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The problem involves calculating the correct value of the correlation coefficient after identifying and correcting errors in the provided data.

### Given Data:
- \( n = 25 \)
- \( \Sigma x = 125 \)
- \( \Sigma y = 100 \)
- \( \Sigma x^2 = 650 \)
- \( \Sigma y^2 = 460 \)
- \( \Sigma xy = 508 \)

However, the operator incorrectly recorded two pairs of observations:
- Incorrect: \( (x_1 = 6, y_1 = 14), (x_2 = 8, y_2 = 6) \)
- Correct: \( (x_1 = 8, y_1 = 12), (x_2 = 6, y_2 = 8) \)

### To calculate the correct correlation coefficient:

1. **Adjust the sums**:
   We need to first adjust the incorrect values and update the following sums:
   - \( \Sigma x \)
   - \( \Sigma y \)
   - \( \Sigma x^2 \)
   - \( \Sigma y^2 \)
   - \( \Sigma xy \)

### Step-by-Step Corrections:
Let’s calculate the differences caused by the incorrect values and apply the corrections.

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I will now proceed with calculations to adjust the sums and find the correct correlation coefficient.The correct value of the correlation coefficient between \(x\) and \(y\) after adjusting for the errors is approximately \( r = 0.667 \).

Would you like more details on any step, or do you have any further questions?

Here are five related questions that can expand your understanding:
1. How is the correlation coefficient interpreted in statistical analysis?
2. What does a correlation coefficient of 0.667 indicate about the relationship between \(x\) and \(y\)?
3. How can outliers affect the value of the correlation coefficient?
4. What are the assumptions for using the Pearson correlation coefficient?
5. How does the correlation coefficient relate to the concept of covariance?

**Tip:** When computing correlation coefficients, it's essential to verify the data entry, as errors can significantly affect the result.

A computer, while calculating the correlation coefficient between two variables x and y, gave the following information: n = 25, Σx = 125, Σy = 100, Σx² = 650, Σy² = 460, Σxy = 508. It was discovered that the operator had copied down two pairs of observations incorrectly: Incorrect values: (x = 6, y = 14) and (x = 8, y = 6); Correct values: (x = 8, y = 12) and (x = 6, y = 8). Obtain the correct value of the correlation coefficient between x and y.

Learn how to correct the correlation coefficient calculation after discovering errors in two data observations. Given the initial sum values for x and y, this problem guides through the step-by-step process of recalculating the coefficient after adjusting for incorrect entries. Suitable for undergraduate students studying statistics.

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