A soft drink bottler purchases glass bottles from a vendor. The bottles are required to have an internal pressure of at least $150$ pounds per square inch (psi). A prospective bottle vendor claims that its production process yields bottles with a mean internal pressure of $157$ psi and a standard deviation of $3$ psi. The bottler strikes an agreement with the vendor that permits the bottler to sample from the production process to verify the claim. the bottler randomly selects $40$ bottles from the last $10000$ produced, measures the internal pressure of each, and finds the mean pressure for the sample to be $0.6$ psi below the process mean cited by the vendor.

(a) Assuming that the vendor is correct in his claim, what is the probability of obtaining a sample mean this far or farther below the process mean?

(b) If the standard deviation were $3$ psi as claimed, but the mean was $152$ psi, what is the probability of obtaining a sample mean of $156.4$ psi or below?
(c) If the process mean were $157$ psi as claimed, but the standard deviation was $3.4$ psi, what is the probability of obtaining a sample mean of $156.4$ psi or below?