Question

Castaneda v. Partida is an important court case in which statistical methods were used as part of a legal argument. When reviewing this case, the Supreme Court used the phrase "two or three standard deviations" as a criterion for statistical significance. This Supreme Court review has served as the basis for many subsequent applications of statistical methods in legal settings. (The two or three standard deviations referred to by the Court are values of the z statistic and correspond to P-values of approximately 0.05 and 0.0026.) In Castaneda the plaintiffs alleged that the method for selecting juries in a county in Texas was biased against Mexican Americans. For the period of time at issue, there were 181,825 persons eligible for jury duty, of whom 143,300 were Mexican Americans. Of the 869 people selected for jury duty, 338 were Mexican Americans.

(b) Let p be the probability that a randomly selected juror is a Mexican American. The null hypothesis to be tested is H_o: p = p_o. Find the value of p̂ for this problem, compute the z statistic, and find the P-value. What do you conclude? (A finding of statistical significance in this circumstance does not constitute a proof of discrimination. It can be used, however, to establish a prima facie case. The burden of proof then shifts to the defense.) (Use α = 0.01. Round your test statistic to two decimal places and your P-value to four decimal places.)

z
P-value

(c) We can reformulate this exercise as a two-sample problem. Here we wish to compare the proportion of Mexican Americans among those selected as jurors with the proportion of Mexican Americans among those not selected as jurors. Let p₁ be the probability that a randomly selected juror is a Mexican American, and let p₂ be the probability that a randomly selected nonjuror is a Mexican American. Find the z statistic and its P-value. (Use α = 0.01. Round your test statistic to two decimal places and your P-value to four decimal places.)

z
P-value

Answer 1

here,

there were 181,825 persons eligible for jury duty, of whom 143,300 were Mexican Americans.

so the population proportion,

P0 = 143,300 / 181,825 = 0.7881

Of the 869 people selected for jury duty, 338 were Mexican Americans.

so the sample proportion,

P = 338 / 869 = 0.3890

b) Hypothesis test

$\\H_0:p=p_0 \\H_a:p\ne p_0$

we know that the test statistic,

Z = (P - P0) / sqrt(P0(1 - P0) / n) = (0.3890 - 0.7881) / sqrt(0.7881(1 - 0.7881) / 869) = -28.79

P - value = P(Z < -28.79) + P(Z > 28.79) = 0.0000

Since p-value is less than 0.01, reject the null hypothesis. We can conclude that proportion of Mexican Americans has changed.

c) Let p₁ be the probability that a randomly selected juror is a Mexican American.

P1 = 338 / 869 = 0.3890

let p₂ be the probability that a randomly selected nonjuror is a Mexican American.

P2 = (143300 - 338) / (181825 - 869) = 0.7900

we know that test statistic,

Z = (P1 - P2) / sqrt(P1(1-P1)/n1 + P2(1-P2)/n2) = -24.21

P - value = P(Z < -24.21) + P(Z > 24.21) = 0.0000

please revert if any doubts