Chapter 13, Problem 13.82LM

A process is under control and follows a normal distribution with mean 100 and standard deviation 10. In constructing a standard $\bar{x}$-chart for this process, the control limits are set 3 standard deviations from the mean–that is, $100\pm 3\left(10/\sqrt{n}\right)$. The probability of observing an $\bar{x}$ outside the control limits is (.00135 + .00135) = .0027. Suppose it is desired to construct a control chart that signals the presence of a potential special cause of variation for less extreme values of $\bar{x}$. How many standard deviations from the mean should the control limits be set such that the probability of the chart falsely indicating the presence of a special cause of variation is .10 rather than .0027?