The measure we usually associate with "innate intelligence", the so-called "G-factor", is actually defined as the correlation between several different factors (education, grades in several subjects, various standardized tests including IQ, career success, socioeconomic status) that all trend in the same direction. It's basically a PCA on the observed metrics. There's no way to measure intelligence directly or even know for sure that it exists as a separate innate "thing" - all we know is that a large number of observable metrics are positively correlated, and then we give a name to the correlation.
What g really means is that you can't design a meaningful intelligence test that doesn't correlate with all the other intelligence tests - with slightly divergent axes for verbal and spatial/mathematical skills. Thus, for instance, backward digit span correlates with Raven's matrices, even though these tasks have nothing obvious in common. Someone who's good at one will be good at the other.
It's also very hard to produce training/educational effects that show an effect on g, though dual N-back perhaps has some promise:
http://en.wikipedia.org/wiki/N-back
However, these kinds of brain exercises have very little to do with education as we know it. The obvious null hypothesis is that we're looking at a physiological effect, such as the quality of myelin insulation in neurons.
Obviously, we see the same correlation effect in CPU benchmarks - any benchmark at all will reveal that a Xeon is faster than a Celeron. The obvious null hypothesis is that the Xeon has smaller transistors and more of them. The causality behind neurological g is probably something just as crude and straightforward.
One could argue, however, that when we compare Xeon motherboards to Celeron motherboards, we see faster DRAM and the like. Perhaps it's the fast CPU's environment, rather than its lithography, that makes it faster.
But... this isn't an argument anyone would make without a strong prior conviction that all CPUs are created equal. It's unclear where such an idea comes from in the case of the human brain, but it doesn't seem evidentiary in nature.
But the causal link between parents' income and kid's academic success is small to non-existent[1] so even if there are factors influencing IQ, wealth doesn't seem to be one, at least not in this way.
[1] "Low Family Income Not a Major Reason For Poor Student Achievement" http://educationnext.org/low-family-income-not-a-major-reaso...
As an example of how IQ test results aren't too reliable: IQ tests have been rising year on year for decades (the Flynn effect). Either we're all a pile of geniuses (which makes you wonder what score people in the middle ages got), or IQ test is inflenced by external things.