Question

Rhino viruses typically cause common colds. In a test of the effectiveness of​ echinacea, 36 of the 43 subjects treated with echinacea developed rhinovirus infections. In a placebo​ group, 96 of the 109 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.

b. Identify the test statistic

c. P-value

d. Confidence interval

Answer 1

a)
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 ≠ p2

b)
Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.8372-0.8807)/sqrt(0.8684*(1-0.8684)*(1/43 + 1/109))
z = -0.71

c)
P-value Approach
P-value = 0.4777
As P-value >= 0.01, fail to reject null hypothesis.
There is not sufficient evidence to conclude that echinacea has an effect on rhinovirus infections.

d)
Here, , n1 = 43 , n2 = 109
p1cap = 0.8372 , p2cap = 0.8807

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.8372 * (1-0.8372)/43 + 0.8807*(1-0.8807)/109)
SE = 0.0643

For 0.99 CI, z-value = 2.58
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.8372 - 0.8807 - 2.58*0.0643, 0.8372 - 0.8807 + 2.58*0.0643)
CI = (-0.2094 , 0.1224)