Final answer:
A Pearson correlation of r = 0.43 with n = 20 is significant if the t-value exceeds the critical value for the chosen alpha level. Upon checking with statistical tables or software, if the t-value is higher for α = 0.05 than for α = 0.01, the correlation is significant at the 0.05 level but not at the 0.01 level. Option b. the correlation is significant with α = .05 but not with α = .01 is the correct answer.
Step-by-step explanation:
When determining the significance of the correlation coefficient, we use a hypothesis test to decide if the observed correlation in a sample is significantly different from zero for a given population. The null hypothesis (H0) in this test is typically that there is no correlation in the population (the population correlation coefficient, ρ, is zero), while the alternative hypothesis (H1) suggests that there is a significant correlation (the population correlation coefficient, ρ, is not zero).
In the question at hand, a researcher obtains a Pearson correlation of r = 0.43 with a sample size of n = 20 participants. To determine the significance of this correlation, one would typically reference either a table of critical values for the correlation coefficient or conduct a t-test. In both cases, we are interested to see if the value of r falls within a range that would be considered significantly different from zero for a specified alpha level (α).
Upon checking a table of critical values (or using statistical software), we can decide to either accept or reject the null hypothesis. Given that n = 20, we would have degrees of freedom df = n - 2 = 18. We need to find the critical value for df = 18 at α = 0.05 and α = 0.01 for a two-tailed test. If the absolute value of r exceeds the critical value, the correlation is significant.
When n = 20 and r = 0.43, we refer to statistical tables or software to find the t-value corresponding to our obtained r-value. If the calculated t-value is greater than the critical t-value for α = 0.05 but less than that for α = 0.01, then the correlation is significant at α = 0.05 but not at α = 0.01, matching option (b) the correlation is significant with α = .05 but not with α = .01.