Chapter 10.9, Problem 78E

A manufacturer of hard safety hats for construction workers is concerned about the mean and the variation of the forces its helmets transmit to wearers when subjected to a standard external force. The manufacturer desires the mean force transmitted by helmets to be 800 pounds (or less), well under the legal 1000-pound limit, and desires σ to be less than 40. Tests were run on a random sample of n = 40 helmets, and the sample mean and variance were found to be equal to 825 pounds and 2350 pounds², respectively.

1. a If µ = 800 and σ = 40, is it likely that any helmet subjected to the standard external force will transmit a force to a wearer in excess of 1000 pounds? Explain.
2. b Do the data provide sufficient evidence to indicate that when subjected to the standard external force, the helmets transmit a mean force exceeding 800 pounds?
3. c Do the data provide sufficient evidence to indicate that σ exceeds 40?