Question

( Castaneda v Partida: frequentist treatment An interesting and celebrated example of application of a Z-test for a proportion is contained in a verdict of the U.S. Supreme Court. The beginning of the story is told in a document of Partida seeks habeas corpus relief contending that he was denied due process and equal pro- tection of law because the grand jury of Hidalgo County, Teras, which indicted him, was a lower court unconstitutionally underrepresented by Mezican-Americans. This went to the U.S. Supreme court which ruled: CASTANEDA .PARTIDA, Decided March 23, 1977 ry discrimination issue, such as the sta- tistical disparities (the county population u as 79% Menan-American, but, over an 11-year period, only 39% of those summoned for grind jury service were Merican-American), the method of jury selection, and any other relevant testimony as to the manner in which the se- lection process twas implemented, the proof offered by respondent was sufficient to demonstrate a prima facie case of intentional discrimination in grand jury selection, facie case of discrimination in grand jury selection. Under this standard, the proof offered by respondent was sufficient to demonstrate a prima e statistical evidence is given in a footnote. We again cite the original text and we will then teproduce the reasoning using our mathematical notation. l Footnote ǐ지 Ifthe jurors wore drawn randomly from tho general population, thon the numbr of Maican-Americans in the sample could be modeled by a binomial distribution. See Finkelsteln, The Application of Statistical Decision Theory to the Jary Discrimination Cases, 80 Harv. L. Rev. 338, 353-356 (1966). See generally P. Hoel, Introdaction to Mathematical Statistica 58-61·79.86 (4th ed. 197)i P, Moeleller、R. Rourke, nona, proladility with Saakal Application« 130 146, 270291 (2ded. į970). Givea that79.1%of the popriationis Mexics American,the expect a aumber of Mexican-Americans among the 870 persons summoned to serve aa grand juroes over the 1l-year period is approximately 688. The observed mamber is 339. Of course, is any given drewwlng some Buctuatios from the expected อเ¡mber is predicted. The important point, however, is that the tatatical modal tam tat ge tilla at a tuda drawing aro likely to fali in the vicinity of the expoeted value. Soe F. Mosteller, R. Rourko, &G.Thomas, supes, st 270-290. The measure of [430 U.S. 482, 497 the predicted fluctuations from the expected value is the standard deviation., deined for the hinotaial distribntion as the square root of the prodact of the total amher in the sample (here 870) tiues the probability of selecting a Mexicas-Amerieaa (0.791) times the probability of selecting a noh-MexicasAwerican (0.209). Id, at 213. Thus, in th samples, if the difference between the expected valse and the observed number is greater deviation, theu the hypothesis that the jury drawing was random would be suspect to a social scientist. The 11 is case the staadard deviation is aspproxisuately 12. As a geueral rule for soch Jarge than two or three standard -year data here reflect a diferezce between the expected and observed number of Mexican- Americans of spproximately 29 s varse deviations. A detailed calculation reveals that the likelhood that such a substantial departure from the expectesd occur by chance is less than 1 in 10140. The data for the 2 1/2-year period during which the State District selection process similarly support the inference that the excluxion of Mexicsu-Americans did not. occur by chance. Of 220 persous called to serve as grand jurors, only 100 were MMexiean-Americans. The expected Mexicau-American representation is approximately 174 and the standard deviation, as caleulated fron the binonial nodel, is approximately six. The discrepancy between the expocted and obscrved valuces is more than 12 standard ttp://cases justia.com/us-court-of-appeals/F2/524/481/

Answer 1

Uhen the pro hahỉlity of the number of JuYoYs being a pe cPied number of slondax deviations owa fm the expecte nombe, the statistics used at (he Proubility acua eutccme exceeds expedteu outcome + Specified nombex x tanda devialions aisoning Jke noll hygroftiesis o Stedus vos hold via Scme 3ven Status oo hold via Some iven Jistbi botion, This stadistic is the p-value. tis s used to vahdale the ag uments in he