Question

in a recent court case it was found that during a period of 11 years 897 people were selected for grand jury duty and 42% of them were from the same ethnicity. among the people eligible for grand jury duty, 80.7% were of this ethnicity. use a 0.01 significance level to test the claim that the selection process is biased against allowing this ethnicity to sit on the grand jury. identify the null hypothesis, alternative hypothesis, test statistic, p - value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. use the p - value method and the normal distribution as an approximation to the binomial distribution.
(round to three decimal places as needed.)
what is the conclusion on the null hypothesis?
reject the null hypothesis because the p - value is greater than the significance level, α.
reject the null hypothesis because the p - value is less than or equal to the significance level, α.
fail to reject the null hypothesis because the p - value is less than or equal to the significance level, α.
fail to reject the null hypothesis because the p - value is greater than the significance level, α.
does the jury selection system appear to be fair?
a. there is sufficient evidence to support the claim that the selection process is biased against allowing this ethnicity to sit on the grand jury. the jury selection process appears to be unfair.
b. there is not sufficient evidence to support the claim that the selection process is biased against allowing this ethnicity to sit on the grand jury. the jury selection process appears to be unfair.
c. there is sufficient evidence to warrant rejection of the claim that the selection process is biased against allowing this ethnicity to sit on the grand jury. the jury selection process appears to be fair.
d. there is not sufficient evidence to warrant rejection of the claim that the selection process is biased against allowing this ethnicity to sit on the grand jury. the jury selection process appears to be fair.

Explanation:

Step1: State the null and alternative hypotheses

The null hypothesis \(H_0: p = 0.807\) (no bias, proportion is equal to the eligible proportion). The alternative hypothesis \(H_1: p<0.807\) (biased against, proportion is less than the eligible proportion).

Step2: Calculate the sample proportion \(\hat{p}\)

\(\hat{p}=\frac{x}{n}\), where \(x = 0.42\times897=376.74\approx377\), \(n = 897\). So \(\hat{p}=\frac{377}{897}\approx0.420\)

Step3: Calculate the test - statistic \(z\)

The formula for \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Substitute \(p = 0.807\), \(\hat{p}=0.420\), \(n = 897\)
\(z=\frac{0.420 - 0.807}{\sqrt{\frac{0.807\times(1 - 0.807)}{897}}}\)
First, calculate the denominator: \(\sqrt{\frac{0.807\times0.193}{897}}=\sqrt{\frac{0.155751}{897}}\approx\sqrt{0.0001736}\approx0.0132\)
Then \(z=\frac{- 0.387}{0.0132}\approx - 29.32\)

Step4: Find the P - value

For a left - tailed test with \(z\approx - 29.32\), the P - value is extremely small (close to \(0\))
Since the P - value \(<\alpha=0.01\)

Step5: Conclusion on the null hypothesis

Reject the null hypothesis because the P - value is less than or equal to the significance level, \(\alpha\)

Step6: Final conclusion about the original claim

There is sufficient evidence to support the claim that the selection process is biased against allowing this ethnicity to sit on the grand jury. The jury selection process appears to be unfair.

Answer:

• Conclusion on the null hypothesis: Reject the null hypothesis because the P - value is less than or equal to the significance level, \(\alpha\)
• Final conclusion: A. There is sufficient evidence to support the claim that the selection process is biased against allowing this ethnicity to sit on the grand jury. The jury selection process appears to be unfair.