Reason and religion are not incompatible; never have been, and never will be. They are simply two aspects of experience, two lenses through which the inner and outer world are apprehended. Reason and logic are wonderful, but will BY DEFINITION remain forever unable to explain or account for - much less accept- visceral experience lacking a material stimulus. The numinous (which included religion, spiritualism, ooga-booga stuff, etc) is also wonderful, but beyond its neurophysiological correlates during the experience itself, has nothing to do BY DEFINITION with logico-deductive neo-cerebral processes. This debate cannot ever have a winner, because the notional conflict is spurious.
Graeme says: "Reason and logic are wonderful, but will BY DEFINITION remain forever unable to explain or account for - much less accept- visceral experience lacking a material stimulus."
Your defense only works if and only if the "visceral experience" remains undefined and if you accept that all religions are equally right. The minute that a person defines this "visceral experience" as coming from a certain god, then your argument falls to pieces because this "visceral experience" is the same on a psychological level for believers in Allah, Jesus, or Zeus. By definition, these religious traditions are mutually exclusive. They each claim monopoly on "truth."
So, on a semantic level, your claim that "Reason and religion are not incompatible" is false because religion is an attempt to codify and crystallize (therefore injecting reason) into the otherwise elusive, ephemeral and irrational spiritual experience. You can claim that reason and a generic spirituality are compatible, but you cannot claim the same for reason and religion.
And besides, spirituality reduces to psychology so even it too can be analyzed within a rational and reasoned framework. I think you may wish to re-evaluate just who here is drawing the false dichotomies.
<<Paul Carr says: It strikes me that a much, much simpler explanation is that the putative deity does not exist, and so prayer is not efficacious. >>
Super late response!
Besides that, the bible explicitly claims that with an infinitesimal amount of faith we can do things more miraculous than those depicted in the bible, and that we can do basically anything. If that's not open to being scientifically evaluated...