You work for a manufacturing company as a statistical process analyst. Your job is to analyze processes and make sure they are in statistical control. In on process, a machine cuts wood boards to a thickness of 25 millimeters with acceptable margin of error of +/- 0.6 millimeter. (Assume this process can be approximated by a normal distribution.) So, the acceptable range of thicknesses for the boards is 24.4 millimeters to 25.6 millimeter, inclusive.
Because of machine vibrations and other factors, the setting of the wood-cutting machine “s” from 25 millimeters. To check that the machine is cutting the boards to the correct thickness, you select at random three samples of four boards and find the mean thickness (in millimeters) of each sample. A coworker asks you why you take three samples of size 4 and the mean instead of randomly choosing and measuring 12 boards individually to check on the machine’s setting. (Note: Both samples are chosen without replacement.)

Exercises
1. Sampling Individuals
You select on board and measure its thickness. Assume the machine shifts and is cutting boards with a mean thickness of 2534 millimeters and a stand deviation of 0.2 millimeter.
(a) What is the probability that you select a board that is not outside the acceptable range (in other words, you do not detect that the machine has shifted)? (See figure.)
(b) You randomly select 12 boards. What is the probability that you elect at least one board that is not outside the acceptable range?
2. Sampling Groups of Four
You select four boards and find their mean thickness. Assume the machine shifts and is cutting boards with a mean thickness of 25.4 millimeters and a standard deviation of 0.2 millimeter.
(a) What is the probability that you select a sample of four boards that has a mean that is not outside the acceptable rang? (See figure.)
(b) You randomly select three samples of four boards. What is the probability that you select at least one sample of four boards that has a mean that is not outside the acceptable range?
(c) What is more sensitive to change – an individual measure or the mean?
3. Writing an Explanation
Wire a paragraph to your coworker explaining why you take three samples of size 4 and find the mean of each sample instead of randomly choosing and measuring 12 boards individually to check the machine’s settings.