To determine the critical value for the linear correlation coefficient ($r$), you can use the following process:

Step 1: Determine the degrees of freedom (df)

The degrees of freedom are calculated as: $df = n - 2$ For $n = 15$: $df = 15 - 2 = 13$

Step 2: Find the critical value for $r$

The critical value for $r$ at a given significance level ($\alpha$) is based on the $t$-distribution. The formula for converting $t$-values to $r$-values is: $r = \pm \sqrt{\frac{t^2}{t^2 + df}}$ Here, $t$ is the critical value from the $t$-distribution table for $df = 13$ and $\alpha = 0.01$ (two-tailed test).

Using a $t$-table or software, the critical $t$-value for $df = 13$ and $\alpha = 0.01$ (two-tailed) is: $t = 3.012$

Substitute $t$ and $df$ into the formula: $r = \pm \sqrt{\frac{3.012^2}{3.012^2 + 13}}$

Step 3: Calculate $r$

$r = \pm \sqrt{\frac{9.072}{9.072 + 13}} = \pm \sqrt{\frac{9.072}{22.072}} = \pm \sqrt{0.411} = \pm 0.641$

Final Answer:

The critical value for $r$ is approximately: $\pm 0.641$

Let me know if you'd like further clarification or assistance!

Here are 5 related questions:

1. How do critical $r$-values change with larger sample sizes?
2. What happens to the critical value if $\alpha$ changes to 0.05?
3. How does the formula for $r$ relate to hypothesis testing for correlation?
4. Can you calculate the $p$-value for a given sample correlation coefficient?
5. How does a one-tailed test affect the critical $r$-value?

Tip: Always double-check if the test is one-tailed or two-tailed, as it directly impacts critical values.