91Ó°ÊÓ

The\(c\)chart for the number of defectives in a unit has lower and upper control limits given by

\(\begin{array}{l}LCL = \bar x - 3\sqrt {\bar x} \\UCL = \bar x + 3\sqrt {\bar x} ,\end{array}\)

and the center line is\(\bar x\). When\(LCL \le 0\)it is set to zero.

The center line, from the fact that

\(\sum\limits_{i = 1}^k {{{\hat x}_i}} = 92\)

is given by

\(\bar x = \frac{{92}}{{24}} = 3.833\)

The\(LCL\)is

\(\begin{array}{l}LCL = \bar x - 3\sqrt {\bar x} = 3.833 - 3 \times \sqrt {3.833} \\ = 0\end{array}\)

set it to zero because it is lower than zero.

The\(UCL\)is

\(\begin{array}{l}UCL = \bar x + 3\sqrt {\bar x} = 3.833 + 3 \times \sqrt {3.833} \\ = 9.7\end{array}\)

From the \(c\) chart one can notice that the process has an out-of-control point on \({22^{nd}}\) place.

\({\rm{L C L = 0 ; U C L = 9}}{\rm{.7}}\)