91Ó°ÊÓ

The grand jurors were chosen from a population that is 79.1 percent Mexican American, but only 100 grand jurors were Mexican American. The expected value of a binomial random variable X with

parameters 220 and 0.791 is

\(\begin{aligned}E\left( X \right) = 220 \times 0.791\\ = 174.02\end{aligned}\).

This is much larger than the observed value of X =100. After all, there is positive probability of\(X = x\)for all x = 0,.., 220. Let P stand for the proportion of Mexican Americans among all grand jurors that would be chosen under the current system being used. The court assumed that X had the binomial distribution with parameters n =220 and p, conditional on\(P = p\). We should then be interested in whether P is substantially less than the value 0.791, which represents impartial juror choice.

For example, suppose that one define discrimination to mean

that\(P \le 0.8 \le 0.791 = 0.6328\).

One would like to compute the conditional probability

of\(P \le 0.8\)given\(P \le 0.8\)given.\(X = 100\)

Suppose that the distribution of P prior to observing Xwas the beta distribution

with parameters\(\alpha \)and\(\beta \).

After choosing values of\(\alpha \)and\(\beta \)we could compute\({\rm P}\left( {p \le 0.6328|X = 100} \right)\)and

decide how likely it is that there was discrimination. One will see how to choose \(\alpha \) and \(\beta \) after we learn how to compute the expected value of a beta random variable.

A statistical model consists of an identification of random variables of interest (both observable and only hypothetically observable),a specification of a joint distribution for the observable random variables, the identification of any parameter of those distribution that are assumed unknown and possibly hypothetically observable, and a specification for a joint distribution for a unknown parameters.

From the given data and the definition of statistical model:

The random variables of interest are the observable number X of Mexican-American grand jurors and the hypothetically observable (parameter) P.

The conditional distribution of X given\(P = p\)is the binomial distribution with parameters 220 and p. Also, P has the beta distribution with parameters\(\alpha \)

and \(\beta \), which have not yet been specified.