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Finally, part 4:
PLAYING WITH THE GAP
We turn now to the questions posed previously. We asked for the probability that a young adult white American will be faster than his African American counterpart, both being selected randomly. The question reduces to finding the probability that the white has a higher sprinting AQ. The probability that the black has an AQ between y and y + dy is PB(y)dy. Let g(y) be the probability that the white has an AQ > y. That is,
The conditional probability that the white has the higher AQ given that the black has an AQ between y and y + dy is g(y)PB(y)dy. Integration over all values of y gives the probability, p, that the white has the higher AQ. That is,
Using Δ = 0.82 in (6) returns the value 0.28 as the probability that the white is faster. Thus, a random white has a reasonable chance of out-sprinting a random black. However, as competition gets stiffer, athletes move into the high-end tail of the AQ distribution, where whites are badly outnumbered.
Next we looked at a hypothetical integrated high school with 2000 students, 25 percent black and 75 percent white. We asked for the most likely racial composition of the starting five on the basketball team. Assuming half the students are male, we will fill 5 slots in rank order of basketball AQ from a pool of 750 whites and 250 blacks. Being a bit more general, suppose NS slots are to be filled in order of AQ from a pool of NB blacks and NW whites. All those with an AQ equal to or greater than some value, λ, will be selected. The pool sizes, NB and NW, the number of available slots, NS, and the AQ gap, Δ, determine the cutoff AQ. The variables are related as follows:
The first term on the left side of (7) represents the number of blacks who make the cutoff; the second term is the corresponding number of whites. The two terms add up to the number of slots, NS. The cutoff AQ, λ, is obtained by solving (7) numerically. We then evaluate each term the left side of (7) to get the number of blacks and whites who make the cutoff. For the case at hand, NB = 250, NW = 750, and NS = 5. For Δ we used the value 0.87 SD obtained from the analysis of the NBA data, and found λ = 2.14 SD. That is, the cutoff AQ is 2.14 SD above the black mean. The first term on the left of (7) evaluates to 4.03, the second to 0.97. Consequently, the most probable makeup of the starting five would be 4 blacks and 1 white. If we perform the same analysis for a half white school of the same size, the starting five would most probably be all black.
The next question asked to what size the NBA would need grow to make it half white? The question may be rephrased as: What value of NS will make the 2 terms on the left-hand side of (7) equal to each other? Doing the arithmetic, we find the NBA would have to grow to approximately 860,000 and field almost 72,000 twelve-man teams to obtain a league half white and half black.
Finally, we asked for the most probable racial makeup of the Olympic Dream Team. This is like the high school starting five question with different input. The pools, NB and NW, are the respective numbers of black and white American men between the ages of 20 and 35. The number of slots, NS is 12. Again using the basketball AQ gap of 0.87 we found the most probable Dream Team would have 11 blacks and 1 white. The team selected in January 2000 had precisely this composition. It is worth noting that all the Dream Teamers were high up on the list of 150 high scorers used to assess the gap. We also note that the lone white on the team ranked last among his teammates on this list.
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We hope that readers will use the techniques developed here to explore other aspects of human biodiversity. If you do, please share your insights with La Griffe du Lion.
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