K In a recent court case it was found that during a period of 11 years 871 people were selected for grand jury duty and $41 \%$ of them were from the same ethnicity. Among the people eligible for grand jury duty, $80.1 \%$ were of this ethnicity. Use a 0.05 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 Incorrect: 1 Friat is the conciusion on ule Inin nypunesise Fail to reject the null hypothesis because the $\mathrm{P}$-value is greater than the significance level, $\alpha$. Reject the null hypothesis because the P-value is less than or equal to the significance level, $\alpha$. Fail to reject the null hypothesis because the P-value is less than or equal to the significance level, $\alpha$ Reject the null hypothesis because the P-value is greater than the significance level, $\alpha$ Does the jury selection system appear to be fair? A. 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: B. 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. 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.
Solution
Step 1 ：Define the null hypothesis as the selection process is not biased, meaning the proportion of people from this ethnicity selected for grand jury duty is equal to the proportion of people from this ethnicity in the population eligible for grand jury duty.
Step 2 ：Define the alternative hypothesis as the selection process is biased, meaning the proportion of people from this ethnicity selected for grand jury duty is not equal to the proportion of people from this ethnicity in the population eligible for grand jury duty.
Step 3 ：Calculate the sample proportion, which is the number of successes divided by the sample size. In this case, the sample size is 871 and the number of successes is \(0.41 \times 871\), so the sample proportion is 0.41.
Step 4 ：Calculate the standard error, which is the square root of the product of the population proportion, the complement of the population proportion (1 - population proportion), and the reciprocal of the sample size. In this case, the population proportion is 0.801, so the standard error is approximately 0.0135.
Step 5 ：Calculate the test statistic, which is the difference between the sample proportion and the population proportion, divided by the standard error. In this case, the test statistic is approximately -28.90.
Step 6 ：Calculate the P-value, which is the probability of observing a test statistic as extreme as the one we calculated, assuming the null hypothesis is true. In this case, the P-value is approximately 0.
Step 7 ：Compare the P-value to the significance level. If the P-value is less than the significance level, reject the null hypothesis. If the P-value is greater than the significance level, fail to reject the null hypothesis. In this case, the P-value is less than the significance level of 0.05, so we reject the null hypothesis.
Step 8 ：Conclude that 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.
Step 9 ：Final Answer: \(\boxed{\text{B. There is sufficient evidence to support the claim that the selection process is biased against allowing this ethnicity to sit on the grand jury. The jury selection process appears to be unfair.}}\)
