If `N=50,SigmaX=75,SigmaY=80,SigmaX^(2)=130,SigmaY^(2)=140, SigmaXY=120` find the value of the correlation coefficient between the variables X and Y.

Answer

To find the correlation coefficient \( r \) between the variables \( X \) and \( Y \), we can use the formula:

\[ r = \frac{N \Sigma XY - \Sigma X \Sigma Y}{\sqrt{(N \Sigma X^2 - (\Sigma X)^2)(N \Sigma Y^2 - (\Sigma Y)^2)}} \]

Given:
- \( N = 50 \)
- \( \Sigma X = 75 \)
- \( \Sigma Y = 80 \)
- \( \Sigma X^2 = 130 \)
- \( \Sigma Y^2 = 140 \)
- \( \Sigma XY = 120 \)

### Step 1: Calculate the numerator

First, we calculate the numerator of the formula:

\[ N \Sigma XY - \Sigma X \Sigma Y \]

Substituting the values:

\[ = 50 \times 120 - 75 \times 80 \]

Calculating each term:

\[ = 6000 - 6000 = 0 \]

### Step 2: Calculate the denominator

Next, we calculate the denominator:

\[ \sqrt{(N \Sigma X^2 - (\Sigma X)^2)(N \Sigma Y^2 - (\Sigma Y)^2)} \]

Calculating each part:

1. **For \( X \)**:

\[ N \Sigma X^2 - (\Sigma X)^2 = 50 \times 130 - 75^2 \]

Calculating:

\[ = 6500 - 5625 = 875 \]

2. **For \( Y \)**:

\[ N \Sigma Y^2 - (\Sigma Y)^2 = 50 \times 140 - 80^2 \]

Calculating:

\[ = 7000 - 6400 = 600 \]

Now, substituting these into the denominator:

\[ \sqrt{875 \times 600} \]

Calculating the product:

\[ = \sqrt{525000} = 750 \]

### Step 3: Calculate the correlation coefficient \( r \)

Now substituting the values back into the correlation coefficient formula:

\[ r = \frac{0}{750} = 0 \]

### Final Result

Thus, the value of the correlation coefficient \( r \) is:

\[ \boxed{0} \]