A particular manufacturing process requires steel rods that are at least 3 meters in length. The rods are purchased in lots of 50,000. To determine whether the lot meets the required quality standards, 100 rods are randomly sampled from each incoming lot and the mean length of rods in the sample is calculated. The quality manager has decided to accept lots whose sample mean is 3.005 meters or more. Assume that the standard deviation of the rod lengths in a lot is 0.03 meter.
a. If in fact each lot has a mean length of 3 meters, what percentage of the lots received by the manufacturer will be returned to the vendor (i.e., the supplier)?
b. If in fact all of the rods in all of the lots received by the manufacturer are between 2.999 and 3.004 meters in length, what percentage of the lots will be returned to the vendor?