Question

In a recent court case it was found that during a period of 11 years 888 people were selected for grand jury duty and 39% of them were from the same ethnicity. Among the people eligible for grand jury​ duty, 79.2​% 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.

What is the test​ statistic?

zequals=nothing

​(Round to two decimal places as​ needed.)

What is the​ P-value?

​P-valueequals=nothing

​(Round to four decimal places as​ needed.)

What is the conclusion on the null​ hypothesis?

RejectReject

the null hypothesis because the​ P-value is

less than or equal toless than or equal to

the significance​ level,

alphaα.

RejectReject

the null hypothesis because the​ P-value is

greater thangreater than

the significance​ level,

alphaα.

Fail to rejectFail to reject

the null hypothesis because the​ P-value is

greater thangreater than

the significance​ level,

alphaα.

Fail to rejectFail to reject

the null hypothesis because the​ P-value is

less than or equal toless than or equal to

the significance​ level,

alphaα.

Does the jury selection system appear to be​ fair?

A.

There

isis

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 notis 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

isis

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 notis 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.

Answer 1

To test the claim of bias in grand jury duty selection, we set up a hypothesis test. The test statistic is calculated using the z-score for proportions, and the resulting P-value is compared to the significance level to determine if we should reject the null hypothesis.

Step-by-step explanation:

To address the claim that the selection process for grand jury duty is biased against a particular ethnicity, we will conduct a hypothesis test using a normal distribution as an approximation to the binomial distribution because of the large sample size.

Hypotheses

The null hypothesis (H0): The selection process is fair, and the proportion of the selected jurors of the ethnicity in question is equal to the proportion in the population eligible for jury duty (p = 0.792).

The alternative hypothesis (Ha): The selection process is biased against the ethnicity, implying the proportion of selected jurors of the ethnicity is less than the proportion in the population (p < 0.792).

Test Statistic

To calculate the test statistic, we use the formula for the z-score in a proportion test:
z = (p-hat - p) / sqrt(p(1-p)/n), where p-hat is the sample proportion, p is the population proportion, and n is the sample size.

Thus, the test statistic is z = (0.39 - 0.792) / sqrt(0.792(1-0.792)/888), which should be calculated and rounded to two decimal places.

P-value

The P-value can be found by looking up the calculated z-score in a z-table or using a statistical software. It should be compared to the significance level (alpha = 0.01).

Conclusion

If the P-value is less than the significance level, we reject the null hypothesis, indicating evidence of bias. If the P-value is greater, we fail to reject the null hypothesis, indicating insufficient evidence of bias.