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 mm and a standard deviation of 0.07 mm. suppose that the process will be monitored by selecting a random sample of 15 drives from each shift's production and determining x, the mean coating thickness for the sample. (round all your intermediate calculations to four decimal places. round the answers to four decimal places.)

(a) what are the mean and standard deviation of the x sampling distribution? μx = σx =
(b) when no unusual circumstances are present, we expect x to be within 3σx of 3 mm, the desired value. an x value farther from 3 than 3σx is interpreted as an indication of a problem that needs attention. compute 3 ± 3σx. (enter solutions from smallest to largest. round your answers to four decimal places.)
(c) referring to part (b), what is the probability that a sample mean will be outside 3 ± 3σx 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 3.07 mm. what is the probability that a problem will be detected when the next sample is taken? (hint: this will occur if x > 3 + 3σx or x < 3 - 3σx when μ = 3.07.)