I am having trouble isolating these two concepts.
If the sample correlation, using Pearson’s r, is 0 and statistically significant then we can conclude the data are not linearly related - there is no correlation.
If the sample correlation, whatever it may be, is statistically insignificant then we fail to reject the null hypothesis that the population correlation is 0. So essentially we are saying the population correlation is 0.
How are these two observation different?