To solve this problem, we need to determine whether the correlation coefficient $r = 0.767$ is statistically significant given the sample size $n = 25$ and a significance level of 0.05.

Step 1: Determine the critical value of $r$

The critical value for $r$ depends on the sample size $n$ and the chosen significance level (in this case, $\alpha = 0.05$). For small samples, the critical value of $r$ can be found using a statistical table for the Pearson correlation coefficient.

For a sample size of $n = 25$, we look up the critical values for $r$ at $\alpha = 0.05$ (two-tailed). Based on the table, the critical value for $n = 25$ is approximately:

$r_{\text{critical}} = \pm 0.396$

Step 2: Compare the given $r$ value to the critical value

• The given correlation coefficient is $r = 0.767$.
• Since $r = 0.767$ is greater than the critical value $0.396$, this suggests that the correlation is statistically significant at the 0.05 level.

Conclusion

Since $r = 0.767$ is greater than $r_{\text{critical}} = \pm 0.396$, we reject the null hypothesis and conclude that there is a significant linear correlation.

Thus, the correct answer is: D. Critical values: $r = \pm 0.396$, significant linear correlation

Would you like further details on how to find critical values or how the correlation test works?

Here are 5 related questions:

1. How do you find the critical value of $r$ for different sample sizes?
2. What is the interpretation of the linear correlation coefficient?
3. How does changing the significance level affect the critical value?
4. What are the assumptions for using Pearson's correlation coefficient?
5. How does sample size impact the reliability of $r$?

Tip: Larger sample sizes lead to smaller critical values, making it easier to detect significant correlations.