8.4 14 *9. Suppose that we are using an T-chart with subgroups of size n -…

8.4 14 *9. Suppose that we are using an T-chart with subgroups of size n – 5 in an idealized setting in which the data is Normally distributed with known mean μ and standard deviation σ. We know that σ,-sd(X)-4. Use also the facts that and Assume that the process is under control and that subgroups are indepen- dent. Suppose that 100 subgroup means are plotted onto the chart over the course of a day. (a) What is the probability that at least one subgroup mean lies outside 2-sigina control limits? Repeat for 3-signa control limits. Control Charts (b) What is (1) the distribution, and (Il) the expected val ue of the number of subgroup means that will lie outside 2-sigma control limits Repeat for 3-sigma control limits. Suppose now that the process goes out of control in that the mean shifts from μ to μ + 0.5 . (The variability remains unchanged.) *(c) Show that the probability that the next subgroup mean lies outside the 2-sigma control limits is 0.19 This is the probability that the control chart signals the change with the lirst pust-change subgroup. (d) What is the probability that at least one of the next & subgroup means (e) Repeat (c) and (d) but for 3-sigma limits. (f) What have you learned from the calculations in this question’? les outside the 2-sigma control limits?