A manufacturer produces safety jackets for competitive fencers. These jackets are rated by the minimum force, in newtons, that will allow a weapon to pierce the jacket. When this process is operating correctly, it produces jackets that have ratings with an average of 840 newtons and a standard deviation of 15 newtons. FIE, the international governing body for fencing, requires jackets to be rated at a minimum of 800 newtons. To check whether the process is operating correctly, a manager takes a sample of 50 jackets from the process, rates them, and calculates x̅, the mean rating for jackets in the sample. She assumes that the standard deviation of the process is fixed but is worried that the mean rating of the process may have changed.

a. What is the sampling distribution of x̅ if the process is still operating correctly?

b. Suppose the manager's sample has a mean rating of 830 newtons. What is the probability of getting an x̅ of 830 newtons or lower if the process is operating correctly?

c. Given the manager's assumption that the standard deviation of the process is fixed, what does your answer to part b suggest about the current state of the process (i.e., does it appear that the mean jacket rating is still 840 newtons)?

d. Now suppose that the mean of the process has not changed, but the standard deviation of the process has increased from 15 newtons to 45 newtons. What is the sampling distribution of x̅ in this case? What is the probability of getting an x̅ of 830 newtons or lower when x̅ has this distribution?