A factory machine produces a certain copper tubing component in a refrigeration unit. The tubing components produced by the manufacturer have a mean diameter of 0.75 inch with a standard deviation of 0.004 inch. The quality control inspector takes a random sample of 30 components once each week and calculates the mean diameter of these components. If the mean is either less than 0.748 inch or greater than 0.752 inch, the inspector concludes that the machine needs an adjustment.

(a) Describe the sampling distribution of

x, the sample mean diameter, for a random sample of 30 such components.

(b) What is the probability that, based on a random sample of 30 such components, the inspector will conclude that the machine needs an adjustment when, in fact, the machine is correctly calibrated?

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