Oh no. That's not true. God has answered a lot of my prayers and quite visibly. You can expect the same. But it doesn't happen that way suddenly. It takes some practice to hear the voice of God. It's as hard as science.
So if your god answers your prayers more often than, say, a jug of milk can (i.e., chance), and he would do the same for me (or presumably anyone and everyone else who believed in him the right way), then the results should be, as you say, "quite visible," statistically. This conflicts with your initial claim that the existence of your god cannot be "shown to another," i.e., demonstrated through the statistical effects of his activity on behalf of his True Followers.
Could you accept a god that would reveal himself only to the sincere seeker, one who expects you to take a leap of faith before revealing himself to you?
No. Why? Because there would be no way for me to tell, even in principle, that I'm believing in the right god. The "sincere seeker" thing is just an invitation to engage confirmation bias.
So, completely impossible then?
Isn't that what you've been telling us? If it's not completely impossible, then your proposed methodology of "leap of faith + confirmation bias" would not be necessary. We would be able to sort out which god/s (if any) is/are the real one(s) by observing the effects of their existence, which would make reality differ from the anticipated consequences of a naturalistic model with no supernatural activity.
If it is
completely impossible, and you can offer no observable anticipated consequences of your god's existence that differ from the anticipated consequences of the naturalistic model, then you're back to living in the same godless Universe we do, and we have no reason not to trim your god away with Occam's Razor.
So which is it?
No. I just think things get awfully muddled as in all non-controlled experiments. For example, Christians tend to be less educated than atheists (kudos to you). This can skew results. Many people come to God after experiencing some hardship, which also skews things. The religious have more children thus spreading resources a little thinner (and socio-economic class is related to so many of the things you'd want to measure).
There are false effects that can bias results in the other direction. For example, Mormons, with a lay clergy, get additional opportunities to speak and lead. We send our kids on missions where they often learn a second language and a little about the world. We fast, which provides health benefits. Even independent of God, prayer looks a lot like meditation, which has positive effects. Belonging to a community is healthy, etc.
Then you set up your study to account for such variables, the same way it's done when testing to see if taking aspirin reduces heart attacks. Your god is a more powerful phenomenon than aspirin, isn't he? The main thing here is that you're offering an explanation of why things look exactly the way they would if your god does not exist.
There is no reason to assume at the outset that a Universe governed and manipulated by Invisible Magic Persons of whatever sort (gods, goddesses, angels, demons, djinn, sprites, whatever) would behave the same as an atheistic or deistic Universe governed solely by natural regularities.
Imagine a person who wakes up from a coma suffering total amnesia. This person doesn't know anything about Universe or how it works. This person is given a Bible and/or a Book of Mormon and told that Universe works like what may be seen in the pages of the Book. Apart from knowledge that Universe in fact works differently, this person would expect to find all sorts of miracles and magic. Even if Yahweh decides to play hide and seek with people, that wouldn't stop Egyptian sorcerers from turning their staffs into serpents or water into blood. It wouldn't stop a statue of Ganesh from sipping milk from spoons, or the Blessed Virgin Mary from weeping tears of blood, or the Fair Folk from crossing over through the local barrow and abducting people to their mystic kingdom. People always and everywhere have believed in all sorts of supernatural manifestations--until they discovered tools that made it possible to discover, objectively, how reality actually works.
Then, all of a sudden, every IMP in Universe developed an urge to play hide and seek. Suspicious, no?
For heaven's sake, how could you expect to draw some conclusions about the inner, potentially meta-physical workings of the human mind when we can't even decide on a food guide pyramid!?
Because, oh, I dunno, wouldn't an omnipotent god, or a sorta-nearly-omnipotent Devil, or whole pantheons of mighty gods and goddesses be a bit more powerful
than the distinction between one portion of vegetables a week and two portions of vegetables a week? You're anticipating
that your god's activity (and that of all of his minions and enemies put together) is so weak
that it would vanish into statistical noise and error bars. Which is to say, you're anticipating
, before any such study can be designed or attempted, that the results would validate atheistic naturalism. You live in the same godless Universe we do, and you know it
One other point. It would be a mistake to think that religious principles are primarily for helping us live carefree, easy lives here and now. "Take up thy cross," said Jesus. "He that loseth his life for my sake shall find it," etc. Religious principles are primarily revelations about how to live more consistent with who you are spiritually--teaching men to "come to themselves." Sometimes that means taking one for the team in the here-and-now.
And once again, we have another "explanation" of why your god's existence just happens to be exactly equivalent to your god's non-existence. The straw man argument about religious principles being primarily for helping people live carefree lives is irrelevant. The effects of IMP existence would be just as apparent if the True Believers tended to have more difficult, harrowing lives than everyone else. If Mormons could be shown to have higher rates of random tragedy (car accidents not their fault, fires, animal attacks, disease, whatever) due to Satanic attack or "testing" from god, were more likely to be driven to shuddering insanity by meeping tentacled Things from the Abyss when they foolishly let their Spiritual Warfare Prayer Shieldstm
down in a moment of carelessness, or in any other way were genuinely more harrowed by supernaturally-caused calamity than non-Mormons, this would show
Instead, you appear to be claiming that all of this supernatural activity (your god's actions, those of his holy angels, and whatever Satan, his demons, and any other IMPs you believe in are doing independently and/or in opposition to your god) just happens to exactly balance out in a way that is perfectly indistinguishable from no supernatural activity at all!
How...convenient! Seriously, would you give any credence at all to someone who believed the Wizarding World of the Harry Potter stories exists, if they gave the excuse that, "Of course
there's no evidence for magic or wizards! The wizards use more magic
magic and dragons and trolls and Dementors and all the rest from us Muggles! They cooperate to do this perfectly, even when they're massacring each other in all-out wars
! It says so right there in the Books!"