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 in. and a standard deviation of 1.0 in. (based on anthropometric survey from Gordon, Churchill, et al.).
a. If one male is randomly selected, find the probability that his head breadth is less than 6.2 in.
b. The Safeguard Helmet company plans an initial production run of 100 helmets. Find the probability that 100 randomly selected men have a mean head breadth less than 6.2 in.
c. The production manager sees the result from part (b) and reasons that all helmets should be made for men with head breadths less than 6.2 in., because they would fit all but a few men. What is wrong with that reasoning?
m = 6, s = 1
(a) z = (x - m)/s
z = (6.2 - 6)/1 = 0.2
P(x < 6.2) = P(z < 0.2) = 0.5793
(b) z = (x - m)/(s/ sqrt n)
z = (6.2 - 6)/(1/ sqrt 100) ...
The solution uses statistics to determine sizes of helmets for a mans head. Reasoning is given.