Question

10.  HYPOTHESIS ABOUT DIFFERENCE IN PROPORTIONS

Suppose the Acme Drug Company develops a new drug, designed to prevent colds. The company states that the drug is less effective on seniors than on children. To test this claim, they choose a simple random sample of 80 seniors and 70 children. At the end of a whole year, 16 of the seniors caught a cold; and 11 of the children caught a cold. Based on these findings, can we conclude that the drug is less effective for seniors than for children? Use a= 0.05 level of significance.

Answer 1

Ho:   p1 - p2 =   0          
Ha:   p1 - p2 >   0          
                  
sample #1   ----->              
first sample size,     n1=   80          
number of successes, sample 1 =     x1=   16          
proportion success of sample 1 , p̂1=   x1/n1=   0.2000          
                  
sample #2   ----->              
second sample size,     n2 =    70          
number of successes, sample 2 =     x2 =    11          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.1571          
                  
difference in sample proportions, p̂1 - p̂2 =     0.2000   -   0.1571   =   0.0429
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.1800          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.06288          
Z-statistic = (p̂1 - p̂2)/SE = (   0.043   /   0.0629   ) =   0.6816
                  
z-critical value , Z* =        1.6449   [excel function =NORMSINV(α)]      
p-value =        0.2477   [excel function =NORMSDIST(-z)]      
decision :    p-value>α,Don't reject null hypothesis               
                  
Conclusion:   There is not enough evidence that the drug is less effective for seniors than for children   

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