ction 8-2
Part 2 of 5

In a recent court case it was found that during a period of 11 years 896 people were selected for grand jury duty and of them were from the same ethnicity. Among the people eligible for grand jury duty, 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. B.

C.
D.

E.

F.

What is the test statistic?

(Round to two decimal places as needed.)

Ask by Gray Garrett. in the United States

Mar 21,2025

Question

ction 8-2
Part 2 of 5

In a recent court case it was found that during a period of 11 years 896 people were selected for grand jury duty and of them were from the same ethnicity. Among the people eligible for grand jury duty, 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. B.

C.
D.

E.

F.

What is the test statistic?

(Round to two decimal places as needed.)

Ask by Gray Garrett. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

To conduct the hypothesis test, we need to follow these steps:

1. **Identify the null and alternative hypotheses**:
   - The null hypothesis ($$ H_0 $$) states that the proportion of the selected grand jury members from the specific ethnicity is equal to the proportion of eligible members from that ethnicity, which is $$ 0.789 $$.
   - The alternative hypothesis ($$ H_1 $$) states that the proportion of the selected grand jury members from the specific ethnicity is less than $$ 0.789 $$, indicating a bias against this ethnicity.

Therefore, the hypotheses are:
   - $$ H_0: p = 0.789 $$
   - $$ H_1: p < 0.789 $$

This corresponds to option **E**.

2. **Calculate the test statistic**:
   - We can use the formula for the test statistic for a proportion:
   $$
   z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
   $$
   where:
   - $$ \hat{p} $$ is the sample proportion,
   - $$ p_0 $$ is the population proportion under the null hypothesis,
   - $$ n $$ is the sample size.

Given:
   - $$ n = 896 $$
   - $$ \hat{p} = \frac{0.35 \times 896}{896} = 0.35 $$
   - $$ p_0 = 0.789 $$

Now, we can calculate the test statistic $$ z $$. Let's compute it step by step:

1. Calculate $$ \hat{p} $$:
   $$
   \hat{p} = \frac{0.35 \times 896}{896} = 0.35
   $$

2. Calculate the standard error:
   $$
   SE = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.789(1 - 0.789)}{896}}
   $$

3. Substitute into the $$ z $$ formula:
   $$
   z = \frac{0.35 - 0.789}{SE}
   $$

Let's calculate the standard error and then the test statistic $$ z $$.
Calculate the value by following steps:
- step0: Calculate:
  $$0.35-0.789$$
- step1: Subtract the numbers:
  $$-0.439$$
Calculate or simplify the expression $$ \sqrt((0.789*(1-0.789))/896) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{\frac{\left(0.789\left(1-0.789\right)\right)}{896}}$$
- step1: Remove the parentheses:
  $$\sqrt{\frac{0.789\left(1-0.789\right)}{896}}$$
- step2: Subtract the numbers:
  $$\sqrt{\frac{0.789\times 0.211}{896}}$$
- step3: Multiply the numbers:
  $$\sqrt{\frac{0.166479}{896}}$$
- step4: Divide the terms:
  $$\sqrt{\frac{166479}{896000000}}$$
- step5: Use the properties of radicals:
  $$\frac{\sqrt{166479}}{\sqrt{896000000}}$$
- step6: Simplify the expression:
  $$\frac{\sqrt{166479}}{8000\sqrt{14}}$$
- step7: Multiply by the Conjugate:
  $$\frac{\sqrt{166479}\times \sqrt{14}}{8000\sqrt{14}\times \sqrt{14}}$$
- step8: Multiply the numbers:
  $$\frac{\sqrt{2330706}}{8000\sqrt{14}\times \sqrt{14}}$$
- step9: Multiply the numbers:
  $$\frac{\sqrt{2330706}}{112000}$$
Calculate or simplify the expression $$ (-0.439)/(0.013631) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(-0.439\right)}{0.013631}$$
- step1: Remove the parentheses:
  $$\frac{-0.439}{0.013631}$$
- step2: Convert the expressions:
  $$\frac{-\frac{439}{1000}}{\frac{13631}{1000000}}$$
- step3: Multiply by the reciprocal:
  $$-\frac{439}{1000}\times \frac{1000000}{13631}$$
- step4: Reduce the numbers:
  $$-439\times \frac{1000}{13631}$$
- step5: Multiply:
  $$-\frac{439\times 1000}{13631}$$
- step6: Multiply:
  $$-\frac{439000}{13631}$$
Now we have the necessary calculations:

1. **Standard Error (SE)**:
   $$
   SE \approx 0.013631
   $$

2. **Difference in proportions**:
   $$
   0.35 - 0.789 \approx -0.439
   $$

3. **Test Statistic (z)**:
   $$
   z = \frac{-0.439}{0.013631} \approx -32.21
   $$

Thus, the test statistic $$ z $$ is approximately $$ -32.21 $$.

### Summary of Results:
- **Null Hypothesis**: $$ H_0: p = 0.789 $$
- **Alternative Hypothesis**: $$ H_1: p < 0.789 $$
- **Test Statistic**: $$ z \approx -32.21 $$

Next, we would typically find the P-value associated with this test statistic to draw a conclusion about the null hypothesis. Given that the test statistic is extremely low, we can expect the P-value to be very small.

Would you like me to calculate the P-value for this test statistic?
**Hypothesis Test:** - **Null Hypothesis ($$ H_0 $$)**: The proportion of selected grand jury members from the specific ethnicity is equal to the proportion of eligible members ($$ p = 0.789 $$). - **Alternative Hypothesis ($$ H_1 $$)**: The proportion of selected grand jury members from the specific ethnicity is less than the proportion of eligible members ($$ p < 0.789 $$). **Test Statistic ($$ z $$)**: Approximately $$ -32.21 $$ **Conclusion**: The test statistic $$ z \approx -32.21 $$ is significantly less than the critical value at a 0.01 significance level, leading to the rejection of the null hypothesis. This indicates that there is strong evidence to support the claim that the selection process is biased against allowing this ethnicity to sit on the grand jury.

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Statistics Questions