You would be disappointed as your example omits the genetic contribution to the baseline IQ which you already accepted as part of the equation. Further, you would have to study a large enough sampling of families to avoid cherry picking the data.
Black children in poor neighborhoods are likely in that situation due to low IQ/low achievement parentage. Poor nutrition isn't going to help their situation. Neither is a school environment where classrooms are gang battlezones that no decent teacher will approach. As observed by many teachers, success starts in the home. Bonehead parents aren't going to help a child succeed. Intelligent parents may well help a child succeed in spite of low quality classroom instruction.
Taking the observation a step further, the forced integration by busing and other means was the liberal ideal of "equal" education. It was a tremendous failure. Learning is not an osmotic process. Mixing students of greatly disparate ability results in either 1) holding the classroom back to the lowest common denominator 2) leaving the low performers twisting in the wind to be perennial failures in life. We have had to hear the annual wailing about the "achievement gap" between blacks and whites for the last 50 years. It is the same story every year. The same remedy of "more funding" is demanded. The problem never gets solved because it ignores the reality of the IQ differences between the measured groups.
My solution? Stop trying to fix an unfixable reality. Administer IQ tests and sort the kids by ability. You'll get some of those black kids on the high end of the IQ range into a place where they can be maximally successful. A classroom full of students with similar ability is much easier to handle in the K-12 world. I structured my college courses in a self-paced fashion. The high end students assisted me with the lower performers. In turn, I rewarded the high performers with more challenging instruction in techniques that went well beyond the course requirements.
I have no reason to think that the distribution of genes that contribute to intelligence are significantly different by race. While there may be some population/sub-population differences in the allelic frequency of intelligence-related genes, I expect that all genes that contribute to intellect are present in all populations. This is only speculation, since we (the scientific community) have not identified the genes related to intellect, nor determined how they interact with one another.
On the other hand, environmental and nutritional influences have been shown to have a huge effect on IQ. So do infectious diseases. A kid who spends most of his energy trying to survive various illnesses does not use that energy developing his brain. All of these factors add up; once a child ages past the crucial early years of brain development, the opportunity to maximize IQ potential is lost.
Black poverty culture today does *not* embrace the factors that enable children to develop to their full potential. Furthermore, that culture actively censures any exercise of intellect. Until that culture is eradicated, I do not think we can say anything about the genetics of IQ in black populations. As I said above, I have no reason to think that black people lack any intellect genes--it's just that their poverty culture negates the effect of those genes.
I should say that much of white liberal culture embraces the same anti-intellect values as the poverty culture...