The following table shows the age and selling price of cars sold recently at a used car lot. Calculate the correlation coefficient, r, and determine whether r is statistically significant at the 0.01 level of significance. Round your answer to the nearest thousandth.

Critical Values of the Pearson Correlation Coefficient

Car Ages and Selling Prices
Car Age	2	4	5	6	7	9	10	11
Price ($)	19,959	30,599	11,805	16,225	8229	9989	7299	4905

To calculate the correlation coefficient $ r $ and determine its statistical significance, we need to:

1. Compute the Pearson correlation coefficient $ r $.
2. Compare the result with the critical value for the Pearson correlation at the 0.01 level of significance for a given degree of freedom $ df $.

### Steps:

1. **List the data**:
   - Car Age (X): 2, 4, 5, 6, 7, 9, 10, 11
   - Price (Y in $): 19,959, 30,599, 11,805, 16,225, 8229, 9989, 7299, 4905

2. **Formula for Pearson correlation coefficient $ r $**:
   \[
   r = \frac{n(\sum{XY}) - (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} - (\sum{X})^2][n\sum{Y^2} - (\sum{Y})^2]}}
   \]
   where:
   - $ n $ is the number of data points,
   - $ X $ is the car age,
   - $ Y $ is the selling price.

3. **Check for statistical significance**:
   - Use the degrees of freedom $ df = n - 2 $ where $ n $ is the number of pairs.
   - At the 0.01 level of significance, we check whether $ |r| $ exceeds the critical value based on $ df $.

Let me calculate the correlation coefficient for you.The Pearson correlation coefficient $ r $ is approximately $ -0.779 $.

### Next Step: Statistical Significance
To determine if this correlation is statistically significant at the 0.01 level of significance, we need to compare it to a critical value for the Pearson correlation based on the degrees of freedom $ df = n - 2 $, where $ n = 8 $ (since there are 8 data points).

Thus, $ df = 8 - 2 = 6 $.

Let's check the critical value for $ r $ at $ df = 6 $ and the 0.01 significance level.For $ df = 6 $ and a significance level of 0.01, the critical value of $ r $ from the Pearson correlation table is approximately $ \pm 0.834 $.

### Conclusion:
- The calculated correlation coefficient $ r = -0.779 $ does not exceed the critical value $ -0.834 $.
- Therefore, $ r $ is **not statistically significant** at the 0.01 level.

Would you like further details or clarifications?

Here are 5 related questions to consider:
1. How does the sign of the correlation coefficient affect the relationship interpretation?
2. What would happen if the significance level was 0.05 instead of 0.01?
3. How would adding more data points affect the degrees of freedom and significance?
4. What is the relationship between car age and selling price based on this correlation?
5. How can outliers influence the correlation coefficient?

**Tip**: Correlation measures only linear relationships, so a low correlation doesn't always imply no relationship—it could be nonlinear.

The following table shows the age and selling price of cars sold recently at a used car lot. Calculate the correlation coefficient, r, and determine whether r is statistically significant at the 0.01 level of significance. Round your answer to the nearest thousandth.

Car Ages and Selling Prices
Car Age: 2, 4, 5, 6, 7, 9, 10, 11
Price ($): 19,959, 30,599, 11,805, 16,225, 8229, 9989, 7299, 4905

Learn how to calculate the Pearson correlation coefficient (r) for car ages and selling prices. This solution also covers determining statistical significance at the 0.01 level with detailed steps and real-world data points. Suitable for Grades 11-12 and introductory college students.

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