Chapter 8.2, Problem 22E

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 $\bar{x}$, the mean coating thickness for the sample.

a. Describe the sampling distribution of $\bar{x}$ for a random sample of size 16.

b. When no unusual circumstances are present, we expect $\bar{x}$ to be within $3{\sigma }_{\bar{x}}$ of 2 mm, the desired value. An $\bar{x}$ value farther from 2 mm than $3{\sigma }_{\bar{x}}$ is interpreted as an indication of a problem that needs attention. Calculate $2\pm 3{\sigma }_{\bar{x}}$.

c. Referring to Part (b), what is the probability that a sample mean will be outside $2\pm 3{\sigma }_{\bar{x}}$ 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? (Hint: This will occur if $\bar{x}>2+3{\sigma }_{\bar{x}}$ or $\bar{x}<2-3{\sigma }_{\bar{x}}$ when μ = 2.05.)