A process for producing electronic circuits has achieved very high yield levels. An average of only 10 defective parts per million is currently produced.
1. What are the upper and lower control limits for a sample size of 100?
2. Recompute the upper and lower control limits for a sample size of 10,000?
3. Which of these two sample sizes would you recommend? Explain.
Management has reconsidered the method of quality control and has decided to use process control by variables instead of attributes. For variables control a circuit voltage will be measured based on a sample of only five circuits. The past average voltage for samples of size 5 has been 3.1 volts, and the range has been 1.2 volts.
1. What would the upper and lower control limits be for the resulting control charts (average and range)?
2. Five samples of voltage are taken with the results in the table below. What action should be taken if any?
3. Discuss the pros and cons of using this variables control chart versus the control chart discussed in the first part of the assignment. Which do you prefer?
Ex. 1 2 3 4 5
x 3.6 3.3 2.6 3.9 3.4
R 2.0 2.6 0.7 2.1 2.3
To answer the questions related to control limits for production processes and quality control, statistical calculations will be used. Let's begin by explaining how to calculate the upper and lower control limits for a sample size.
1. Upper and lower control limits for a sample size of 100:
To calculate the control limits, we need to know the average defect rate and the sample size. In this case, the average defect rate is given as 10 defective parts per million (10 ppm).
The first step is to convert the defect rate into a fraction, which is 10/1,000,000 (since 1 ppm = 1/1,000,000). Then, multiply this fraction by the sample size (100) to get the average number of defective parts in the sample.
Upper Control Limit (UCL) = Average Defect Rate + 3 * Square Root of (Average Defect Rate * (1 - Average Defect Rate) / Sample Size)
Lower Control Limit (LCL) = Average Defect Rate - 3 * Square Root of (Average Defect Rate * (1 - Average Defect Rate) / Sample Size)
Plug in the values to calculate the control limits.
2. Recompute the upper and lower control limits for a sample size of 10,000:
Repeat the same calculation using a sample size of 10,000.
3. Which sample size would you recommend?
To decide which sample size is recommended, consider the trade-off between the precision of the control limits and the cost of sampling. Larger sample sizes provide more precise estimates of the process average and better control limits. However, they also increase the cost of sample collection and analysis. Therefore, it's important to balance statistical accuracy and practicality.
For the second part of the assignment:
1. Control limits for variable control charts:
For variable control charts, two types of charts are used: an average chart and a range chart.
For average chart:
UCL = Average Voltage + 3 * Average Range
LCL = Average Voltage - 3 * Average Range
For range chart:
UCL = D4 * Range
LCL = D3 * Range
D4 and D3 are constants obtained from statistical tables based on the sample size.
2. Action based on sample results:
To determine if any action is required, compare the measured values (x) and the range (R) from the sample with the calculated control limits. If any value falls outside the control limits, it indicates an out-of-control condition, and appropriate action should be taken to investigate and correct the process.
3. Pros and cons of variables control chart vs. attributes control chart:
- Variables control charts provide more detailed information about the process, including the average and variation, compared to attributes control charts, which only focus on pass/fail or defect/non-defect.
- Variables control charts require measuring and recording numerical data, which may be more time-consuming and expensive compared to attributes control charts.
- Variables control charts are suitable for processes where measurements are essential, such as circuit voltage, while attributes control charts are more suitable for quality characteristics that can be easily categorized or counted.
- The choice between variables and attributes control charts depends on the nature of the process, the availability of measurement data, and the specific quality characteristics being monitored.
In summary, the answers to the questions depend on the specific data provided for calculations and analysis. To answer the questions accurately, the given data and formulas need to be applied properly.