If the regression coefficient of Y on X is `-6`, and the correlation coefficient between X and Y `-(1)/(2),` then the regression coefficient of X on Y would be

Options

A. `(1)/(24)`
B. `-(1)/(24)`
C. `-(1)/(6)`
D. `(1)/(6)`

Explanation

To solve the problem, we need to find the regression coefficient of X on Y, denoted as \( b_{xy} \), given the regression coefficient of Y on X, \( b_{yx} = -6 \), and the correlation coefficient \( r = -\frac{1}{2} \).

### Step-by-Step Solution:

1. **Identify the given values**:
- Regression coefficient of Y on X: \( b_{yx} = -6 \)
- Correlation coefficient: \( r = -\frac{1}{2} \)

2. **Use the formula relating the correlation coefficient and regression coefficients**:
The formula that relates the regression coefficients and the correlation coefficient is:
\[ r^2 = b_{yx} \cdot b_{xy} \]
where \( b_{xy} \) is the regression coefficient of X on Y.

3. **Calculate \( r^2 \)**:
\[ r^2 = \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \]

4. **Substitute the known values into the formula**:
\[ \frac{1}{4} = (-6) \cdot b_{xy} \]

5. **Solve for \( b_{xy} \)**:
Rearranging the equation gives:
\[ b_{xy} = \frac{1}{4} \div (-6) = \frac{1}{4} \cdot \left(-\frac{1}{6}\right) = -\frac{1}{24} \]

6. **Final answer**:
The regression coefficient of X on Y is:
\[ b_{xy} = -\frac{1}{24} \]