Final answer:
True, to detect a small correlation as statistically significant we often need a large sample size to increase the statistical power of the test, enabling us to differentiate the correlation from zero with more certainty.
Step-by-step explanation:
True. If we want to detect a small correlation as statistically significant, we often need a large sample size. This is because smaller correlations can be harder to distinguish from zero purely by chance. A larger sample provides more data points, which can help to ensure that any detected correlation is not just a result of random variation in the data. In statistical terms, a large sample size gives us more power to detect a statistically significant correlation, assuming one exists.
In the context of hypothesis testing for correlations, the null hypothesis typically states that there is no correlation (p = 0). The alternative hypothesis (Ha) is that there is a significant correlation (p is not equal to zero). To conclude that a correlation coefficient is significant, we need enough evidence to reject the null hypothesis. A larger sample size can increase the statistical power of the test, thereby improving the likelihood of detecting a small but true correlation.