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 180,950 persons eligible for jury duty, of whom 143,150 were Mexican Americans. Of the 881 people selected for jury duty, 346 were Mexican Americans.

(a) What proportion of eligible voters were Mexican Americans? Let this value be p_o. (Round your answer to four decimal places.)

(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

From the given data

Total number of eligible voters for jury duty = 180,950

Total number of eligible voters for jury duty were Mexican Americans = 143,150

(a)

The proportion of eligible voters were Mexican Americans

(b)

Given the sample size, n = 881 are the eligible voters selected for the jury duty

Of the selected, total number of eligible voters selected for the jury duty were Mexican Americans = 346

Sample proportion (probability) of randomly selected juror is a Mexican American is

Note that we use z-test for single proportion to test the hypothesis

Null Hypothesis

versus

Alternative Hypothesis

The test statistic is

P-value = p(Z < z) < .00001

Since the p-value < 0.01 (level of significance), we reject null hypothesis stated and infer that the circumstance does not constitute a proof of discrimination for selecting juries in a county in Texas

(c)

Please provide details of sample size n1, n2 to solve the problem