Solution Found!
Selecting the best wafer-slicing machine. Silicon wafer
Chapter 13, Problem 18E(choose chapter or problem)
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 mean bow measurements are normally distributed.
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?
b. If the process is in control, what is the probability that 3 of 67 mean bow measurements fall above the upper control limit?
Questions & Answers
QUESTION:
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 mean bow measurements are normally distributed.
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?
b. If the process is in control, what is the probability that 3 of 67 mean bow measurements fall above the upper control limit?
ANSWER:Step 1 of 2
(a)
Find the probability that a mean bow measurement for a randomly selected hour will fall above the upper control limit.
The observations are unusual when the observations lie outside the upper control limit. In addition, the 
Since the normal distribution is symmetric,
Thus, the probability that a mean bow measurement for a randomly selected hour will fall above the upper control limit is 