Given `r= sqrt((3)/(4)), b_(yx)= (1)/(2)`, variance of x=12, then standard deviation of y is

Options

A. 2
B. 3
C. 4
D. 9

Answer

To find the standard deviation of \( y \), we will use the given information and the relationship between the correlation coefficient, regression coefficient, and standard deviations of \( x \) and \( y \).

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Correlation coefficient \( r = \sqrt{\frac{3}{4}} \)
- Regression coefficient \( b_{yx} = \frac{1}{2} \)
- Variance of \( x \) is given as \( 12 \).

2. **Calculate the Standard Deviation of \( x \):**
- The variance of \( x \) is given by \( \sigma_x^2 = 12 \).
- Therefore, the standard deviation of \( x \) is:
\[ \sigma_x = \sqrt{12} = 2\sqrt{3} \]

3. **Use the Formula for Regression Coefficient:**
- The regression coefficient \( b_{yx} \) is related to the correlation coefficient \( r \) and the standard deviations of \( x \) and \( y \) as follows:
\[ b_{yx} = r \cdot \frac{\sigma_y}{\sigma_x} \]
- Rearranging this formula to find \( \sigma_y \):
\[ \sigma_y = \frac{b_{yx} \cdot \sigma_x}{r} \]

4. **Substitute the Known Values:**
- Substitute \( b_{yx} = \frac{1}{2} \), \( \sigma_x = 2\sqrt{3} \), and \( r = \sqrt{\frac{3}{4}} \) into the equation:
\[ \sigma_y = \frac{\frac{1}{2} \cdot 2\sqrt{3}}{\sqrt{\frac{3}{4}}} \]

5. **Simplify the Expression:**
- First, simplify the numerator:
\[ \sigma_y = \frac{\sqrt{3}}{\sqrt{\frac{3}{4}}} \]
- Now simplify the denominator:
\[ \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]
- Therefore:
\[ \sigma_y = \frac{\sqrt{3}}{\frac{\sqrt{3}}{2}} = \sqrt{3} \cdot \frac{2}{\sqrt{3}} = 2 \]

6. **Final Result:**
- The standard deviation of \( y \) is:
\[ \sigma_y = 2 \]

### Conclusion:
The standard deviation of \( y \) is \( 2 \).