If the regression coefficient of x and y on'x are `-1/2` and `-1/8` respectively, then what is the correlation coefficient between x and y?

Options

A. `-1/4`
B. `-1/16`
C. `1/16`
D. `1/4`

Answer

To find the correlation coefficient between x and y given the regression coefficients, we can use the relationship between the regression coefficients and the correlation coefficient.

### Step-by-Step Solution:

1. **Identify the given regression coefficients**:
- The regression coefficient of x on y (denoted as \( b_{xy} \)) is given as \( -\frac{1}{2} \).
- The regression coefficient of y on x (denoted as \( b_{yx} \)) is given as \( -\frac{1}{8} \).

2. **Use the formula for correlation coefficient**:
The correlation coefficient \( r \) can be calculated using the formula:
\[ r = \sqrt{b_{xy} \cdot b_{yx}} \]

3. **Substitute the values into the formula**:
\[ r = \sqrt{\left(-\frac{1}{2}\right) \cdot \left(-\frac{1}{8}\right)} \]

4. **Calculate the product**:
\[ r = \sqrt{\frac{1}{2} \cdot \frac{1}{8}} = \sqrt{\frac{1}{16}} = \frac{1}{4} \]

5. **Determine the sign of the correlation coefficient**:
Since both regression coefficients \( b_{xy} \) and \( b_{yx} \) are negative, the correlation coefficient \( r \) will also be negative. Therefore, we have:
\[ r = -\frac{1}{4} \]

6. **Conclusion**:
The correlation coefficient between x and y is \( -\frac{1}{4} \).

### Final Answer:
The correlation coefficient \( r \) is \( -\frac{1}{4} \).

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