A machine that cuts corks for wine bottles operates in such

A machine that cuts corks for wine bottles operates in such a way that the distribution of the diameter of the corks produced is well approximated by a normal distribution with mean 2 cm and standard deviation 0.1 cm. The specifications call for corks with diameters between 1.85 and 2.15 cm. What proportion of corks produced by this machine are defective?

Solution

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    1.85      
x2 = upper bound =    2.15      
u = mean =    2      
          
s = standard deviation =    0.1      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -1.5      
z2 = upper z score = (x2 - u) / s =    1.5      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.066807201      
P(z < z2) =    0.933192799      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.866385597      

Thus, those outside this interval is the complement = 0.133614403   [ANSWER, PROPORTION OF DEFECTIVE]