A computer while calculating the correlation coefficient between the variables x and y obtained the following results :
`n=25,sumx_(i)=125,sumy_(i)=100,sumx_(i)^(2)=650,sumy_(i)^(2)=460,sumx_(i)y_(i)=508` It was however later discovered at the time of checking that it has copied down two pairs of obervations as (6,14) and (8,6) where as values were (8,12) and (6,8) .Calculate the correct correlation coefficient of x and y.

Answer

To calculate the correct correlation coefficient between the variables \( x \) and \( y \), we need to adjust the given sums based on the corrections to the observations. Let's go through the steps systematically.

### Given Data:
- \( n = 25 \)
- \( \sum x_i = 125 \)
- \( \sum y_i = 100 \)
- \( \sum x_i^2 = 650 \)
- \( \sum y_i^2 = 460 \)
- \( \sum x_i y_i = 508 \)

### Observations to Correct:
- Incorrect pairs: \( (6, 14) \) and \( (8, 6) \)
- Correct pairs: \( (8, 12) \) and \( (6, 8) \)

### Step 1: Adjust \( \sum x_i \)
1. Remove the incorrect \( x \) values:
\[ \sum x_i = 125 - 6 - 8 = 111 \]
2. Add the correct \( x \) values:
\[ \sum x_i = 111 + 8 + 6 = 125 \]
(This shows that the sum remains unchanged.)

### Step 2: Adjust \( \sum y_i \)
1. Remove the incorrect \( y \) values:
\[ \sum y_i = 100 - 14 - 6 = 80 \]
2. Add the correct \( y \) values:
\[ \sum y_i = 80 + 12 + 8 = 100 \]
(This shows that the sum remains unchanged.)

### Step 3: Adjust \( \sum x_i^2 \)
1. Remove the squares of the incorrect \( x \) values:
\[ \sum x_i^2 = 650 - 6^2 - 8^2 = 650 - 36 - 64 = 550 \]
2. Add the squares of the correct \( x \) values:
\[ \sum x_i^2 = 550 + 8^2 + 6^2 = 550 + 64 + 36 = 650 \]
(This shows that the sum remains unchanged.)

### Step 4: Adjust \( \sum y_i^2 \)
1. Remove the squares of the incorrect \( y \) values:
\[ \sum y_i^2 = 460 - 14^2 - 6^2 = 460 - 196 - 36 = 228 \]
2. Add the squares of the correct \( y \) values:
\[ \sum y_i^2 = 228 + 12^2 + 8^2 = 228 + 144 + 64 = 436 \]
(This shows that the sum remains unchanged.)

### Step 5: Adjust \( \sum x_i y_i \)
1. Remove the products of the incorrect pairs:
\[ \sum x_i y_i = 508 - (6 \cdot 14) - (8 \cdot 6) = 508 - 84 - 48 = 376 \]
2. Add the products of the correct pairs:
\[ \sum x_i y_i = 376 + (8 \cdot 12) + (6 \cdot 8) = 376 + 96 + 48 = 520 \]

### Step 6: Calculate the Correct Correlation Coefficient \( r \)
The formula for the correlation coefficient \( r \) is given by:
\[ r = \frac{n \sum x_i y_i - \sum x_i \sum y_i}{\sqrt{[n \sum x_i^2 - (\sum x_i)^2][n \sum y_i^2 - (\sum y_i)^2]}} \]

Substituting the values:
- \( n = 25 \)
- \( \sum x_i y_i = 520 \)
- \( \sum x_i = 125 \)
- \( \sum y_i = 100 \)
- \( \sum x_i^2 = 650 \)
- \( \sum y_i^2 = 436 \)

Calculating the numerator:
\[ n \sum x_i y_i - \sum x_i \sum y_i = 25 \cdot 520 - 125 \cdot 100 = 13000 - 12500 = 500 \]

Calculating the denominator:
\[ \sqrt{[n \sum x_i^2 - (\sum x_i)^2][n \sum y_i^2 - (\sum y_i)^2]} \]
Calculating each part:
1. \( n \sum x_i^2 - (\sum x_i)^2 = 25 \cdot 650 - 125^2 = 16250 - 15625 = 625 \)
2. \( n \sum y_i^2 - (\sum y_i)^2 = 25 \cdot 436 - 100^2 = 10900 - 10000 = 900 \)

Now, the denominator becomes:
\[ \sqrt{625 \cdot 900} = \sqrt{562500} = 750 \]

Finally, substituting back into the formula for \( r \):
\[ r = \frac{500}{750} = \frac{2}{3} \approx 0.667 \]

### Final Answer:
The correct correlation coefficient \( r \) is approximately \( 0.667 \).