Statistics Engineers must consider the breadths of male head

Statistics

Engineers must consider the breadths of male heads when designing motorcycle helmets. Men have head breadths that are normally distributed with a mean of 6.0 inches and a standard deviation of 1.0 inches. due to financial constraints, the helmet will be designed to fit all men EXCEPT those with head breadths that are between 4.0 inches and 8.0 inches. What percentage of the male population does this include?

Solution

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    4      
x2 = upper bound =    8      
u = mean =    6      
          
s = standard deviation =    1      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -2      
z2 = upper z score = (x2 - u) / s =    2      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.022750132      
P(z < z2) =    0.977249868      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.954499736      

Thus, those outside this interval is the complement =    0.04550026 OR 4.55% [ANSWER]