The thickness (in millimeters) of the coating applied to hard 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 x̅ for a random sample of size 16.

b. When no unusual circumstances are present, we expect x̅ to be within 3σ_x̅ of 2 mm, the desired value. An x̅ value farther from 2 mm than 3σ_x̅ is interpreted as an indication of a problem that needs attention. Calculate 2 ± 3σ_x̅.

c. Referring to Part (b), what is the probability that a sample mean will be outside 2 ± 3σ_x̅ 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? This will occur if x̅ > 2 + 3 σ_x̅ or x̅ < 2 – 3σ_x̅ when µ = 2.05.