Even the rate improved due to prayer, atheist will not give the credit to God but to some other reasons. The rising rate of survival is not a new feature. What is your interpretation?
If prayer-at-a-distance can be positvely correlated to survival, or the sponteanous regrowing of amputated limbs, or the immediate and miraculous remission of cancer, that would be good evidence that a deity is indeed interfering in the world. You're just making excuses - can you give any real reason why the experiment would fail? If God does indeed answer the occasional prayer with a 'yes', and intervenes in the world to heal someone who, had they not been prayed for, wouldn't be healed... if this common claim of intercessory prayer is true, why
wouldn't the test result in a positive correlation? What physical, mechanical, statistical, etc, reason do you believe would prevent the test from working?
"God can't be tested" isn't a reason. "God kills anyone who tries to test him" is a reason. Go fish.
Science can always give a model to something or some feature, natural or supernatural. They are all models anyway. It is cheap. What is your "scientific" model to the feature called the sixth sense, if you care to give one?
If the sixth sense is some sort of psychic remote viewing, place a piece of paper in a sealed box and get the purported psychic to tell us what's written on it.
If the sixth sense is NDEs, place a code on top of a cabinet in the operating theatre, which the disembodied individual can tell us.
If the sixth sense is the ability to see dead people (as per the titular film), get the psychic to retrieve information that can be independently corroborated.
If the sixth sense is a sort of early warning system (akin to Spiderman's spidey-sense), set up a series of trials where they are preternaturally aware of some event that hasn't happened yet. Crudely, we could pelt them with BB bullets from one of two guns, and they get to press one of two buttons which would jam one gun or the other. If they can consistently jam the gun that's about to fire each time, that shows they can indeed react with psychic powers.
In general, if you purport to have the psychic ability to acquire information that would otherwise not be attainable, the test is quite simple: get them to acquire the information. Such things aren't untestable, not at all, it's just that people don't
want them to be tested, because they know, in their heart of hearts, that rigours scientific trials would reveal their heartfelt beliefs to be wrong. Dowsing, homoeopathy, astrology, remote viewing, mediumship, seances, ouija boards, all can be easily disproven, and whenever a psychic offers herself to be tested, she is always disproven.
It is an example on the power of prayer. Is that what the OP is asking?
Indeed, but unless you purport that repeating Elijah's actions would cause the spontaneous combustion of such an altar, I don't see the relevance. Recall also that drjean implied this was proof of some sort - proof of what? That the Bible makes claims about God?