If this probability is equal to or lower than a low-probability cutpoint (usuall
ID: 3255282 • Letter: I
Question
If this probability is equal to or lower than a low-probability cutpoint (usually .05), then reject the null hypothesis as it is unlikely to be correct, and favor the alternative hypothesis. If this probability is higher than the low-probability cutpoint, retain the null hypothesis.
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, the defense asserted that 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: According to Alabama law, the jury was selected from a panel of 100 persons. The panel included eight black men, none of whom served on the jury because they were removed through peremptory challenges by the prosecution. The Supreme Court ruled that the presence of eight blacks on the panel showed that “the overall percentage disparity has been small and reflects no studied attempt to include or exclude a specified number of (blacks).”
In 1965 in Alabama, only men over the age of 21 were eligible for jury duty. Of 16,000 such men in Talladega County, 26% were black.
The Supreme Court effectively ruled that the small proportion of blacks included in the original panel of prospective jurors was representative of the larger population of blacks in Talladega County, Alabama—even though the proportion of blacks in the panel was not identical to the proportion in the population. What the Supreme Court did not do was to evaluate its ruling statistically. This is what you are going to do. Answer the following questions:
1. State the null and alternative hypotheses in both words and symbols.
2. Calculate the sample statistic that will serve as the estimate of the parameter specified in your null/alternative hypothesis statements and its standard error.
3. Calculate the appropriate test statistic, state your decision to reject or retain the null hypothesis and the rationale for your decision.
4. What do you conclude about the Supreme Court’s ruling? Does the ruling reflect sound statistical reasoning? Should the Supreme Court have ruled differently, or do the results of your statistical test support the Court’s original ruling? Explain.
Explanation / Answer
Let's denote the population proportion of black men with p and that of panel by p'
H0: p' - p = 0
Ha: p' - p < 0
N = 100
p' = 8/100 = 0.08
p = 0.26
Standard error, SE = (p*(1-p)/N)0.5 = (0.26*(1-0.26)/100)0.5 = 0.0439
Calculating test statistic:
z = (p'-p)/SE = (0.08-0.26)/0.0439 = -4.1
Critical z-score for 95% confidence for a left tailed test is:
zc = -1.64
Since z<zc, so H0 is rejected in favor of Ha.
Thus the decisio of supreme court is wrong statistically and it should have ruled differently.