Answers without complete justification are not correct. Answer all questions usi

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Answers without complete justification are not correct. Answer all questions using complete sentences and applicable vocabulary. Be sure to define all symbols and to include relevant tables, plots and hypothesis tests. All conclusions should be backed up with statistical arguments 1. In 1965, the U.S. Supreme Court decided the case of Swain vs. Alabama. Swain, a black man, was convicted in Talladega County, Alabama, of raping a white woman. He was sentenced to death. The case was appealed to the Supreme Court on the grounds that there were no blacks on the jury; moreover, no black "within the memory of persons now living has ever served on any petit jury in any civil or criminal case tried in Talladega County, Alabama." The Supreme Court denied the appeal, on the following grounds. As provided by Alabama law, the jury was selected from a panel of about 100 persons. There were 8 blacks on the panel. (They did not serve on the jury because they were "struck," or removed, through a maneuver called peremptory challenges by the prosecution. Such challenges were until quite recently constitutionally protected.) The Supreme Court ruled that the presence of 8 blacks on the panel showed The overall percentage disparity has been small and reflects no studied attempt to include or exclude a specified number of blacks." That is, the Supreme Court ruled that the fact that there were only 8 blacks in the panel of 100 was within reasonable chance. At that time in Alabama, only men over the age of 21 were eligible for jury duty. There were 16,000 such men in Talladega County, of whom about 26% were black. What do you conclude about the Supreme Court's opinion?

Explanation / Answer

Out of 16000 eligible men, 26% were black, but in an actual panel of 100 only 8 were black, which apparently is not commensurate with 26%. Our job is to test statistically if this phenomenon is by chance only, as observed by the Supreme Court.

Let X = Number of blacks in the panel of 100.

Then, X ~ B(n, p), where n = sample size and p = probability that a black is included in the panel, which is also equal to the proportion of blacks eligible in the population.

Claim :

8 blacks in a panel of 100 is just by chance.

Hypotheses:

Null H0 : p = p0 = 0.26 Vs HA : p < 0.26

Test Statistic:

Z = (pcap - p0)/{p0(1 - p0)/n} where pcap = sample proportion and n = sample size.

Calculations:

pcap = 8/100 = 0.08, n = 100. So,

Z = (0.08 – 0.26)/{(0.26 x 0.74)/100}

   = - 1.8/{(0.26 x 0.74)

= - 1.8/0.4386

= - 4.104

Distribution, Critical Value and p-value:

Under H0, distribution of Z can be approximated by Standard Normal Distribution, provided

np0 and np0(1 - p0) are both greater than 10.

So, given a level of significance of %, Critical Value = lower % of N(0, 1), and p-value = P(Z < Zcal)

Using Excel Functions of N(0, 1), and taking = 5%, Critical Value = - 1.645 and p-value = P(Z < - 4.104) = 0.00002.

Decision Criterion (Rejection Region):

Reject H0, if Zcal < Zcrit or if p-value < .

Decision:

Since Zcal < Zcrit, p-value < , H0 is rejected.

Conclusion :

There is not enough evidence to suggest that the claim is valid. => 8 out 100 cannot be taken to be by chance if the actual proportion is 26%. ANSWER

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