A manufacturer of hard 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 desired  to be less then 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.
a. If  = 800 and  = 40, how likely is it that any helmet subjected to the standard external force will transmit a force to the wearer in excess of 1000 pounds?
b. Test the hypothesis that, when subjected to the standard external force, the helmets transmit a mean force exceeding 800 pounds, and determine the p-value.
c. Test the hypothesis that  exceeds 40.
The solution contains a testing of hypothesis problem and the determination of P-value.