Q$2)$ 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 $250$ records typed by these operators has been noted, as follows.

a$)$ Establish $38$ upper and lower control limits$.(10$ points$)$

b$)$ Why can the lower control limit not be a negative number? $(10$ points$)$

c$)$ The industry standards for the upper and lower control limits are $0\mathrm{.}10$ and $0\mathrm{.}01,$ respectively. What does this imply about Birmingham Bank's own standards? $(15$ points$)$

a$)$

$P=$ total number of defective items $P=\frac{\text{t}\text{o}\text{t}\text{a}\text{l}\text{n}\text{u}\text{m}\text{b}\text{e}\text{r}of\text{o}\text{b}\text{s}\text{e}\text{r}\text{v}\text{a}\text{t}\text{i}\text{o}\text{n}\text{s}}{?}$

${}_p=\sqrt[\phantom{2}]{\frac{(\frac{?}{\text{b}\text{a}\text{r}}(p))(1-(\frac{?}{\text{b}\text{a}\text{r}}(p)))}{n}}$

UCL$\frac{?}{b}=ar(p)+z{}_p$

$LCL\frac{?}{b}=ar(p)-z{}_p$