Question

Rhino viruses typically cause common colds. In a test of the effectiveness of​ echinacea, 37 of the 42 subjects treated with echinacea developed rhinovirus infections. In a placebo​ group, 84 of the 99 subjects developed rhinovirus infections. Use a 0.05 significance level to test the claim that echinacea has an effect on rhinovirus infections.

Answer 1

Ho:p1 - p2 =0

Ha:p1 - p2 <0

first sample size,                                  n1=42

number of successes, sample 1 =    x1=37

proportion success of sample 1 , p̂1=x1/n1=0.881

second sample size,                                  n2 = 99

number of successes, sample 2 =    x2 = 84

proportion success of sample 1 , p̂ 2=x2/n2   = 0.848

difference in sample proportions, p̂1 - p̂2 =  0.03247

pooled proportion , p =(x1+x2)/(n1+n2)=0.8582

std error ,SE = =SQRT(p*(1-p)*(1/n1+ 1/n2)=0.0642

Z-statistic = (p̂1 - p̂2)/SE =  0.5054

p-value = 0.6933    [excel function: =normsdist(z) ]

decision : p-value>α,Don't reject null hypothesis

so, there is no enough evidence to claim that echinacea has an effect on rhinovirus infections at α=0.05