Question

Rhino viruses typically cause common colds. In a test of the effectiveness of​ echinacea, 34 of the 41 subjects treated with echinacea developed rhinovirus infections. In a placebo​ group, 94 of the 107 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?

Identify the test statistic.

Identify the​ P-value.

What is the conclusion based on the hypothesis​ test?

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

The 99% confidence interval is

What is the conclusion based on the confidence​ interval?

c. Based on the​ results, does echinacea appear to have any effect on the infection​ rate?

Answer 1

Answer:

Given,

Ho : p1 = p2

Ha : p1 != p2

p1 = x1/n1 = 34/41 = 0.8293

p2 = x2/n2 = 94/107 = 0.8785

p = (x1+x2)/(n1+n2)

= (34+94)/(41+107)

= 0.8649

test statistic z = (p1 - p2)/sqrt(p(1-p)(1/n1+1/n2))

substitute values

= (0.8293 - 0.8785)/sqrt(0.8649(1-0.8649)(1/41 + 1/107))

= -0.78

P value = 0.4353909 [since from z table]

= 0.4354

Here we observe that, p value > alpha, so we fail to reject Ho.

So we don't have sufficient evidence.

Now consider,

Here at 99% CI, z value = 2.58

99% CI = (p1 - p2) +/- z*sqrt(p(1-p)(1/n1+1/n2))

substitute values

= (0.8293 - 0.8785) +/- 2.58*sqrt(0.8649(1-0.8649)(1/41 + 1/107))

= -0.0492 +/- 0.1616

= (-0.2108 , 0.1124)