# Statistical Quality Control p-chart

A company manufactures small metal brackets. They are packaged in containers of 1000 brackets each. At the unloading facility, 10 containers have arrived and 36 brackets are selected at random from each container. The fraction non-confirming in each sample is $0.0,\ 0.0,\ 0.0,\ 0.01,\ 0.02,\ 0.02, \ 0.06,\ 0.0,\ 0.0 \ and\ 0.0$.

(i) Do the data from this shipment indicate statistical control?

(ii) What is the minimum sample size that would give a positive lower control limit for this chart?

I tried to attempt this question by using the p-chart to determine statistical quality control.

(i) Here, $n = 36$ and so, $\overline p = \frac{{\sum {{p_i}} }}{{10}} = \frac{{0.2}}{{10}} = 0.02$

The $3\sigma$ control limits for the p-chart are given by:

$\overline p \pm 3\sqrt {\frac{{\overline p (1 - \overline p )}}{n}}$

Hence, we have:
Lower Control Limit (LCL) = -0.05 $\approx$ 0 (as LCL cannot be negative)
Control Line (CL) = 0
Upper Control Limit (UCL) = 0.09

We see that all the points lie within these control limits. So, we can say that the shipment is in statistical control.

For the (ii) part of the question, we need:
$LCL\ > \ 0$
$\Rightarrow \overline p > 3\sqrt {\frac{{\overline p (1 - \overline p )}}{n}}$
Taking $\overline p = 0.02$ and solving the above inequality I finally got:
$n > 441$ So, the minimum sample size should be $442$ for the LCL to be positive.

Can someone please tell me if my approach is correct or not?

As long as the sample size is consistent, you should be using an $np$ chart instead of a $p$ chart.
There is the $huge$ problem that with 36 samples, the numbers should be $0.00, 0.00, 0.00, 0.03, 0.03, 0.03, 0.06, 0.00, 0.00$ and $0.00$. $\frac{1}{36}=0.02\overline{7}$ and $\frac{2}{36}=0.0\overline{5}$. The values provided in the sample problem are impossible.