Final answer:
Whether a correlation coefficient is significant depends on the sample size, degrees of freedom, and comparison to critical values or p-values for a two-tailed test with a set alpha level. Larger samples increase the chance of finding significant results.
Step-by-step explanation:
When determining if a correlation coefficient (r) indicates a significant relationship between two variables for a given sample size (n), we utilize hypothesis testing and compare the resulting test statistic to critical values or the associated p-value. For a two-tailed test with an alpha level (α) of 0.05, we reject the null hypothesis if the test statistic is beyond the critical value for our degrees of freedom (df).
Part A: Sample of n=10
For the initial sample of n=10 with r=0.60, the degrees of freedom (df) is n-2, which is 8. Consulting a t-table or using the formula t=r√((n-2)/(1-r2)), we can determine the significance of r. Here, we'd compare our t value to the critical value associated with df=8 at the 0.05 alpha level.
Part B: Sample of n=25
With a larger sample size of n=25 and the same correlation of r=0.60, the degrees of freedom increase to 23. The critical value for a two-tailed test at df=23 and α=0.05 would be lower, making it easier to achieve significance with the same correlation coefficient. Thus, with a larger sample size, the likelihood of determining a significant correlation increases.
To conclude, both scenarios require comparison to critical values or p-values to decide on the significance of the correlation coefficient, with larger sample sizes generally increasing the potential for statistical significance.