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I'm stuck with this problem on quality control and p-charts.

There is a given process, monitored periodically taking samples of 100 items each and counting the number that are defective. In the last 50 samples we found a total of 622 defective items.

I'm OK for the 3-sigma limits of the p-chart and in particular: $$\bar{p} = \frac{622}{50*100} = 0.1244$$ $$\text{UCL} = \bar{p} + 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}} = 0.1244+ 3\sqrt{\frac{0.1244 (1-0.1244)}{100}} = 0.2234$$ $$\text{LCL} = \bar{p} - 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}} = 0.1244+ 3\sqrt{\frac{0.1244 (1-0.1244)}{100}} = 0.0254$$ where $\bar{p}$ represents the estimated proportion of defects from the process, used to calculate the control limits.

Now, how am I supposed to find the average number of samples before having a point falling outside the control limits (signaling an of out-of-control), given an actual shift of the estimated proportion of defects from $\bar{p}$ to $p' = \bar{p} + 0.06 = 0.1844$ due, for example, to something new that is now influencing the process?

I think this has to do with the average run length $\text{ARL}_1 = \frac{1}{1-\beta}$, but I'm not quite sure on the steps leading to its value.

Scortchi - Reinstate Monica
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emain90
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  • I don't know much about quality control & control / p-charts, but I can't follow your question. Can you explain this more fully? Eg, what does it mean for a signal to go "out of control"? I suspect that isn't a well defined term. – gung - Reinstate Monica Jul 04 '14 at 13:36
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    Thanks for the reply gung. Yes, I think the question could be rephrased as: given that something in the process changed and the proportion of defects shifts from p = 622/(100*50) = 0.1244 to the new p' = 0.1244 + 0.06 = 0.1844, after how many samples (on average) will a point on the chart fall outside the control limits? I thinks that the answer has to do with the ARL1 (average run length), but I'm not quite sure. – emain90 Jul 04 '14 at 14:09
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    So $p$ is the estimated proportion of defects from some process, used to calculate the control limits (assuming independence); while $p'$ is the true proportion of defects produced following some change in the process - is that right? Please edit the question to clarify rather than adding comments. There aren't very many industrial statisticians active on this site, so unless you make the question generally intelligible through context & definition it's likely to be closed as unclear, despite there likely being many readers who could answer. – Scortchi - Reinstate Monica Jul 04 '14 at 14:44

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