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.