If 30 samples of 100 items are tested for nonconformity, and 95 of the 3,000 items are defective, find the upper and lower control limits for a p-chart.
To find the upper and lower control limits for a p-chart, we first need to calculate the proportion of defects (p-hat) in each sample of 100 items.
Step 1: Calculate p-hat
p-hat = (Number of defects / Total number of items in the sample)
= (95 / 3000)
= 0.0317
Step 2: Calculate standard deviation (σ)
σ = sqrt[(p-hat * (1 - p-hat)) / n]
where n is the number of items in the sample, which is 100 in this case.
σ = sqrt[(0.0317 * (1 - 0.0317)) / 100]
= sqrt[(0.0317 * 0.9683) / 100]
= sqrt(0.03070411 / 100)
= sqrt(0.0003070411)
≈ 0.017514
Step 3: Calculate the control limits
Upper Control Limit (UCL) = p-hat + 3σ
Lower Control Limit (LCL) = p-hat - 3σ
UCL = 0.0317 + 3(0.017514)
= 0.0317 + 0.052542
≈ 0.0842
LCL = 0.0317 - 3(0.017514)
= 0.0317 - 0.052542
≈ -0.0219
However, since a negative lower control limit doesn't make sense, we adjust it to zero.
So, the upper control limit (UCL) for the p-chart is approximately 0.0842, and the lower control limit (LCL) is 0.