If `sigma_(x) = 3, sigma_(y) = 4`, and `b_(xy) = (1)/(3)`, then the value of r is

Options

A. `(1)/(4)`
B. `(9)/(4)`
C. `(4)/(9)`
D. 4

Answer

To find the value of \( r \) given the values of \( \sigma_x \), \( \sigma_y \), and \( b_{xy} \), we can use the relationship between the regression coefficient \( b_{xy} \) and the correlation coefficient \( r \).

### Step-by-Step Solution:

1. **Understand the relationship**: The regression coefficient \( b_{xy} \) is given by the formula:
\[ b_{xy} = r \frac{\sigma_x}{\sigma_y} \]
where \( r \) is the correlation coefficient, \( \sigma_x \) is the standard deviation of \( x \), and \( \sigma_y \) is the standard deviation of \( y \).

2. **Substitute the known values**: We know that:
- \( \sigma_x = 3 \)
- \( \sigma_y = 4 \)
- \( b_{xy} = \frac{1}{3} \)

Plugging these values into the formula gives:
\[ \frac{1}{3} = r \frac{3}{4} \]

3. **Solve for \( r \)**: To isolate \( r \), we can multiply both sides of the equation by \( \frac{4}{3} \):
\[ r = \frac{1}{3} \cdot \frac{4}{3} \]

Simplifying this, we get:
\[ r = \frac{4}{9} \]

4. **Final Result**: Therefore, the value of \( r \) is:
\[ r = \frac{4}{9} \]