Men have head circumferences that are normally distributed w

Men have head circumferences that are normally distributed with a mean of 23.0 inches and a standard deviation of 0.82 inches (based on data from the National Health and Nutrition Examination Survey). If a company wants to produce hats to fit all men except for those with the smallest 3% and the largest 3% head circumference, what head circumferences should be accommodated? PLEASE HELP BY SHOWING WORK

Solution

Normal Distribution
Mean ( u ) =23
Standard Deviation ( sd )=0.82
Normal Distribution = Z= X- u / sd ~ N(0,1)                  

P ( Z < x ) = 0.03
Value of z to the cumulative probability of 0.03 from normal table is -1.881
P( x-u/s.d < x - 23/0.82 ) = 0.03
That is, ( x - 23/0.82 ) = -1.88
--> x = -1.88 * 0.82 + 23 = 21.4576  
              
P ( Z > x ) = 0.03
Value of z to the cumulative probability of 0.03 from normal table is 1.88
P( x-u/ (s.d) > x - 23/0.82) = 0.03
That is, ( x - 23/0.82) = 1.88
--> x = 1.88 * 0.82+23 = 24.5424                  

It is in b/w 21.46 to 24.54