Rhino viruses typically cause common colds. In a test of the effectiveness of echinacea, 43 of the 49 subjects treated with echinacea developed rhinovirus infections. In a placebo group, 92 of the 109 subjects developed rhinovirus infections. Use a 0.05 significance level to test the claim that echinacea has an effect on rhinovirus infections. Complete parts (a) through (c) below.

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**Title: Testing the Effectiveness of Echinacea on Rhinovirus Infections**

**Introduction:**
109 subjects were divided into two groups to test the effectiveness of echinacea in treating rhinovirus infections. In the echinacea group, 43 of the 49 subjects developed rhinovirus infections. In the placebo group, 92 of the 109 subjects developed rhinovirus infections. The following analysis uses a 0.05 significance level to evaluate the claim that echinacea affects the rate of rhinovirus infections.

**Steps and Analysis:**

1. **Compute the Test Statistic (z):**
   \[
   z = \_\_\_\_
   \]
   (Round to two decimal places as needed.)

2. **Identify the P-value:**
   \[
   \text{P-value} = \_\_\_\_
   \]
   (Round to three decimal places as needed.)

3. **Conclusion Based on the Hypothesis Test:**
   The P-value is \(\_\_\_\_\) the significance level of \(\alpha = 0.05\), so \(\_\_\_\_\) the null hypothesis. There \(\_\_\_\_\) sufficient evidence to support the claim that echinacea treatment has an effect.

4. **Construct the Confidence Interval:**
   The 95% confidence interval is:
   \[
   \_\_\_\_\ < (p_1 - p_2) < \_\_\_\_\ 
   \]
   (Round to three decimal places as needed.)

5. **Conclusion Based on the Confidence Interval:**
   Because the confidence interval limits \(\_\_\_\_\) 0, there \(\_\_\_\_\) appear to be a significant difference between the two proportions. There \(\_\_\_\_\) evidence to support the claim that echinacea treatment has an effect.

6. **Final Conclusion:**
   c. Based on the results, does echinacea appear to have any effect on the infection rate?

   **Options:**
   - A. Echinacea does not appear to have a significant effect on the infection rate.
   - B. Echinacea does appear to have a significant effect on the infection rate. There is evidence that it increases the infection rate.
   - C. Echinacea does appear to have a significant effect on the infection rate. There is evidence that it lowers the infection rate.
   - D. The results
Transcribed Image Text:**Title: Testing the Effectiveness of Echinacea on Rhinovirus Infections** **Introduction:** 109 subjects were divided into two groups to test the effectiveness of echinacea in treating rhinovirus infections. In the echinacea group, 43 of the 49 subjects developed rhinovirus infections. In the placebo group, 92 of the 109 subjects developed rhinovirus infections. The following analysis uses a 0.05 significance level to evaluate the claim that echinacea affects the rate of rhinovirus infections. **Steps and Analysis:** 1. **Compute the Test Statistic (z):** \[ z = \_\_\_\_ \] (Round to two decimal places as needed.) 2. **Identify the P-value:** \[ \text{P-value} = \_\_\_\_ \] (Round to three decimal places as needed.) 3. **Conclusion Based on the Hypothesis Test:** The P-value is \(\_\_\_\_\) the significance level of \(\alpha = 0.05\), so \(\_\_\_\_\) the null hypothesis. There \(\_\_\_\_\) sufficient evidence to support the claim that echinacea treatment has an effect. 4. **Construct the Confidence Interval:** The 95% confidence interval is: \[ \_\_\_\_\ < (p_1 - p_2) < \_\_\_\_\ \] (Round to three decimal places as needed.) 5. **Conclusion Based on the Confidence Interval:** Because the confidence interval limits \(\_\_\_\_\) 0, there \(\_\_\_\_\) appear to be a significant difference between the two proportions. There \(\_\_\_\_\) evidence to support the claim that echinacea treatment has an effect. 6. **Final Conclusion:** c. Based on the results, does echinacea appear to have any effect on the infection rate? **Options:** - A. Echinacea does not appear to have a significant effect on the infection rate. - B. Echinacea does appear to have a significant effect on the infection rate. There is evidence that it increases the infection rate. - C. Echinacea does appear to have a significant effect on the infection rate. There is evidence that it lowers the infection rate. - D. The results
### Testing the Effectiveness of Echinacea on Rhino Virus Infections

Rhino viruses typically cause common colds. In a test of the effectiveness of echinacea, 43 of the 49 subjects treated with echinacea developed rhinovirus infections. In a placebo group, 92 of the 109 subjects developed rhinovirus infections. Use a 0.05 significance level to test the claim that echinacea has an effect on rhinovirus infections. Complete the parts (a) through (c) below:

#### a. Test the claim using a hypothesis test.

Consider the first sample to be the sample of subjects treated with echinacea and the second sample to be the sample of subjects treated with a placebo. What are the null and alternative hypotheses for the hypothesis test?

- **Options:**
  - A. \( H_0: p_1 \neq p_2 \); \( H_1: p_1 = p_2 \)
  - B. \( H_0: p_1 = p_2 \); \( H_1: p_1 \neq p_2 \)
  - C. \( H_0: p_1 \leq p_2 \); \( H_1: p_1 \neq p_2 \)
  - D. \( H_0: p_1 \geq p_2 \); \( H_1: p_1 \neq p_2 \)
  - E. \( H_0: p_1 = p_2 \); \( H_1: p_1 < p_2 \)
  - F. \( H_0: p_1 = p_2 \); \( H_1: p_1 > p_2 \)

#### Identify the test statistic.

\( z = \) [Blank for student's response]
(Round to two decimal places as needed.)

#### Identify the P-value.

P-value = [Blank for student's response]
(Round to three decimal places as needed.)

#### What is the conclusion based on the hypothesis test?

_The P-value is_ [Dropdown selection] _the significance level of \(\alpha = 0.05\), so we_ [Dropdown selection] _the null hypothesis. There_ [Dropdown selection] _sufficient evidence to support the claim that echinacea treatment has an effect._

#### b. Test the claim by constructing an appropriate confidence interval.

_The
Transcribed Image Text:### Testing the Effectiveness of Echinacea on Rhino Virus Infections Rhino viruses typically cause common colds. In a test of the effectiveness of echinacea, 43 of the 49 subjects treated with echinacea developed rhinovirus infections. In a placebo group, 92 of the 109 subjects developed rhinovirus infections. Use a 0.05 significance level to test the claim that echinacea has an effect on rhinovirus infections. Complete the parts (a) through (c) below: #### a. Test the claim using a hypothesis test. Consider the first sample to be the sample of subjects treated with echinacea and the second sample to be the sample of subjects treated with a placebo. What are the null and alternative hypotheses for the hypothesis test? - **Options:** - A. \( H_0: p_1 \neq p_2 \); \( H_1: p_1 = p_2 \) - B. \( H_0: p_1 = p_2 \); \( H_1: p_1 \neq p_2 \) - C. \( H_0: p_1 \leq p_2 \); \( H_1: p_1 \neq p_2 \) - D. \( H_0: p_1 \geq p_2 \); \( H_1: p_1 \neq p_2 \) - E. \( H_0: p_1 = p_2 \); \( H_1: p_1 < p_2 \) - F. \( H_0: p_1 = p_2 \); \( H_1: p_1 > p_2 \) #### Identify the test statistic. \( z = \) [Blank for student's response] (Round to two decimal places as needed.) #### Identify the P-value. P-value = [Blank for student's response] (Round to three decimal places as needed.) #### What is the conclusion based on the hypothesis test? _The P-value is_ [Dropdown selection] _the significance level of \(\alpha = 0.05\), so we_ [Dropdown selection] _the null hypothesis. There_ [Dropdown selection] _sufficient evidence to support the claim that echinacea treatment has an effect._ #### b. Test the claim by constructing an appropriate confidence interval. _The
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