A machine manufactures bolts that are supposed to be 3 inches in length. Each day a quality engineer selects a random sample of 50 bolts from the day's production, measures their lengths, and performs a hypothesis test of H0 :μ = 3 versus H1 :μ = 3, where μ is the mean length of all the bolts manufactured that day. Assume that the population standard deviation for bolt lengths is 0.1 in. If H0 is rejected at the 5% level, the machine is shut down and recalibrated.
a. Assume that on a given day, the true mean length of bolts is 3 in. What is the probability that the machine will be shut down? (This is called the false alarm rate.)
b. If the true mean bolt length on a given day is 3.01 in., find the probability that the equipment will be recalibrated?