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 $300$ records typed by these operators has been$$ noted, as$$ follows:

$$

Sample No$.$

No$.$ Defectives

Sample No$.$

No$.$ Defectives

Sample No$.$

No$.$ Defectives

$1$

$6$

$11$

$6$

$21$

$17$

$2$

$5$

$12$

$6$

$22$

$13$

$3$

$18$

$13$

$15$

$23$

$6$

$4$

$9$

$14$

$3$

$24$

$8$

$5$

$10$

$15$

$10$

$25$

$14$

$6$

$8$

$16$

$9$

$26$

$10$

$7$

$12$

$17$

$11$

$27$

$15$

$8$

$8$

$18$

$3$

$28$

$7$

$9$

$6$

$19$

$18$

$29$

$11$

$10$

$14$

$20$

$15$

$30$

$3$

Part $2$

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

UCL Subscript p $=$

$$

$.064($enter your response as a number between $0$ and$1,$ rounded to three decimal$$ places$).$

Part $3$

LCL Subscript p $=$

$$

$.002($enter your response as a number between $0$ and$1,$ rounded to three decimal$$ places$).$

Part $4$

$$b$)$ Why can the lower control limit not be a negative$$ number?