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 -chart.

To find the upper and lower control limits for an -chart, we need to calculate the average number of nonconformities per sample and the standard deviation of the nonconformities.

First, we calculate the average number of nonconformities per sample:
Number of nonconforming items = 95
Number of samples = 30
Number of items per sample = 100

Average number of nonconformities per sample = (Number of nonconforming items) / (Number of samples * Number of items per sample)
= 95 / (30 * 100)
= 0.0317

Next, we calculate the standard deviation of the nonconformities:
Number of nonconforming items = 95
Number of samples = 30
Number of items per sample = 100

To calculate the standard deviation, we need to know the proportion of nonconforming items for each sample. Assuming a binomial distribution, we can calculate the standard deviation using the formula:

Standard deviation = sqrt((p * (1 - p)) / (n-1))

Where:
p = Proportion of nonconforming items for each sample
n = Number of items per sample

To calculate p, we divide the number of nonconforming items by the total number of items:
p = (Number of nonconforming items) / (Number of items per sample)
= 95 / 100
= 0.95

Substituting these values into the formula, we get:
Standard deviation = sqrt((0.95 * (1 - 0.95)) / (100-1))
= sqrt(0.0475 / 99)
= 0.0220

Now, we can calculate the upper and lower control limits using the following formulas:
Upper control limit (UCL) = Average number of nonconformities per sample + 3 * Standard deviation
Lower control limit (LCL) = Average number of nonconformities per sample - 3 * Standard deviation

Let's substitute the values we calculated earlier:
Average number of nonconformities per sample = 0.0317
Standard deviation = 0.0220

UCL = 0.0317 + 3 * 0.0220
UCL = 0.0977

LCL = 0.0317 - 3 * 0.0220
LCL = -0.0322

Note that the lower control limit cannot be negative in this case since we are dealing with nonconformities, so we should set it to zero. Therefore, the lower control limit would be 0.

So, the upper control limit (UCL) for the -chart is 0.0977, and the lower control limit (LCL) is 0.