Question

Rhino viruses typically cause common colds. In a test of the effectiveness of​ echinacea, 33 of the 40 subjects treated with echinacea developed rhinovirus infections. In a placebo​ group, 76 of the 90 subjects developed rhinovirus infections. Use a 0.01 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 equal to) p 2 Upper H 1​: p 1 (not equal)p
2 B. Upper H 0​: p 1=p 2 Upper H 1​: p 1 (not equals) p 2
C. Upper H 0​: p 1 (less than or equal to)p 2 Upper H 1​: p 1 (not equals) p 2
D. Upper H 0​: p 1 (not equals) p 2 Upper H 1​: p 1 (equals) p 2
E. Upper H 0​: p 1=p 2 Upper H 1​: p 1<p 2
F. Upper H 0​: p 1=p 2 Upper H 1​: p 1>p 2

Identify the test statistic. z=____ ​(Round to two decimal places as​ needed.)

Identify the​ P-value. ​P-value=______ ​(Round to three decimal places as​ needed.)

What is the conclusion based on the hypothesis​ test?
The​ P-value is ▼ greater than less than the significance level of alphaequals0.01​, so ▼ reject fail to 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 99​% confidence interval is _____<(p 1 minus p 2 )< _____. ​(Round to three decimal places as​ needed.)

What is the conclusion based on the confidence​ interval?

Because the confidence interval limits ▼ include do not include ​0, there ▼ does not does appear to be a significant difference between the two proportions. There ▼ is not is 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 lowers the infection rate.
B. Echinacea does not appear to have a significant effect on the infection rate.
C. Echinacea does appear to have a significant effect on the infection rate. There is evidence that it increases the infection rate.
D. The results are inconclusive.

Answer 1

Given,

X1 = 33, n1 = 40 and P1^ = 33/40 = 0.825

X2 = 76, n2 = 90 and P2^ = 76/90 = 0.844

P bar = (X1+X2)/(n1+n2) = (33+76)/(40+90) = 0.8385

alpha = 0.01

a)

Hypothesis:

H 0​: p 1 = p 2

H 1​: p 1 not = p 2

Test:

Z stat = (P1^ - P2^)/SQRT(Pbar(1-Pbar)(1/n1+1/n2))

= (0.825-0.844)/SQRT(0.8385*(1-0.8385)*(1/40+1/90))

= -0.27

P value = 0.787

P value > 0.01, Do not reject H0

The​ P-value is greater than the significance level of 0.01​, so fail to reject the null hypothesis. There is not sufficient evidence to support the claim that echinacea treatment has an effect.

b)

99% CI

alpha = 0.01

Zc = 2.5758(Use Z table)

CI = (P1^-P2^)+/-Zc*SQRT(P1^(1-P1^)/n1 + P2^(1-P2^)/n2)

= (0.825-0.844)+/-2.5758*SQRT((0.825*(1-0.825)/40)+(0.844*(1-0.844)/90))

CI = (-0.2025, 0.1644)

Because the confidence interval limits include do, 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)

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