To find the critical value of $r$, we use the following process:

1. Determine the degrees of freedom (df): $\text{df} = n - 2$ For $n = 12$: $\text{df} = 12 - 2 = 10$

2. Look up the critical value of $r$ for $\alpha = 0.01$ and $\text{df} = 10$ in a table of critical values for Pearson's correlation coefficient (or use statistical software).

The critical value for $r$ at $\alpha = 0.01$ and $\text{df} = 10$ is approximately:

$\pm 0.708$

Interpretation:

If the calculated $r$-value is greater than $0.708$ or less than $-0.708$, there is sufficient evidence to conclude that there is a significant linear correlation at the $\alpha = 0.01$ level.

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Would you like further explanation on this process or a visual representation of the $r$-critical table? Here are some related questions to consider:

1. How is the $r$-critical value table constructed?
2. What happens if the $n$-value changes to 15 or 20? How does it affect $r_{\text{critical}}$?
3. Can you explain how to compute the test statistic $r$ from a dataset?
4. What is the relationship between $\alpha$ and the $r_{\text{critical}}$?
5. What are the implications of a two-tailed vs. one-tailed test for the critical $r$?

Tip: Always ensure that the sample size $n$ is large enough for reliable correlation results, as small samples may lead to inaccurate conclusions.