If the regression coefficient of x on y and y on x are `-(1)/(2) and -(1)/(8)` respectively, then what is the correlation coefficient between x and y?`-(1)/(4)`

Options

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

Explanation

To find the correlation coefficient between x and y given the regression coefficients, we can follow these steps:

### Step 1: Identify the given regression coefficients
We are given:
- The regression coefficient of x on y, denoted as \( b_{xy} = -\frac{1}{2} \)
- The regression coefficient of y on x, denoted as \( b_{yx} = -\frac{1}{8} \)

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

### Step 3: Substitute the values into the formula
Substituting the values of \( b_{xy} \) and \( b_{yx} \):
\[ r = \sqrt{\left(-\frac{1}{2}\right) \cdot \left(-\frac{1}{8}\right)} \]

### Step 4: Calculate the product
Calculating the product inside the square root:
\[ r = \sqrt{\frac{1}{2} \cdot \frac{1}{8}} = \sqrt{\frac{1}{16}} = \frac{1}{4} \]

### Step 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:
\[ r = -\frac{1}{4} \]

### Final Answer
Thus, the correlation coefficient between x and y is:
\[ r = -\frac{1}{4} \]

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