for a sample of n=20 how large a Pearson correlation is necessary to be statistically significant for a two tailed test
To determine the necessary correlation magnitude for statistical significance in a two-tailed test, you would typically look at the critical value or p-value associated with the test. The critical value or p-value depends on your pre-defined significance level (often denoted as α). By convention, a significance level of 0.05 is commonly used, which implies that you are willing to accept a 5% chance of making a Type I error (incorrectly rejecting a true null hypothesis).
To find the critical value or p-value associated with a two-tailed test for a given significance level and sample size, you would typically refer to a statistical table or use statistical software. However, I can provide you with general guidelines:
1. Calculate the degrees of freedom (df) for your sample. For a Pearson correlation test, the degrees of freedom is n - 2, where n is the sample size. In your case, the degrees of freedom would be 20 - 2 = 18.
2. Look up the critical value (tα/2) or p-value associated with your chosen significance level (α) and degrees of freedom (df) in a t-distribution table or use statistical software. For example, at a 5% significance level (α = 0.05) and 18 degrees of freedom, the critical value of a two-tailed test would be approximately ±2.101.
3. Alternatively, you can calculate the p-value associated with your correlation coefficient using statistical software. The p-value represents the probability of observing a correlation coefficient as extreme as the one computed, assuming the null hypothesis is true (no correlation). If the calculated p-value is less than your chosen significance level (e.g., p < 0.05), then you can conclude that the correlation is statistically significant.
Keep in mind that these guidelines are general, and relying on specific statistical tables or using software is recommended for precise and accurate results.