Do you mean "subset"? As in, all logical thought is rational, but not all rational thought is necessarily logical? I'd say so, but I'll leave it to our philosophers. "Scientific thought" is a broad idea, so while it qualifies as rational thought, it wouldn't necessarily be a subset of logic only. The scientific process relies occasionally on irrational stabs in the dark, so I'm not sure you could determine that all scientific thought was dependent even on rationality.

The scientific method is a process by which concepts are tested against reality. The scientific method itself is fully rational.

However, the concepts themselves may have been arrived at by irrational "intuition," or desperation, or those "eureka!" moments that are so commonly portrayed in popular culture but very rarely happen in real life. For instance, psychic research is the rational application of the scientific method to an irrational precept (that the human mind exhibits extra-body abilities).  The "science" part is perfectly valid, as long as it is conducted strictly to the scientific method. It's the subject that is irrational.

"Irrational concept" does not mean the concept is incorrect. It just means the concept has no supporting evidence, and is based on something other than strict logic. Sometimes this is because we haven't observed the supporting evidence; other times, the supporting evidence doesn't exist (in which case, the concept is wrong).

This is all part of scientific thought, which is distinct from the scientific method. Basically, the best scientists are creative, with active imaginations, as well as logical and careful and rational.

So science is fully rational but can be used to test irrational thoughts against observed reality... But I guess this is trivial as it is only the scientific thought that is in question.

So what is irrational thought? Is it illogical? Unscientific? An umbrella term for them both?

Ali, science is simply the description of how knowledge of the material universe can be obtained.  It is a progression of steps, just like logic.  Also just like logic, steps can be done correctly or incorrectly.  If they're done incorrectly, your results are unreliable, and just like logic, we can literally say nothing about the results.  They may be true or false, and there's no way to know.

Correctly executed science is based on completely valid logic.

Rational and irrational describe a relationship to logic.  For logic to be invoked, there must be an argument.  So, rational and irrational, in the end, refer to a relationship with a conclusion.  In other words, if I hold up a rock and say, "This rock," my statement isn't particularly rational or irrational, outside of the fact that it appears to correspond to a known language and syntax structure.  However, if I say, "This rock has an elephant standing inside of it," and all available evidence indicates that there is no elephant standing inside of the rock, my statement is irrational with respect to the truth of its claim.  However, if you had previously asked me to speak a nonsense sentence, and that was my response, it's a perfectly rational response.

When I say, "There is an elephant standing in this rock," and I mean for you to believe it, there is an implied argument.  I am saying, "I have reasonable justification for saying there is an elephant in this rock."  Since there is clearly no evidence for my statement's truth, that premise is false:

Premise: I have evidence that there is an elephant in the rock

Conclusion: Therefore, I am justified in telling you that there is an elephant in this rock.

Invalid premise leads to irrational argument.

A lot of the problems with the words rational and irrational are that they can be used in different ways.  When I say, "That girl is being irrational.  She should dump her asshole boyfriend," I mean that from my perspective, there is no good reason for her to stay with her boyfriend.  However, that doesn't mean she is irrational, or for that matter, that even if by my criteria she is irrational, that her decision is actually irrational.  If someone believes they have accurate information, and makes a decision based on it, that decision is rational.  To an outside observer who knows that the information is inaccurate, the decision is irrational.  In other words, rational and irrational apply locally to arguments.  They are not inherent qualities, like "green" to a plant.  A belief, or a statement, or an action is only rational or irrational within a given context.

When we say that theism is irrational, what we mean is that the logic that leads to the conclusion that God exists is flawed, and the conclusion is irrational as a result.  This doesn't mean that every theist on the planet knows that the logic is bad.

As Hamby has pointed out, there is no distinction betwen "scientific thought" and just, "thought." The only distinction comes to the scientific ontology, which is the knowledge gained from the application of the scientific method to any concept.

"Scientific thought" is only distinct from other thoughts in our ability to apply the scientific method. We can have many ideas, but those ideas are only rational if we are able to sift them through the crucible of empiricism. An "irrational" concept is one that cannot be supported by the current ontology. That doesn't make it automatically incorrect. It simply means it isn't supported by current knowledge.

Rationality parallels scientific thought in its ability to be tested against reality. There are rational concepts that cannot currently be tested; further, there are irrational concepts that may turn out to be true. Their only metric is their congruence with reality.

"Irrational thought" is merely thought that is based on something other than empirical knowledge. Logic may apply, but it might be applied to something unobserved but intuited. This doesn't make it right; neither is it necessarily wrong. If it is testable against reality, then it is science. Otherwise, it is supposition, or worse, metaphysics.

Reality is the ultimate judement. If something is congruent with observed reality, it is ultimately rational. If it is incongruent with observed reailty, it is irrational. The only point of contention should concern what is "observed."

That's a great question.  Let me give you an idea of how to go about designing your experiment.

First, you're going to have to decide on a critical distinction.  Are you going to pray for things that are impossible without divine intervention, or are you going to pray for things that might happen anyway?  If you're going to pray for things that are impossible, you're not going to need all that much correlation.  Even one hit would be pretty impressive.

For instance, why don't you pray for Shelleymtjoy?  She's missing bones from from her legs.  If she woke up in the morning and had perfectly normal bones, that would be truly miraculous.  I'd be convinced by just that.  Bones don't come back overnight when they've been removed.  The classic example is amputees.  Why don't you get a thousand amputees together and pray for all of them to grow their limbs back.  If one of them does, that would pretty much prove that prayer had something to do with it.

Honestly, if you want to prove that prayer works, you need to pray for stuff that can't happen.  However, if you don't trust your god to actually do a real miracle, you could still get some impressive results with things that could happen anyway.  It's going to be a LOT harder, though.  First, you have to establish a control and an experimental group, and you need to pray for a very specific kind of result.  There actually was a study done just this way, by the Templeton Foundation.  That's a Christian organization that was trying to prove that prayer is effective for sick people.  It turns out, prayer was not only completely unhelpful, but among one group who knew they were being prayed for, there was apparently some performance anxiety, because they did slightly worse than people who were not being prayed for.  So, if you want to know how to design your test, you could use that as an example.  It was unfortunate for the Christians, because they designed the test well, and it showed them that prayer actually doesn't work.

In general, when we look at the reliability of a study, we look at the results, and try to figure out what the odds are that the results could be generated randomly.  If you get a result that's really impressive, say there's only a 0.01% chance that the result could be chance, you're still not home free.  Strictly speaking, no matter how uncertain an event is, a one time occurrence is not a reliable result.  That's why we make repeatability such a big deal.  So, here's the second problem you're going to run into with a test of things that might happen anyway.  You're going to need to repeat the test, and get a similar result.  You'll need to do this at least twice, maybe more, depending on the probability that random chance could be responsible for your first result.  Consider, if there's a 1 in 10 chance that the results are random, there would only be a 1 in 100 chance that two studies in a row would yield the same result.  In general, we want much better chance of real results... like 1 in 1000 or 10000, and then we want two or three consistent tests.  That way, we know with good probability that there's only about a one in a billion or so chance that it's a false result.