Hypothesis testing question

Question

The article “Statistical evidence of discrimination” (J. Ameri. Stat. Assoc., 1982, 773-83) discussed the court case of Swain v Alabama (1965), in which it was alleged that there was discrimination against blacks in grand jury selection. Census data suggested that 25% of those eligible for grand jury service were black, yet a random sample of 1050 individuals called to appear for possible duty yielded only 177 blacks. Given this observation the court believed that there wasn’t enough evidence to establish a prima facie (without further evidence) case. Would you agree with the court’s decision?

Answer 1

I suppose the interesting question is whether or not the sample of people in the jury room one morning fairly represented the population of eligible jurors from which it was drawn. The proportion of blacks in the sample is specifically of interest.

We naively expect the number of blacks in the sample to be "close to" $0.25 \times 1050 = 262.5$. Since we observed only 177 blacks in the sample, we only need determine the probability of observing that many or fewer of blacks in a sample of size 1050 under the hypothesis that the sampling method used was unbiased.

We're given:

Sample size, n = 1050. Number of blacks in sample, k = 177. Proportion of blacks in population (of those eligible for jury service) p = 25%.

Strictly speaking, drawing a sample of people for jury service, necessarily involves selection without replacement. Sampling without replacement implies that the number of blacks in the sample would have a hypergeometric distribution. Unfortunately, however, we are not given the total number of people eligible for jury service nor are we given the total number of blacks eligible for jury service (although, I assume the census data would have this info) so we cannot use the hypergeometric distribution directly. Happily, it is probably safe to assume that the total number of eligible people and the total number of eligible blacks are both large compared to the sample size. Under these conditions (and noting that p is not close to either 0 nor close to 1) the binomial distribution or even the standard normal can be used to approximate the hypergeometric.

Assume that k has a (approximately) binomial distribution with n = 1050 and p = 0.25 $k \sim BIN(n=1050, p = 0.25)$ . In that case, the probability of drawing an unbiased simple random sample of size 1050 that contained only 177 blacks would be exceedingly small ($p \approx 1.3561\times10^{-10}$) and we would conclude that the sampling method was NOT fair.

The issue is further complicated however, by the fact that otherwise seemingly eligible jurors can "negatively self select".