One that I saw a paper get wrong a while ago was one which claimed to find an inverse correlation between two different traits that individually improved intelligence. The catch? The sample population was chosen from university students at no name school X. But people who did well on both traits would have done well enough to go to a better school and so were underrepresented in the population sample!
(I forget the paper, but pay attention and you'll find lots of other examples...)
Suppose that acting ability and attractiveness are independently normally distributed with mean 0 and standard deviation 1. Further suppose that to be a successful movie star, an individual must have an acting ability plus attractiveness of 6.
Then among people with the necessary attributes to be movie stars, attractiveness and acting ability will be negatively correlated. In this example, we might expect to see movie stars with (intelligence, attractiveness) around (3,3) or (2,4) or (4,2), but it's much more unlikely that we see many people around (4,4).
See http://bentilly.blogspot.com/2010/02/what-is-intelligence.ht... for some of my thoughts on that from a few years back.
A better case might be for students at big name university's where GPA (proxy work etc) and intelligence (proxy SAT score etc) need to be over some threshold for admittance.
If that's the case bankers and others that want both a big name school and high GPA are actually negatively selecting for intellect. Which may account for a lot of fairly dumb behavior at banking institutions etc. As they might have a lot of people that can talk about statistics without actually understanding it.
On what basis do you believe it is unlikely?
As an example I submit that test preparation courses like Kaplan do nothing but improve how well you'll do on a certain type of test without improving your general intelligence.
As another example I am quite aware of how much of an advantage I gain on tests from my ability to relax in a situation where other people tense up. I've described this advantage before with, "Comparing me to a normal person based on the resulting test score is like starting with two runners, taking one out back and beating on him for a while, then expecting them to run a fair race."
Now within a given range IQ rang of say 95 to 105 the correlation between IQ and test taking ability might be tiny. However, that's unlikely to hold up as you keep stretching the IQ range from say 50 to 150.
PS: IQ tests where initially more about testing the low end of the scale than the high end and in that context there not that bad. The early assumptions where also looking at the correlation with IQ and things like reflexes when that failed people started looking into mental retardation etc.
But I think the point remains that a certain amount of what goes into the IQ score is something we don't think of as intelligence. And this means that IQ doesn't measure intelligence nearly as directly as most of us would naively think.
If you compare say: 0-65, 66-85, 86-115, 116-135, 135+ you find plenty of significant differences. Generally 85 vs 86 is meaningless, but 85 vs 116 is not. Which means any hard cutoff is going to exclude people close to the cutoff on a fairly arbitrary basis.
But people selected by success in life tend to on average have decent but not outstanding IQ scores. And people selected for their outstanding IQ tend to have decent but not outstanding success in life.
P(A|B,C) is "Probability of A, given both B and C".
or even more precedence dubious: P(A|B|C) ?