In a quality control study, a P$-$chart is used to monitor the proportion of defective items in a production process. The average proportion of defects is found to be $0\mathrm{.}05.$

a$)$ Calculate the three$-$sigma lower and upper control limits for a P$-$chart with a sample size of $100,$ assuming the process is in statistical control. $($Ans: $0$ and $0\mathrm{.}115)$

b$)$ If a sample of $100$ items yields $8$ defective items, determine if the process is in control based on the P$-$chart limits calculated in part $($a$).$