An important manufacturing process is manufacturing cylinder parts for the automotive industry. Parts of the process with an average diameter of 5.0 millimeters are extremely important. The relevant engineer predicts the mass average as n = 5.0 mm. For the experiment in which 100 pieces are randomly selected from the process and the diameter of each piece is measured, the average diameter is 5.027 mm. The mass standard deviation is known to be 0.1.

a) If the mass mean is 5.0, what is the probability that the X sample mean deviates more than 0.027 mm? If 500 pieces were selected, how many would not meet the engineer's 5.0mm standard?

b) What is the maximum value of the sample mean for the lower 5% of the materials produced?