In order to compare failure rates for two brands of tires a

In order to compare failure rates for two brands of tires, a sample of 500 tires of Brand A was taken, of which 35 failed. For a sample of 700 tires of Brand B, 31 failed. Can we conclude at the 0.05 level that there is a difference in failure rates?

Solution

Formulating the hypotheses          
Ho: p1 - p2   =   0  
Ha: p1 - p2   =/=   0  

Here, we see that pdo =    0   , the
hypothesized population proportion difference.  
          
Getting p1^ and p2^,          
          
p1^ = x1/n1 =    0.07      
p2 = x2/n2 =    0.044285714      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.013808098      
          
Thus,          
          
z = [p1 - p2 - pdo]/sd =    1.862261307      
          
As significance level =    0.05   , then the critical z is  
          
zcrit =    1.959963985      
          
Also, the p value is          
          
P =    0.062566267      
          
As |z| < 1.96, and P > 0.05, then we    FAIL TO REJECT THE NULL HYPOTHESIS.      

Thus, there is no signficant difference in failure rates. [CONCLUSION]