To determine whether the linear relationship between the two variables is statistically significant at the [tex]\( \alpha = 0.05 \)[/tex] level, let's follow these steps:
1. Given Values:
- Sample size ([tex]\( n \)[/tex]) = 9
- Pearson correlation coefficient ([tex]\( r \)[/tex]) ≈ -0.677
- Level of significance ([tex]\( \alpha \)[/tex]) = 0.05
2. Critical Value:
- From the provided critical values table for [tex]\( n = 9 \)[/tex] and [tex]\( \alpha = 0.05 \)[/tex], the critical value is 0.666.
3. Absolute Value of [tex]\( r \)[/tex]:
- Calculate the absolute value of the Pearson correlation coefficient:
[tex]\[
|r| = |-0.677| = 0.677
\][/tex]
4. Compare [tex]\( |r| \)[/tex] with the Critical Value:
- [tex]\( |r| = 0.677 \)[/tex]
- Critical value = 0.666
We see that [tex]\( |r| = 0.677 \)[/tex] is greater than the critical value of 0.666.
5. Conclusion:
- Since [tex]\( |r| \)[/tex] is greater than the critical value, the relationship between the variables is statistically significant at the [tex]\( \alpha = 0.05 \)[/tex] level.
Therefore, the correct conclusion is:
Yes, the linear relationship between the variables is statistically significant at the 0.05 level of significance, because [tex]\( |r| > \)[/tex] critical value.
