Alright, let's go through the problem step-by-step to determine if the given linear correlation coefficient [tex]\( r = 0.843 \)[/tex] represents a significant linear correlation for a sample size of [tex]\( n = 5 \)[/tex] at a significance level of 0.05.
### Step-by-Step Solution:
1. Determine the critical values of [tex]\( r \)[/tex]:
- For a significance level ([tex]\(\alpha\)[/tex]) of 0.05 in a two-tailed test, we split the significance level between the two tails. Thus, each tail has an area of [tex]\(0.025\)[/tex].
- Calculate the degrees of freedom (df) as [tex]\( n - 2 \)[/tex]. For [tex]\( n = 5 \)[/tex], the df is [tex]\( 5 - 2 = 3 \)[/tex].
- Look up the critical t-value from the t-distribution table for [tex]\( df = 3 \)[/tex] at the 0.025 significance level in each tail. This critical t-value is approximately 3.182.
- Using the t-value to find the critical value of [tex]\( r \)[/tex]:
[tex]\[
r_{\text{critical}} = \frac{t_{\text{critical}}}{\sqrt{t_{\text{critical}}^2 + \text{df}}}
\][/tex]
Plugging in our values:
[tex]\[
r_{\text{critical}} = \frac{3.182}{\sqrt{3.182^2 + 3}}
\][/tex]
[tex]\[
r_{\text{critical}} \approx 0.878
\][/tex]
2. Compare the absolute value of the given [tex]\( r \)[/tex] to [tex]\( r_{\text{critical}} \)[/tex]:
- We found that [tex]\( r_{\text{critical}} \)[/tex] is approximately [tex]\( 0.878 \)[/tex].
- The absolute value of the given [tex]\( r \)[/tex] is [tex]\( |0.843| = 0.843 \)[/tex].
3. Determine if [tex]\( r \)[/tex] represents a significant linear correlation:
- For [tex]\( r \)[/tex] to represent a significant linear correlation, its absolute value should be greater than the critical value of [tex]\( r \)[/tex].
- Here, [tex]\( |r| = 0.843 \)[/tex] is less than [tex]\( r_{\text{critical}} = 0.878 \)[/tex].
Since [tex]\( 0.843 < 0.878 \)[/tex], the given [tex]\( r \)[/tex] does not represent a significant linear correlation at the 0.05 significance level.
### Conclusion:
- Critical values: [tex]\( r = \pm 0.878 \)[/tex]
- Significance of given [tex]\( r \)[/tex]: No significant linear correlation
Therefore, the correct choice is:
Critical values: [tex]\( r = \pm 0.878 \)[/tex], no significant linear correlation
