Question

1. If the sample correlation between two random variables is small in magnitude (say 0.1), does that necessarily imply that the test for the significance of the correlation will have a large p-value (> 0.05)?

Answer 1

This is a false statement. If the sample correlation between two random variables is small in magnitude (say 0.1), it does not necessarily imply that the test for the significance of the correlation will have a large p-value (> 0.05)

A small sample correlation between two random variables for large sample size may have low p-value for the test for the significance of the correlation

For example, let's say we get the sample correlation between two random variables as 0.1 from 1000 observations (n = 1000).

Test statistic, t = = 3.18

Degree of freedom = n-2 = 1000 - 2 = 998

For two-tail test, (Ha: 0) ,

p-value = 2 * p(t > 3.18) = 0.0015

Thus, we have get low p-value (< 0.05) for small sample correlation between two random variables.