In a bivariate data , if `b_(yx) = -1.5` and `b_(xy) = -0.5`, then the coefficient of correlation is

Options

A. `sqrt(3)/(2)`
B. `3/(4)`
C. `-3/(4)`
D. `-sqrt(3)/(2)`

Answer

To find the coefficient of correlation given the values of \( b_{yx} \) and \( b_{xy} \), we can follow these steps:

### Step 1: Understand the relationship between the coefficients
The coefficient of correlation \( r \) can be calculated using the formula:
\[ r = \sqrt{b_{yx} \cdot b_{xy}} \]

### Step 2: Substitute the given values
We are given:
- \( b_{yx} = -1.5 \)
- \( b_{xy} = -0.5 \)

Substituting these values into the formula:
\[ r = \sqrt{(-1.5) \cdot (-0.5)} \]

### Step 3: Calculate the product
Now, calculate the product:
\[ (-1.5) \cdot (-0.5) = 0.75 \]

### Step 4: Take the square root
Now, we take the square root of the product:
\[ r = \sqrt{0.75} \]

### Step 5: Simplify the square root
We can simplify \( \sqrt{0.75} \):
\[ \sqrt{0.75} = \sqrt{\frac{75}{100}} = \frac{\sqrt{75}}{10} = \frac{5\sqrt{3}}{10} = \frac{\sqrt{3}}{2} \]

### Step 6: Determine the sign of the correlation coefficient
Since both \( b_{yx} \) and \( b_{xy} \) are negative, the coefficient of correlation \( r \) will also be negative. Therefore:
\[ r = -\frac{\sqrt{3}}{2} \]

### Final Answer
Thus, the coefficient of correlation is:
\[ r = -\frac{\sqrt{3}}{2} \]

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