Model Answer
0 min readIntroduction
Statistical Process Control (SPC) is a method of quality control which employs statistical methods to monitor and control a process. Control charts are a key tool in SPC, visually displaying process data over time and helping to identify variations that may indicate a process is out of control. A p-chart, specifically, is used to monitor the proportion of defective items in a sample. Establishing control limits, typically at ±3 standard deviations from the mean, allows for the identification of assignable causes of variation and ensures process stability. This question requires the construction and interpretation of a p-chart to assess the stability of a manufacturing process.
(A) Constructing the Control Chart
The given data represents the number of defective units and the total units sampled for 10 samples. To construct a p-chart, we need to calculate the sample proportion defective (p̄) and the control limits (UCL and LCL).
1. Calculate the Sample Proportion Defective (pi) for each sample:
pi = (Number of Defective Units) / (Total Units Sampled)
In this case, pi = 150/1000 = 0.15 for all 10 samples as the data is the same for all samples.
2. Calculate the Average Proportion Defective (p̄):
p̄ = (Σpi) / n, where n is the number of samples.
p̄ = (10 * 0.15) / 10 = 0.15
3. Calculate the Standard Deviation of the Sample Proportions (σp):
σp = √[p̄(1-p̄)/n]
σp = √[0.15(1-0.15)/10] = √[0.15 * 0.85 / 10] = √0.01275 ≈ 0.1129
4. Calculate the Control Limits (UCL and LCL):
UCL = p̄ + 3σp = 0.15 + 3 * 0.1129 = 0.15 + 0.3387 = 0.4887
LCL = p̄ - 3σp = 0.15 - 3 * 0.1129 = 0.15 - 0.3387 = -0.1887
Since the LCL cannot be negative, it is set to 0.
Therefore, the control chart has a UCL of 0.4887 and an LCL of 0.
(B) Demonstrating Process Control
Now, let's plot the proportion defective for the next 5 samples (3, 4, 2, 0, and 7 defective units, assuming a constant sample size of 1000) and see if the process is under control.
1. Calculate the Proportion Defective for the next 5 samples:
- Sample 11: p11 = 3/1000 = 0.003
- Sample 12: p12 = 4/1000 = 0.004
- Sample 13: p13 = 2/1000 = 0.002
- Sample 14: p14 = 0/1000 = 0.000
- Sample 15: p15 = 7/1000 = 0.007
2. Graphical Representation:
Imagine a graph with sample number on the x-axis and proportion defective on the y-axis. The UCL is at 0.4887, the center line (p̄) is at 0.15, and the LCL is at 0. All five points (0.003, 0.004, 0.002, 0.000, 0.007) fall well within the control limits (0 to 0.4887).
Conclusion: Since none of the points fall outside the control limits, the process is considered to be under control. The observed variations are likely due to random chance and are within the expected range for a stable process.
Conclusion
In conclusion, we successfully constructed a p-chart with UCL at 0.4887 and LCL at 0, based on the initial data. The subsequent analysis of the next five samples demonstrated that all data points fell within the established control limits, indicating that the manufacturing process remains stable and under statistical control. Continuous monitoring using control charts is crucial for maintaining quality and identifying potential issues before they escalate.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.