Two different types of polishing solutions are being evaluat

Two different types of polishing solutions are being evaluated for possible use in a tumble-polish operation for manufacturing interocular lenses used in the human eye following cataract surgery. Three hundred lenses were tumble polished using the first polishing solution, and of this number 255 had no polishing-induced defects. Another 300 lenses were tumble-polished using the second polishing solution, and 193 lenses were satisfactory upon completion.

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 =    0.85      
p2 = x2/n2 =    0.643333333      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.034494229      
          
Thus,          
          
z = [p1 - p2 - pdo]/sd =    5.991340324      
          
As significance level =    0.01   , then the critical z is  
          
zcrit =    2.575829304      
          
Also, the p value is          
          
P =    2.08119*10^-9 = 0.0000 [ANSWER, P VALUE]

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As P < 0.01, then there IS significant difference in the fraction of polishing-induced defects produced by the two polishing solutions at the 0.01 level of significance. [ANSWER, IS]