Well, this is only really a problem as far as I can see if the experimenters’ belief or disbelief affected the outcome of the experiment, so that’s our hypothetical hypothesis.
I can’t come up with any good examples, because I’m just too abstract like that. I’m sure you get the idea anyway.
I can see this possibly being GD territory; mods, do what you will.
Placebos.
You test them by giving one group a placebo and not giving anything to a control group. If the group who received the placebo does better than the group who got nothing, then their belief had an effect.
On a more “magical” topic, I saw a video where they had essentially a pachinko machine; where a series of balls would be dropped down through a forest of thin columns and fall into any one of several slots at the bottom. Left to itself the balls would generally form into a bell curve with most balls ending up in the middle slot. But then they would sit a human down in front of the machine and the person was asked to simply think “left left left left” at the machine for a few hours. In this instance, the bell curve would shift slightly to the left. As I recall it they said that the amount of shift was on average something like 0.01%–so there’s one answer for how much human’s faith can effect the world around them. 
Yes, placebos, but what if the belief of those administering the experiment also affected it?
That has an effect on most studies from drug trials and quantum mechanics, to the power of prayer. Placebo control groups, double-blind trials, and careful experimental design are the best we have.
With a placebo, there is also no real difference between belief in the administer and belief in the pill. Belief in the administer is in fact what you are testing, no?
In a netshell, you can’t.
Suppose you want to test for the efficacy of prayer (a belief). You divide randomly-selected subjects into two groups, intended to be controls and experimental, then you tell the praying parties to pray for only the B group.
Regardless of the outcome, how can you tell that[ul][li]Someone didn’t pray for the B group[]Someone in the praying group didn’t pray at all[]Someone else in the world didn’t pray for one group or the other[/ul]?[/li]
Each one of these possibilities can theoretically affect the outcome, some in a positive direction, some negative. So you will always have an excuse, but never have an answer.
Which is so little as to be statistically insignificant. Statistically, that is the same as zero. I’ll bet if you ran some more trials, you might get some that registered .01% in the opposite direction. What would that mean? Negative PSI?
Nonsense. If a belief in something can really affect the outcome, all the scientific tests in the world, everywhere, all the time, are suspect and meaningless.
A negative result can always be explained away; you didn’t believe hard enough (and of course this excuse is used all the time by faith healers).
A positive result cannot be explained away if it can be repeated to the point it is statistically significant
So a trial to prove a belief is easy, to disprove it probably impossible
They actually did this experiment with rats. They took a bunch of rats divided them into two different groups and told half of the students that their rats were bred for high intelligence and the other half that thier rats were stupid.
The rats that the students believed were smarter consistantly made it through the race faster.
I believe that they later did the same experiment using teacher and children with similar results.
There are of course several possible causes of these effects. In the case of the rats, it could be a simple as unconscious biases on the part of the students doing the measuring :i.e. both sets of rats were actually just as fast, but the students with ‘smart’ rats tended to round down results, throw out very low runs as ‘outliers’ and so forth, and vice versa with the ‘dumb’ rat students.
It could also be to better care for the ‘smart’ rats (again because of unconscious biases on the students part), making them healthier and therefore faster.
Or it could be that the rats really were identical but mystical energy vibrations from the students caused the ‘smart’ set to go through mazes faster, simply because of ‘belief’.
I know which answer I think is most likely. Science is done by human beings, after all.
How did they get the teacher and children to go down those little mazes? 
You laugh, but negative psi is a well known “phenomenon” in these circles:
http://library.thinkquest.org/C0120993/glossaryfull.html#meta
The so-called Pygmalion effect was tested a number of years ago, I believe in the '60s. (Reported in a book called Pygmalion in the Classroom). Teachers were given identical groups of students but told different things about them. Those classes that were purported to be undiscovered geniuses did better than those that were supposedly below normal. The hypothesis was that expectations that teachers hold of their students affect how they treat those students, and that difference is reflected in such things as grades, test scores, and the like. Unfortunately, although the hypothesis seems to have been supported in a number of arenas over the years, the original research has been called into question, and I think I’ve read that it may never have even been done.
xo, C.
Ottoerotic, CC and Quercus, you are talking about the perception of a result brought about by a subjective observation influenced by a preconceived expectation. I don’t question this phenomena, in fact, it is what must be guarded against and is why double-blinding is used.
That is not the same as belief actually affecting the results, as asked in the OP. If a double-blind test provides a significant difference between one test group that was supposedly prayed for over another that supposedly was not, among the possibilities (but not the best choice) is the one that belief had some influence on the outcome. The question then becomes, “Whose belief and which way did the belief influence it?” which I maintain can never be settled for sure. There could always be some unknown, evil dude in some faraway place (since distance seems to have no measurable influence) who is praying constantly for everyone and screwing up everyone’s experients.
Yep, that is pretty much what I had in mind, although I hesitate to call it a “phenomenon.” (I would call it an “interpretation.”) But if you allow for unknown influences to cause either positive or negative results and claim that such results are directly caused by the unknown influences, you have no valid test at all. This is why the JREF Challenge requires the outcome to be specified in advance, not interpreted afterwards.