For two variables x and y, the two regression coefficients are `b_(yx)=-3//2 and b_(xy)=-1//6.` The correlation coefficient between x and y is :

Options

A. `-1//4`
B. `1//4`
C. `-1//2`
D. `1//2`

Explanation

To find the correlation coefficient \( r \) between two variables \( x \) and \( y \) given the regression coefficients \( b_{yx} \) and \( b_{xy} \), we can use the following steps:

### Step 1: Write down the given regression coefficients
We have:
- \( b_{yx} = -\frac{3}{2} \)
- \( b_{xy} = -\frac{1}{6} \)

### Step 2: Use the formula for 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 \( b_{yx} \):
\[ r = \sqrt{\left(-\frac{1}{6}\right) \cdot \left(-\frac{3}{2}\right)} \]

### Step 4: Calculate the product
Calculating the product inside the square root:
\[ -\frac{1}{6} \cdot -\frac{3}{2} = \frac{1 \cdot 3}{6 \cdot 2} = \frac{3}{12} = \frac{1}{4} \]

### Step 5: Take the square root
Now, take the square root of \( \frac{1}{4} \):
\[ r = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

### Step 6: Determine the sign of the correlation coefficient
Since both regression coefficients \( b_{yx} \) and \( b_{xy} \) are negative, the correlation coefficient \( r \) will also be negative:
\[ r = -\frac{1}{2} \]

### Final Answer
The correlation coefficient between \( x \) and \( y \) is:
\[ r = -\frac{1}{2} \]

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