A good way to prove if whether Allah occasionally answers prayers, would be too check to see if Muslims have a higher cancer survival rate than atheists?
Muslim cancer sufferers would pray more to the monotheistic God than an atheist would, and thus as a whole be more likely to have their prayer answered and survive than a Murtad Kaffir would.
I am sure it would be fairly easy to prove this by comparing the cancer survival statistic of a largely atheist nation (the largest being Sweden which is 80% irreligious, or others like Denmark, Russia or Switzerland) vs. any Muslim nation.
This would prove that prayers do not work (or Allah does not exist).
Although an interesting idea, there are too many contributing effect(s) of confounding variables you would have to control such as genetics, diet (e.g. smokers and high % fat diet), age, environment and health.
Related only until the point of studies exploring effects of religion, you maybe interested in checking out the work of distinguished neuropsychologist Richard Davidson and his work on meditation.
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