A friend is representing a client on appeal, after a criminal trial in which it appears that jury selection was racially biased.
The jury pool consisted of 30 people, in 4 racial groups. The prosecution used peremptory challenges to eliminate 10 of these people from the pool. The number of people and number of actual challenges in each racial group were, respectively:
A: 10, 1 B: 10, 4 C: 6, 4 D: 4, 1 total: 30 in pool, 10 challenges
The defendant was from racial group C and the victims from racial groups A and D, so the concern a priori is whether group C is over-challenged and groups A and D under-challenged. Legally (IIUC; IANAL), the defense does not need to prove racial bias, but merely to show that the data seem to indicate bias, which then puts the burden on the prosecution to explain each challenge non-racially.
Is the following analysis correct in its approach? (I think the calculations are fine.):
There are nCr(30,10) = 30,045,015 distinct sets of 10 pool members. Of these distinct sets, I count that 433,377 sets include both (no more than 2 members of group A and D combined) and (no fewer than 4 members of group C).
Thus the chance of reaching the observed level of apparent bias favoring groups A and D over group C (where favoring means not including in the set of 10 challenges) would be the ratio of these, 433/30045 = 1.44%.
Thus the null hypothesis (no such bias) is rejected at the 5% significance level.
If this analysis is methodologically correct, what would be the most succinct way to describe it to a court, including an academic / professional reference (i.e. not Wikipedia)? While the argument seems simple, how can one most clearly and succinctly demonstrate to the court that it's correct, not shenanigans?
Update: This question was under consideration as a tertiary argument in an appeal brief. Given the technical complexity (from the lawyer's viewpoint) of the discussion here and the apparent lack of legal precedent, the lawyer has chosen not to raise it, so at this point the question is mostly theoretical / educational.
To answer one detail: I believe that the number of challenges, 10, was set in advance.
After studying the thoughtful and challenging answers and comments (thanks, all!), it seems that there are 4 separate issues here. For me, at least, it would be most helpful to consider them separately (or to hear arguments why they are not separable.)
1) Is the consideration of the races of both defendant and victims, in the jury pool challenges, of legal concern a priori? The goal of the appeal argument would merely be to raise reasonable concern, which could lead to a judicial order that the prosecution state the reason for each individual challenge. This does not appear to me to be a statistical question, but rather a social / legal one, which is at the lawyer's discretion to raise or not.
2) Assuming (1), is my choice of an alternative hypothesis (qualitatively: bias against jurors who share the defendant's race, in favor of those who share the victims' races) plausible, or is it impermissibly post hoc? From my lay perspective, this is the most perplexing question -- yes, of course one would not raise it if one did not observe it! The problem, as I understand, is selection bias: one's tests should consider not just this jury pool but the universe of all such jury pools, including all the ones where the defense did not observe a discrepancy and therefore were not tempted to raise the issue. How does one address this? (For example, how does Andy's test address this?) It appears, though I may be wrong about this, that most respondents are not troubled by potentially post-hoc 1-tailed tests for bias solely against the defendant's group. How would it be methodologically different to simultaneously test bias for victim groups, assuming (1)?
3) If one stipulates my choice of a qualitative alternative hypothesis as stated in (2), then what is an appropriate statistic for testing it? This is where I am most puzzled by the responses, because the ratio that I propose seems to be a slightly more conservative analog of Andy's test for the simpler "bias against C" alternative hypothesis (more conservative because my test also counts all cases further out in the tail, not just the exact observed count.)
Both tests are simple counting tests, with the same denominator (same universe of samples), and with numerators corresponding precisely to the frequency of those samples which correspond to the respective alternative hypotheses. So @whuber, why is it not identically as true of my counting test as of Andy's that it "can be based on stipulated null [same] and alternative [as described] hypotheses and justified using the Neyman-Pearson lemma"?
4) If one stipulates (2) and (3), are there references in case law which would convince a skeptical appeals court? From the evidence to date, probably not. Also, at this stage of appeal there's no opportunity for any "expert witness", so references are everything.