Care to post any of your favorite riddles from those sites? I perused a bit, and they seem rather challenging to say the least.
It's been a while since I've actually tried solving them, but off the top of my head...
In the medium section of the Miller site, the "FRUITful Problem" and "Strawberry Fields Forever" are simple but clever classics. I think most people would be able to solve them with some serious thought. A more difficult one, one that I actually discussed in quite a bit of detail with Professor Miller (I was considering Williams at the time), is the "A Penny for Your Thoughts" riddle. That one basically required a rigorous proof and can't be immediately solved with a basic realization. However, it's still solvable with basic math skills. Those should get you started.
For the AoPS site, most of the problems between numbers 13 to 20 in the AMC 12 series are decently challenging. Below that, they're usually trivial, and above that, they're hard as ****. For AIME, up to problems 4 or 5 are usually of decent difficulty. Both series require some algebra 2 and geometry knowledge within the ranges I mentioned. Beyond that, there's still no calculus, but a strong knowledge of number theory and discrete mathematics is almost necessary to solve those problems.
For the Wu site, anything within easy and medium is recommended for start with. You'll also find few that aren't strictly mathematically based. I don't recommend you wasting your time with the Putnam level questions though, unless you're brilliant at math--the median score for the Putnam examination is almost always 0. The Microsoft questions (actual questions asked in interviews for Microsoft) are pretty fun and really put your creativity to the test, haha. Oh, and if you have time, look at the "Pop Quiz" question in the relatively hard section. IIRC, that question was a serious mind-****er, and it seems like there's no real consensus on the answer.
Oh, here's a better (And real life!!!!) example study:
Prayer reduces complications in admitted cardiac patients
Wow, interesting study. I didn't actually read through the paper though--don't have the attention span or patience for it right now. Is it alright to assume that you've given sufficient information about the methods and results in your summary to determine the study's flaws?
Again, I haven't really read through the study, but I'm guessing there was some sort of confirmation bias going on. I realize the study was a double-blind one, but if the researchers compiled their data in such a way that they subconsciously (or maybe even consciously) manipulated it to fit their preconceptions, then that would be a confirmation bias. This is supported by the fact that the index was specifically created for this study.
However, based on your synopsis, I'm not surprised the study was accepted. It actually seems somewhat rigorous--certainly enough to provoke legitimate arguments from both sides in spite of the ostensible absurdity of the matter.
