We are given that some observations were recorded incorrectly, and the task is to find the correct correlation coefficient between $x$ and $y$.

Given Data:

• $n = 25$
• $\Sigma x = 125$
• $\Sigma y = 100$
• $\Sigma x^2 = 650$
• $\Sigma y^2 = 460$
• $\Sigma xy = 508$

Incorrect observations:

• For $x = 6, y = 14$ and $x = 8, y = 6$

Correct observations:

• Corrected values are $x = 8, y = 12$ and $x = 6, y = 8$

Step-by-step process:

1. Adjustment to sums due to incorrect observations:

   First, calculate the incorrect contributions to the sums from the incorrect values: $\text{Incorrect sum of } x = 6 + 8 = 14$ $\text{Incorrect sum of } y = 14 + 6 = 20$ $\text{Incorrect } x^2 = 6^2 + 8^2 = 36 + 64 = 100$ $\text{Incorrect } y^2 = 14^2 + 6^2 = 196 + 36 = 232$ $\text{Incorrect } xy = (6 \times 14) + (8 \times 6) = 84 + 48 = 132$

2. Correct sums for the correct pairs of observations: $\text{Correct sum of } x = 8 + 6 = 14$ $\text{Correct sum of } y = 12 + 8 = 20$ $\text{Correct } x^2 = 8^2 + 6^2 = 64 + 36 = 100$ $\text{Correct } y^2 = 12^2 + 8^2 = 144 + 64 = 208$ $\text{Correct } xy = (8 \times 12) + (6 \times 8) = 96 + 48 = 144$

3. Adjust the original sums: $\Sigma x_{\text{corrected}} = 125 - 14 + 14 = 125$ $\Sigma y_{\text{corrected}} = 100 - 20 + 20 = 100$ $\Sigma x^2_{\text{corrected}} = 650 - 100 + 100 = 650$ $\Sigma y^2_{\text{corrected}} = 460 - 232 + 208 = 436$ $\Sigma xy_{\text{corrected}} = 508 - 132 + 144 = 520$

4. Now, use the formula for the correlation coefficient $r$: [ r = \frac{n \Sigma xy - (\Sigma x)(\Sigma y)}{\sqrt{[n \Sigma x^2 - (\Sigma x)^2][n \Sigma y^2 - (\Sigma y)^2]}} ] Plugging in the corrected values: [ r = \frac{25 \times 520 - (125 \times 100)}{\sqrt{[25 \times 650 - (125)^2][25 \times 436 - (100)^2]}} ] Let's compute this step by step.The correct correlation coefficient $r$ between $x$ and $y$ after making the necessary corrections is approximately $0.667$.

Would you like further clarification on any part of the process or detailed steps on how we reached this result?

Here are some related questions for further understanding:

1. How would the correlation coefficient change if the incorrect observations were more extreme?
2. What is the significance of the correlation coefficient being positive in this case?
3. How can the correlation coefficient be interpreted in real-life situations involving two variables?
4. What is the impact of having incorrect observations on the calculation of correlation coefficients?
5. Can we determine if the relationship between $x$ and $y$ is linear or nonlinear from the correlation coefficient alone?

Tip: Always double-check observations and data before performing any statistical analysis, as incorrect entries can significantly affect the results.