Question

Rhino viruses typically cause common colds. In a test of the effectiveness of​ echinacea, 45 of the 51 subjects treated with echinacea developed rhinovirus infections. In a placebo​ group, 85 of the 102 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. 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? A. Upper H 0 ​: p 1 greater than or equalsp 2Upper H 1 ​: p 1 not equalsp 2 B. Upper H 0 ​: p 1 less than or equalsp 2Upper H 1 ​: p 1 not equalsp 2 C. Upper H 0 ​: p 1 equalsp 2Upper H 1 ​: p 1 less thanp 2 D. Upper H 0 ​: p 1 equalsp 2Upper H 1 ​: p 1 not equalsp 2 E. Upper H 0 ​: p 1 equalsp 2Upper H 1 ​: p 1 greater thanp 2 F. Upper H 0 ​: p 1 not equalsp 2Upper H 1 ​: p 1 equalsp 2 Identify the test statistic. zequals nothing ​(Round to two decimal places as​ needed.) Identify the​ P-value. ​P-valueequals nothing ​(Round to three decimal places as​ needed.) What is the conclusion based on the hypothesis​ test? The​ P-value is ▼ less than greater than the significance level of alpha equals0.05 ​, so ▼ fail to reject reject the null hypothesis. There ▼ is is not sufficient evidence to support the claim that echinacea treatment has an effect. b. Test the claim by constructing an appropriate confidence interval. The 95 ​% confidence interval is nothing less thanleft parenthesis p 1 minus p 2 right parenthesisless thannothing . ​(Round to three decimal places as​ needed.) What is the conclusion based on the confidence​ interval? Because the confidence interval limits ▼ do not include include ​0, there ▼ does not does appear to be a significant difference between the two proportions. There ▼ is is not evidence to support the claim that echinacea treatment has an effect. c. Based on the​ results, does echinacea appear to have any effect on the infection​ rate? A. Echinacea does appear to have a significant effect on the infection rate. There is evidence that it increases the infection rate. B. Echinacea does appear to have a significant effect on the infection rate. There is evidence that it lowers the infection rate. C. Echinacea does not appear to have a significant effect on the infection rate. D. The results are inconclusive.

Answer 1

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The​ P-value is greater than the significance level of alpha 0.05 ​, so fail to reject the null hypothesis. There is not sufficient evidence to support the claim that echinacea treatment has an effect.

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Because the confidence interval limits include ​0, there does not appear to be a significant difference between the two proportions. There is not evidence to support the claim that echinacea treatment has an effect.

c.

C. Echinacea does not appear to have a significant effect on the infection rate.

Answer 2

a. Hypothesis Test: The null hypothesis (H0) states that there is no difference in the proportion of rhinovirus infections between the group treated with echinacea (p1) and the placebo group (p2). The alternative hypothesis (H1) states that there is a difference in the proportion of rhinovirus infections between the two groups.

Null hypothesis (H0): H0: p1 = p2

Alternative hypothesis (H1): H1: p1 ≠ p2

Test Statistic: The test statistic for comparing two proportions is the z-test statistic.

b. Confidence Interval: To construct a confidence interval for the difference in proportions (p1 - p2), we can use the formula:

Confidence Interval = (p1 - p2) ± Z * √[(p1(1 - p1)/n1) + (p2(1 - p2)/n2)]

where Z is the critical value corresponding to the desired confidence level (95% confidence level in this case), p1 and p2 are the sample proportions, and n1 and n2 are the sample sizes for the two groups.

c. Conclusion: Based on the hypothesis test and the confidence interval, we can make the following conclusions:

The hypothesis test suggests that there is insufficient evidence to support the claim that echinacea treatment has a significant effect on the infection rate (since we fail to reject the null hypothesis).

The confidence interval contains zero (0), which means that there is no significant difference between the two proportions (p1 and p2). Therefore, based on the confidence interval, we do not have evidence to support the claim that echinacea treatment has a significant effect on the infection rate.

So, the correct answers are: a. Hypothesis test conclusion: The P-value is greater than the significance level of alpha (0.05), so we fail to reject the null hypothesis. There is not sufficient evidence to support the claim that echinacea treatment has an effect. b. Confidence interval conclusion: Because the confidence interval limits do not include 0, there does not appear to be a significant difference between the two proportions. There is not evidence to support the claim that echinacea treatment has an effect. c. Based on the results, echinacea does not appear to have a significant effect on the infection rate (Option C).