Five data entry operators work at the data processing department of the Birmingham Bank. Each day for $30$ days, the number of defective records in a sample of $250250$ records typed by these operators has been$$ noted, as$$ follows:

$$

Sample No$.$

No$.$ Defectives

Sample No$.$

No$.$ Defectives

Sample No$.$

No$.$ Defectives

$1$

$66$

$11$

$55$

$21$

$1717$

$2$

$66$

$12$

$44$

$22$

$1111$

$3$

$2020$

$13$

$1515$

$23$

$66$

$4$

$99$

$14$

$33$

$24$

$66$

$5$

$1212$

$15$

$1111$

$25$

$1313$

$6$

$99$

$16$

$77$

$26$

$1010$

$7$

$1313$

$17$

$1313$

$27$

$1515$

$8$

$1010$

$18$

$33$

$28$

$55$

$9$

$55$

$19$

$1919$

$29$

$1212$

$10$

$1313$

$20$

$1717$

$30$

$22$

Part $2$

$$a$)$ Establish $3$sigma$\backslash$sigma upper and lower control limits$.$

UCL Subscript pUCLp $=0\mathrm{.}0770\mathrm{.}077($enter your response as a number between $0$ and$1,$ rounded to three decimal$$ places$).$

Part $3$

LCL Subscript pLCLp $=0\mathrm{.}0030\mathrm{.}003($enter your response as a number between $0$ and$1,$ rounded to three decimal$$ places$).$