There is research on this question, the keywords to look for are "lay theories" or "naive theories".

One approach is to measure people's naive theories (e.g., by asking people if they believe that X causes Y), to see if they are correlated with the outcome Y when X is present. Some people hold the belief that X causes Y more strongly than others, and this takes advantage of that variability to test if the strength of the belief is associated with the outcome.

Another approach is to manipulate people's beliefs (e.g. by telling them that scientific research has shown that X is unrelated to Y) and see if that changes the outcome Y (when X occurs).

In some cases, it's possible to make people believe that X occurred when it really did not; in that case researchers can use the same methodology as a placebo control study.