Question

(9 pts) 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 in and a standard deviation of 1.0 in. (The next page has a copy of cdf values for the standard normal distribution.) a) If one male is randomly selected, find the probability that his head breadth is less than 6.2 in.






b) Find the probability that 100 randomly selected men have a mean head breadth that is less than 6.2 in. State any rule/assumption or result to justify your answer.








c) A production manager for Safeguard Helmet Company plans an initial run of 100 helmets. Seeing the result from part b), the manager 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?

Answer 1

a) We are given the distribution here as:

Probability that his head breadth is less than 6.2 is computed here as: P(X < 6.2)

Converting it to a standard normal variable, we have here:

Getting it from the standard normal tables, we have here:

Therefore 0.5793 is the required probability here.

b) For 100 men, the standard deviation for sample mean is obtained as:

The probability thus is computed here as:

Getting it from the standard normal tables, we have here:

Therefore 0.9772 is the required probability here.

c) The reasoning is flawed, because even though the mean value of the breadth would be less than 6.2 inches, from part a), we can see that about 42% of the breadth would be higher than 6.2 inches. Therefore the given reasoning is flawed here, as the helment might not fit for 42% of people.