The thickness (in millimeters) of the coating applied to disk drives is a
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 \(3 \mathrm{~mm}\) and a standard deviation of \(0.05\) \(\mathrm{mm}\).
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 sample of size 16).
b. When no unusual circumstances are present, we expect \(\bar{x}\) to be within
\(3 \sigma_{\bar{x}}\) of \(3 \mathrm{~mm}\), the desired value. An \(\bar{x}\)
value farther from 3 than \(3 \sigma_{\bar{x}}\) is interpreted as an indication
of a problem that needs attention. Compute \(3 \pm 3 \sigma_{\bar{x}}\). (A plot
over time of \(\bar{x}\) values with horizontal lines drawn at the limits \(\mu
\pm 3 \sigma_{\bar{x}}\) is called a process control chart.)
c. Referring to Part (b), what is the probability that a sample mean will be
outside \(3 \pm 3 \sigma_{\bar{x}}\) just by chance (i.e., 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 \(3.05 \mathrm{~mm}\). What is the
probability that a problem will be detected when the next sample is taken?
(Hint: This will occur if \(\bar{x}>3+3 \sigma_{\bar{x}}\) or \(\bar{x}<3-3
\sigma_{\bar{x}}\) when \(\mu=\)
3.05.)
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