Question

Central Limit Theorem

8. Designing Motorcycle Helmets. 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 (based on anthropometric survey data from Gordon, Churchill, et al.)

a) If one male is randomly selected, find the probability that his head breadth is less than 6.2 inches.

b) The Safeguard Helmet Company plans an initial production run of 1200 helmets. Find the probability that 100 randomly selected men have a mean head breadth less than 6.2 inches.

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 inches because they would fit all but a few men. Is this a reasonable argument?

Answer 1

c.

Part b gives the probability that the mean is less than 6.2, not the proportion of men with headsize less than 6.2. The production manager thinks 2.3% of the men would be excluded (P=1-0.9772= 0.0227501= 2.3%), but actually, almost 50% would be excluded if the observed mean were just under 6.2.

Assuming a normal distribution, half of the men in the sample of 100 have a head size greater than 6.2, or 50 men, if the mean were 6.2.

Hence, the argument is not reasonable.