8.22 The thickness (in millimeters) of the coating applied to disk drives is one characteristic that determines the usefulness of the product. When no unusual circumstances are present, the thickness (x) has a normal distribution with a mean of 2 mm and a standard deviation of 0.05 mm. Suppose that the process will be monitored by selecting a random sample of 16 drives from each shift’s production and determining x, the mean coating thickness for the sample.

a. Describe the sampling distribution of x (for a sample of size 16).

b. When no unusual circumstances are present, we expect x to be within 3sx of 2 mm, the desired value.

An x value farther from 2 than 3sx is interpreted as an indication of a problem that needs attention.

Compute 2 6 3sx. (A plot over time of x values with horizontal lines drawn at the limits m 6 3sx is called a process control chart.)

c. Referring to Part (b), what is the probability that a sample mean will be outside 2 6 3sx just by chance

(that is, when there are no unusual circumstances)?

d. Suppose that a machine used to apply the coating is out of adjustment, resulting in a mean coating thickness of 2.05 mm. What is the probability that a problem will be detected when the next sample is taken?