To determine if there is a significant linear correlation between two variables, we need to conduct a hypothesis test and find the critical value for the correlation coefficient [tex]\( r \)[/tex].
Given:
- Number of data pairs ([tex]\( n \)[/tex]) = 8
- Significance level ([tex]\( \alpha \)[/tex]) = 0.05
Here are the steps to find the critical value for this test:
1. Determine the degrees of freedom (df):
The degrees of freedom for testing the correlation is calculated as:
[tex]\[
\text{df} = n - 2
\][/tex]
For our data set with 8 data pairs:
[tex]\[
\text{df} = 8 - 2 = 6
\][/tex]
2. Find the critical t-value:
For a two-tailed test with a significance level [tex]\(\alpha\)[/tex], the critical t-value (t-critical) can be found using statistical tables or software. For [tex]\(\alpha = 0.05\)[/tex] and [tex]\(\text{df} = 6\)[/tex], we need the critical value such that the area in each tail is [tex]\(\alpha/2 = 0.025\)[/tex].
3. Convert the t-value to the r critical value:
The formula to convert the t-value to the critical r-value (correlation coefficient) is:
[tex]\[
r_{\text{critical}} = \sqrt{\frac{t_{\text{critical}}^2}{t_{\text{critical}}^2 + \text{df}}}
\][/tex]
Using this process, the critical values for this test are:
[tex]\[
r_{\text{c.v.}} = \pm 0.707
\][/tex]
Thus, the critical value for this test is:
[tex]\[
r_{\text {c.v. }}= \pm 0.707
\][/tex]
So, the final result is:
[tex]\[
r_{\text {c.v. }}= \pm 0.707
\][/tex]
