Final answer:
A sample correlation coefficient of 0.83 is not significant for a sample size of n = 3 at the α = 0.01 level due to the small sample size, whereas it is significant for a sample size of n = 10 at the same α level.
Step-by-step explanation:
To determine if a sample correlation coefficient (ρ) is significant, we consider both the coefficient value and the sample size (n).
Significance for n = 3
With a small sample size of n = 3, the degrees of freedom (df) would be 1 (df = n - 2 = 3 - 2). Even with a ρ of 0.83, the sample size is too small to reach a solid conclusion for most significance levels, including α = 0.01. It is generally not advised to perform significance tests on sample sizes smaller than 5 as they are unreliable. Thus, it is not significant for n = 3.
Significance for n = 10
Assuming a sample size of n = 10, the degrees of freedom would be 8 (df = n - 2 = 10 - 2). When using Table 12.9 or an equivalent critical value table for α = 0.01, you would find much lower critical values than 0.83 for 8 degrees of freedom. This means that a correlation coefficient of 0.83 is well above those critical values, indicating a significant correlation given the sample size and significance level. Hence, it is significant for n = 10.