Suppose that 25 samples, each of size 200, we selected from what is believed to be an in-control process, and that $\sum \hat { p } _ { i }$ is the fraction of defective items in sample i. The 3-sigma control limits, LCL and UCL, of the p chart for the fraction of defective items are respectively
A) .0236 and .244
B) -.0462 and .0942
C) 0 and .0564
D) 0 and .0942
E) -.0084 and .0564

Question

Quizplus · Accepted Answer

The correct control limits for a p-chart are calculated based on the average fraction defective ($\bar{p}$) and the standard deviation of the proportion ($\sigma_p$). The formula for the 3-sigma control limits is $\bar{p} \pm 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$, where $n$ is the sample size. Negative control limits are not possible for proportions, which eliminates choices B and E. Choices A and D suggest limits that are too wide for typical p-chart applications, especially considering the sample size of 200. Choice C suggests a lower control limit (LCL) of 0, which is plausible since proportions cannot be negative, and an upper control limit (UCL) that is positive and within a reasonable range for the fraction of defective items, making it the most plausible choice.

Suppose That 25 Samples, Each of Size 200, We Selected ∑p^i\sum \hat { p } _ { i }∑p^​i​

Suppose That 25 Samples, Each of Size 200, We Selected $\sum \hat { p } _ { i }$