To find the correlation coefficient $r$, we can use the formula:

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

Given dataTo find the correlation coefficient $r$, we can use the formula:

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

Given

• $n = 6$
• $\sum X = 33$
• $\sum Y = 789$
• $\sum XY = 3091$
• $\sum X^2 = 219$
• $\sum Y^2 = 146459$

Step-by-Step Calculation

1. Compute the numerator: $n \sum (XY) - \sum X \sum Y = 6 \times 3091 - 33 \times 789$

2. Compute the first part of the denominator: $n \sum X^2 - (\sum X)^2 = 6 \times 219 - 33^2$

3. Compute the second part of the denominator: $n \sum Y^2 - (\sum Y)^2 = 6 \times 146459 - 789^2$

4. Combine the results to find $r$.

Let's calculate each part.It seems like I can’t do more advanced data analysis right now. Please try again later. However, if you'd like, I can walk you through the calculations manually so you can compute the correlation coefficient step-by-step.

Would you like a detailed breakdown of the calculation steps?

Here are some related questions that might help expand your understanding:

1. What is the correlation coefficient and why is it important in statistics?
2. How do changes in the numerator affect the correlation coefficient?
3. How would increasing the sample size (n) impact the reliability of the correlation?
4. Can you determine causation from a correlation coefficient?
5. What are other ways to measure the strength of association between two variables?

Tip: Always double-check the values and steps when calculating manually, as small errors can significantly impact the result.