Chapter 13.4, Problem 13.20ACI

13 20 Selecting the best wafer-slicing machine. Silicon wafer slicing is a critical step in the production of semiconductor devices (e.g.. diodes, solar cells, transistors). Yuanpei (China) University researchers used control charts to aid in selecting the best silicon wafer-slicing machine (Computers & Industrial Engineering, Vol. 52, 2007). Samples of n = 2 wafers were sliced each hour for 67 consecutive hours and bow measurements (a measure of precision) were recorded. The resulting $\bar{x}$ -chart for one of the machines tested revealed that the cutting process was out of control on the 19th, 40th, and 59th hours. For each of these 3 hours, the mean bow measurement fell above the upper control limit. Assume the mear bow measurements are normally distributed.

1. a. If the process is in control, what is the probability that a mean bow measurement for a randomly selected hour will fall above the upper control limit?
2. b. If the process is in control, what is the probability that 3 of 67 mean bow measurements fall above the upper control limit?