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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.

A car company wants to investigate how the price of its car models depreciates with age. The manager took a sample of 6 cars and collected the following information on their ages (in years) and prices (in hundreds of dollars). The data is as follows: Age X = [8, 3, 6, 9, 2, 5], Price Y = [45, 210, 100, 33, 267, 134]. Given the summations ΣX = 33, ΣY = 789, ΣXY = 3091, ΣX² = 219, ΣY² = 146459, compute the value of the correlation coefficient.

Learn how to calculate the correlation coefficient for a dataset of car ages and prices to determine depreciation patterns. This example covers key statistical concepts, including the correlation formula and calculation steps, suitable for high school statistics students.

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