Final answer:
The three-sigma control limits for the process producing refrigerators, based on 16 nonconformities in 30 units, are approximately 0 and 2.724.
Step-by-step explanation:
To determine the three-sigma control limits for nonconformities in a process of producing refrigerators, we first need to calculate the mean number of nonconformities per unit and the standard deviation. From the preliminary data, we know that there were 16 nonconformities over 30 refrigerators.
The mean number of nonconformities per refrigerator (\(\bar{x}\)) is calculated by dividing the total number of nonconformities by the total number of units inspected:
\(\bar{x} = \frac{16}{30} \approx 0.5333\)
For a control chart for nonconformities (also known as a u-chart), the control limits are based on the Poisson distribution, where the standard deviation is the square root of the mean. Thus, the standard deviation (\(\sigma\)) is:
\(\sigma = \sqrt{\bar{x}} = \sqrt{0.5333} \approx 0.7303\)
The three-sigma (3\(\sigma\)) control limits are calculated as follows:
But since we cannot have a negative count of nonconformities, the LCL is set to 0 if the calculation gives a negative result.
Therefore, the UCL is:
UCL = 0.5333 + 3(0.7303) \approx 2.7242
And the LCL is:
LCL = max(0, 0.5333 - 3(0.7303)) = 0
Thus, the three-sigma control limits are approximately 0 and 2.724 for the process.