Problem

The average time $($minutes$)$ that a customer has to wait for the arrival of a cab after calling the company has been observed for random samples of size $4.$ The data for $20$ such samples are shown in Table.

$\backslash$table$[[\backslash$table$[[$Sam$],[$ple$]],\backslash$table$[[$Average$],[$Waiting Time$]],$Sample,Average Waiting Time$],[1,8\mathrm{.}4,11,8\mathrm{.}8],[2,6\mathrm{.}5,12,10],[3,10\mathrm{.}8,13,9\mathrm{.}5],[4,9\mathrm{.}7,14,9\mathrm{.}6],[5,9\mathrm{.}0,15,8\mathrm{.}8],[6,9\mathrm{.}4,16,10],[7,10\mathrm{.}2,17,9\mathrm{.}9],[8,8\mathrm{.}1,18,10],[9,7\mathrm{.}4,19,8\mathrm{.}6],[10,9\mathrm{.}6,20,9\mathrm{.}9]]$

a$)$ Identify and construct an appropriate control chart for the above data to help the manager to determine whether the process is in control or not. $(6$M$)$

b$)$ Is the process under control? If not, what could be the special cause for the process to go out of control?$(1$M$)$

c$)$ If the process is not under control, construct a revised chart after bringing it in control? $(2$M$)$

d$)$ What are the $1$sigma and $2$ sigma control limts for the revised chart? $1M$