$$ 18-2 $$ Part 2 of 5 Points: 0 of 1 In a recent court case it was found that during a period of 11 years 896 people were selected for grand jury duty and $$ 35 \% $$ of them were from the same ethnicity. Among the people eligible for grand jury duty, $$ 78.9 \% $$ 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. Which of the following is the hypothesis test to be conducted? A. $$ \mathrm{H}_{0}: \mathrm{p}0.789 $$ B. $$ H_{0}: p=0.789 $$ $$ H_{1}: p=0.789 $$ C. $$ H_{0}: p \neq 0.789 $$ $$ \mathrm{H}_{1}: \mathrm{p}=0.789 $$ D. $$ \mathrm{H}_{0}: \mathrm{p}

Question

$$ 18-2 $$ Part 2 of 5 Points: 0 of 1 In a recent court case it was found that during a period of 11 years 896 people were selected for grand jury duty and $$ 35 \% $$ of them were from the same ethnicity. Among the people eligible for grand jury duty, $$ 78.9 \% $$ 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. Which of the following is the hypothesis test to be conducted? A. $$ \mathrm{H}_{0}: \mathrm{p}>0.789 $$ B. $$ H_{0}: p=0.789 $$ $$ H_{1}: p=0.789 $$ C. $$ H_{0}: p \neq 0.789 $$ $$ \mathrm{H}_{1}: \mathrm{p}=0.789 $$ D. $$ \mathrm{H}_{0}: \mathrm{p}<0789 $$ $$ H_{1} \cdot p=0.789 $$ E. $$ \mathrm{H}_{0}: \mathrm{p}=0.789 $$ $$ \mathrm{H}_{1}, \mathrm{p}<0789 $$ F. $$ H_{0}: \mathrm{p}=0.789 $$ $$ \mathrm{H}_{1}, \mathrm{p} \neq 0.789 $$ What is the test statistic? $$ z= $$ $$ \square $$ (Round to two decimal places as needed.)

UpStudy AI · Accepted Answer

To test the claim that the selection process is biased against allowing a specific ethnicity to sit on the grand jury, we need to set up a hypothesis test.

Given:
- Total number of people selected for grand jury duty: 896
- Percentage of people from the same ethnicity: 35%
- Percentage of people eligible for grand jury duty from this ethnicity: 78.9%

Let's denote:
- $$ p $$ as the proportion of people from the same ethnicity in the grand jury selection process.

The null hypothesis ($$ H_{0} $$) is that the selection process is not biased against allowing this ethnicity to sit on the grand jury, which means the proportion of people from this ethnicity in the grand jury selection process is equal to the proportion of people eligible for grand jury duty from this ethnicity.

The alternative hypothesis ($$ H_{1} $$) is that the selection process is biased against allowing this ethnicity to sit on the grand jury, which means the proportion of people from this ethnicity in the grand jury selection process is not equal to the proportion of people eligible for grand jury duty from this ethnicity.

Therefore, the correct hypothesis test is:
$$ H_{0}: p = 0.789 $$
$$ H_{1}: p \neq 0.789 $$

To calculate the test statistic, we use the formula for the z-test:
$$ z = \frac{\hat{p} - p_{0}}{\sqrt{\frac{p_{0}(1-p_{0})}{n}}} $$

where:
- $$ \hat{p} $$ is the sample proportion (35% or 0.35)
- $$ p_{0} $$ is the null hypothesis proportion (78.9% or 0.789)
- $$ n $$ is the sample size (896)

Substitute the values into the formula to calculate the test statistic.
The test statistic is approximately -32.21 (rounded to two decimal places).

Now, we need to calculate the P-value to determine the significance of the test result. The P-value represents the probability of observing a test statistic at least as extreme as the one we obtained, assuming that the null hypothesis is true.

Since the P-value is not provided, we can conclude that the null hypothesis is rejected at a 0.01 significance level, indicating that the selection process is biased against allowing this ethnicity to sit on the grand jury.
**Hypothesis Test:** - **Null Hypothesis ($$ H_{0} $$):** The selection process is not biased against the ethnicity, so $$ p = 0.789 $$. - **Alternative Hypothesis ($$ H_{1} $$):** The selection process is biased against the ethnicity, so $$ p \neq 0.789 $$. **Test Statistic:** $$ z = -32.21 $$ (rounded to two decimal places) **Conclusion:** The P-value is extremely small, leading to the rejection of the null hypothesis at a 0.01 significance level. This indicates that the selection process is significantly biased against allowing the specified ethnicity to sit on the grand jury.

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