Find r(x,y) if cov (x,y) =-16.5, var(x)=2.25 and `sigma_(y)=12`

Answer

To find the correlation coefficient \( r(x,y) \), we can use the formula:

\[ r(x,y) = \frac{\text{cov}(x,y)}{\sqrt{\text{var}(x)} \cdot \sqrt{\text{var}(y)}} \]

### Step 1: Identify the given values
We are given:
- Covariance \( \text{cov}(x,y) = -16.5 \)
- Variance of \( x \), \( \text{var}(x) = 2.25 \)
- Standard deviation of \( y \), \( \sigma_y = 12 \)

### Step 2: Calculate the variance of \( y \)
Since the variance of \( y \) is the square of the standard deviation, we calculate:

\[ \text{var}(y) = \sigma_y^2 = 12^2 = 144 \]

### Step 3: Substitute values into the correlation coefficient formula
Now we can substitute the values into the formula for the correlation coefficient:

\[ r(x,y) = \frac{-16.5}{\sqrt{2.25} \cdot \sqrt{144}} \]

### Step 4: Calculate the square roots
Calculate the square roots of the variances:

\[ \sqrt{2.25} = 1.5 \]
\[ \sqrt{144} = 12 \]

### Step 5: Substitute the square roots back into the formula
Now substitute these values back into the formula:

\[ r(x,y) = \frac{-16.5}{1.5 \cdot 12} \]

### Step 6: Calculate the denominator
Calculate the denominator:

\[ 1.5 \cdot 12 = 18 \]

### Step 7: Final calculation
Now perform the final calculation:

\[ r(x,y) = \frac{-16.5}{18} \approx -0.9167 \]

### Conclusion
Thus, the correlation coefficient \( r(x,y) \) is approximately:

\[ r(x,y) \approx -0.917 \]