statistics - probability

Quote:

Originally Posted by Jason Bourne

Like usual I leave my homework until the very last minute (ahhhhh), any help is appreciated:

In 1965 the Supreme Court of the USA heard the case of Swain vs. Alabama. Swain was black and was convicted and sentenced to death for the rape of a white woman. He appealed on the grounds that there were no blacks on the jury and that no black `within the memory of persons now living has ever served on any petit jury. . . in Talladega County, Alabama'. The Supreme court rejected the appeal on the following grounds. In accordance with Alabama law the jury was selected from a panel of 100 people. There were 8 blacks on this panel. The Supreme Court ruled that the presence of 8 blacks on the panel showed that `the overall percentage disparity [between proportion of blacks on the panel and that in the population] has been small and reflects no studied attempt to include or exclude a specified number of blacks'. In Alabama at that time, only males over the age of 21 were eligible for jury service. There were about 16,000 such men in Talladega county: 26% of these were black.

(a) Calculate the probability that a random sample of 100 from this population would include 8 or fewer black men.

(b) Comment on the Supreme Court's judgement.

I will treat the county population as sufficiently large that we can treat the jury selection as thought it was sampling with replacement from a population of $16000$ with $26 \%$ black, and that the panel size is sufficiently large to use the normal approximation. The mean number of blacks on a panel of $100$ is $\mu=26$ , the standard deviation is $\sigma=\sqrt{100 \times 0.26 \times 0.74} \approx 4.39$

Then the probability of a panel with $8$ or fewer blacks assuming fair panel selection is:

$p(n_b \le 8) = P\left(\frac{8.5-26}{4.39}\right)\approx P(-3.99)\approx 0.0033 \%$

where $P$ is the cumulative standard normal distribution.

Now the number of blacks on the panel was so low that it would probably be wise to repeat this calculation using the binomial distribution directly, when:

$p(n_b \le 8) = \sum_{r=0}^8 b(r;100,0.26) \approx 0.00047 \%$

either way the probability is so low as to be negligable.

I will leave it to you to comment on the supreme courts judgement

(Note such a probability based argument would be inadmissible in front of a UK court)

RonL