Central limit theorems have important applications in the area of quality control. One such application concerns so-called control charts, and in particular, \(\bar{X}\) charts, which are used to monitor whether the variation in the calculated mean levels of some characteristics of a production process are within acceptable limits. The actual chart consists of plotting calculated mean levels (vertical axis) over time (horizontal axis) on a graph that includes horizontal lines for the actual mean characteristic level of the process, \(\mu\), and for upper and lower control limits that are usually determined by adding and subtracting two or more standard deviations, \(\sigma_{\bar{X}}\), to the actual mean level. If, at a certain time period, the outcome of the calculated mean lies outside the control limits, the production process is considered to be no longer behaving properly, and the process is stopped for appropriate adjustments. For example, if a production process is designed to fill cans of soda pop to a mean level of 12 oz., if the standard deviation of the fill levels is . 1 , and if 100 cans of soda are randomly drawn from the packaging line to record fill levels and calculate a mean fill level \(\bar{x}\), then the control limits on the daily calculated means of the filling process might be given by \(12 \mp 3 \operatorname{std}(\bar{x})=\) \(12 \mp .03\).

(a) Provide a justification for the \(\bar{X}\) chart procedure described above based on asymptotic theory. Be sure to clearly define the conditions under which your justification applies.

(b) Suppose that control limits are defined by adding and subtracting three standard deviations of \(\bar{X}\) to the mean level \(\mu\). In light of your justification of the control chart procedure in (a), what is the probability that the production process will be inadvertently stopped at a given time period, even though the mean of the process remains equal to \(\mu\) ?

(c) In the soda can-filling example described above, if the process were to change in a given period so that the mean fill level of soda cans became \(12.05 \mathrm{oz}\). what is the probability that the control chart procedure would signal a shutdown in the production process in that period?