Find correlation coefficient when, `Sigma x_(i) = 40, Sigma y_(i)= 55, Sigma x_(i)^(2) = 192, Sigmay_(i)^(2)=385, Sigma(x_(i)+y_(i))^(2)=947` and n = 10.

Answer

To find the correlation coefficient \( r_{xy} \), we will use the formula:

\[ r_{xy} = \frac{n \sum x_i y_i - \sum x_i \sum y_i}{\sqrt{n \sum x_i^2 - (\sum x_i)^2} \sqrt{n \sum y_i^2 - (\sum y_i)^2}} \]

### Step 1: Write down the given information
Given:
- \( \sum x_i = 40 \)
- \( \sum y_i = 55 \)
- \( \sum x_i^2 = 192 \)
- \( \sum y_i^2 = 385 \)
- \( \sum (x_i + y_i)^2 = 947 \)
- \( n = 10 \)

### Step 2: Find \( \sum x_i y_i \)
We know that:

\[ \sum (x_i + y_i)^2 = \sum x_i^2 + \sum y_i^2 + 2 \sum x_i y_i \]

Substituting the known values:

\[ 947 = 192 + 385 + 2 \sum x_i y_i \]

Calculating \( 192 + 385 \):

\[ 192 + 385 = 577 \]

So we have:

\[ 947 = 577 + 2 \sum x_i y_i \]

Rearranging gives:

\[ 2 \sum x_i y_i = 947 - 577 = 370 \]

Thus,

\[ \sum x_i y_i = \frac{370}{2} = 185 \]

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

\[ r_{xy} = \frac{10 \cdot 185 - 40 \cdot 55}{\sqrt{10 \cdot 192 - 40^2} \sqrt{10 \cdot 385 - 55^2}} \]

### Step 4: Calculate the numerator
Calculating the numerator:

\[ 10 \cdot 185 = 1850 \]

Calculating \( 40 \cdot 55 \):

\[ 40 \cdot 55 = 2200 \]

Thus, the numerator is:

\[ 1850 - 2200 = -350 \]

### Step 5: Calculate the denominator
Calculating the first part of the denominator:

\[ 10 \cdot 192 = 1920 \]
\[ 40^2 = 1600 \]

So,

\[ 10 \cdot 192 - 40^2 = 1920 - 1600 = 320 \]

Calculating the second part of the denominator:

\[ 10 \cdot 385 = 3850 \]
\[ 55^2 = 3025 \]

So,

\[ 10 \cdot 385 - 55^2 = 3850 - 3025 = 825 \]

Now, the denominator becomes:

\[ \sqrt{320} \cdot \sqrt{825} \]

### Step 6: Final calculations
Calculating \( \sqrt{320} \) and \( \sqrt{825} \):

\[ \sqrt{320} = 8\sqrt{5} \quad \text{and} \quad \sqrt{825} = 5\sqrt{33} \]

Thus, the denominator is:

\[ 8\sqrt{5} \cdot 5\sqrt{33} = 40\sqrt{165} \]

### Step 7: Putting it all together
Now we can write:

\[ r_{xy} = \frac{-350}{40\sqrt{165}} \]

Simplifying gives:

\[ r_{xy} = \frac{-35}{4\sqrt{165}} \]

### Step 8: Approximate the value
Calculating the approximate value:

Using a calculator, we can find that \( \sqrt{165} \approx 12.845 \):

\[ r_{xy} \approx \frac{-35}{4 \cdot 12.845} \approx \frac{-35}{51.38} \approx -0.68 \]

### Final Result
Thus, the correlation coefficient \( r_{xy} \) is approximately:

\[ \boxed{-0.68} \]