Find r(x,y) if Cov (x,y) = -165 , Var( x)= 2.25 and Var(y)= 144.

Answer

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

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

### Step 1: Identify the given values
- Covariance \( \text{Cov}(x,y) = -165 \)
- Variance \( \text{Var}(x) = 2.25 \)
- Variance \( \text{Var}(y) = 144 \)

### Step 2: Substitute the values into the formula
We can substitute the values into the formula:

\[ r(x,y) = \frac{-165}{\sqrt{2.25 \cdot 144}} \]

### Step 3: Calculate the product of the variances
Now we need to calculate \( \sqrt{2.25 \cdot 144} \).

First, calculate \( 2.25 \cdot 144 \):

\[ 2.25 \cdot 144 = 324 \]

### Step 4: Calculate the square root
Next, we find the square root of 324:

\[ \sqrt{324} = 18 \]

### Step 5: Substitute back into the formula
Now we substitute this back into our formula for \( r(x,y) \):

\[ r(x,y) = \frac{-165}{18} \]

### Step 6: Simplify the fraction
To simplify \( \frac{-165}{18} \), we can divide both the numerator and the denominator by 3:

\[ r(x,y) = \frac{-55}{6} \]

### Step 7: Convert to decimal (if needed)
Now, we can convert \( \frac{-55}{6} \) into decimal form:

\[ r(x,y) \approx -9.17 \]

### Final Answer
Thus, the correlation coefficient \( r(x,y) \) is:

\[ r(x,y) \approx -9.17 \]