} = \frac{15}{30} = \frac{1}{2} \]

5. **Find \( r \):**
Taking the square root of both sides:
\[ r = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

6. **Determine the Sign of \( r \):**
Since both \( b_{yx} \) and \( b_{xy} \) are positive, \( r \) will also be positive:
\[ r = \frac{1}{\sqrt{2}} > 0 \]

7. **Compare with Given \( r \):**
The statement claims \( r = \frac{1}{2} \). Since \( \frac{1}{\sqrt{2}} \) is not equal to \( \frac{1}{2} \), the statement is false.

### Conclusion:
The given data is inconsistent with the statement provided. Therefore, the answer is **False**.

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