I am working on a research project in social sciences. My null hypothesis is that there is a high correlation between two variables and my alternative hypothesis is that there is very low to no correlation. Why would I do it this way? I want to show that under certain conditions the mainstream theories that rely on high correlation between the two variables do not work and lead to poor policy choices.
When correlation drops to near zero, so does the p-value, the benchmark measure of statistical significance. What can I use to show that a near-zero correlation is statistically significant and is NOT due to a chance?
Edit: I am doing a time-series cross-country analysis of correlation between two macroeconomic indicators. I stratify my sample to control for other variables that are also measured on a country level. One stratum has, as expected, ~0 correlation, the other has positive correlation around 0.3. The stratum with positive correlation has p < 0.05, the one with ~0 correlation has p closer to 1, even though it has the same number of observations. How do I show that the ~0 correlation in that stratum is not due to a chance?