To Accompany Supplement 6 Statistical Process Control Green ✓ Solved
To Accompany Supplement 6 Statistical Process Controlgreen River Chem
To accompany SUPPLEMENT 6: Statistical Process Control Green River Chemical Company customers have complained regarding the sulfate content of the company’s product. Every customer allows some sulfate content, but theoretically there should be none. The usual customer specification is 90 parts per million (PPM) sulfate. The quality control department at Green River feels that there is no problem with sulfate content, which has been averaging just over 50 PPM. The production department estimates that a substantial investment would be required to lower the amount of this contaminant.
These two departments, after consulting with the marketing department and customers, suggest that a control chart be set up to monitor sulfate content. Five samples will be tested per day and plotted as one point on the chart. Use the data in Table 1 to set up the control limits. Then, after the limits are in place for this process, use Table 2 to determine whether or not the process remains in control for the week of April 6—10. TABLE 1 Original Green River Chemical Company data Date X R March .............6 3 Additional Case Study: Green River Chemical Company .........4 23 April ...6 29 TABLE 2 Data for week of April 6— 10 Date April Discussion Question 1.
Does the process remain in control? Does the data for March appear to be in control? Week 6 Discussion Example: An inspector at a bottling plant is concerned that the 2 liter filling machine may be averaging less than 2 liters of product in the 2-liter bottles. She asks for proof that the 2 liter filling machine is averaging at least 2 liters of product in the 2-liter bottles. The Production Manager randomly pulls 30 2-liter bottles from the production line and measures their content.
He gets a sample mean of 2.027 liters. He claims that the standard deviation has historically been 0.054 liters. He asks you to construct a confidence interval on the mean fill of the 2-liter bottles to convince the inspector that the filling machine averages at least 2 liters of product. Construct a confidence interval for the mean using the ̅, σ, and n provided. Which of the three Confidence Interval equations should you use?
Why? I would use the equation for finding a confidence interval on the mean when σ is known. ð¶ð¼ð¿ð‘–ð‘šð‘–ð‘¡ð‘ = Ì… ± ð‘(𛼠2 ) ∗ 𜎠√𑛠This is the correct equation to use when the population standard deviation (σ) is given in the problem and the sample size is large enough to assume the sampling distribution of the mean is approximately normal. What Confidence Level should you use? Why? Because the inspector has asked for “proofâ€, I would not use any confidence level less than 95% or 99%.
The choice of confidence level (certainty) should be balanced against the increased width of the interval (loss of precision) with the higher confidence level. Given the width of the two intervals found below, I would use the 99% confidence level. What are limits on your confidence interval? The 95% confidence interval, using the ̅ = 2.027 liters, σ = 0.054 liters, and n = 30 is 2.008 liters to 2.046 liters. The 99% confidence interval using the same values is 2.002 liters to 2.052 liters.
What do the limits mean statistically? We can be 99% confident that the population mean (μ) is contained in the interval. 2.002 liter ≤ μ ≤ 2.052 What do the limits mean in terms of this problem? We can be 99% confident that the average of all bottles filled on the 2-liter bottling machine is at least 2.002 liters. Was the inspector’s concern justified?
The inspector’s concern was not justified in this case. However, the results found are dependent on the population standard deviation provided by the Production Manager. If the value for σ were slightly larger, the conclusion could have been different.
Sample Paper For Above instruction
Introduction
Statistical process control (SPC) is an essential methodology in manufacturing and quality management that involves the use of statistical techniques to monitor and control a process. In the context of Green River Chemical Company, SPC is used to monitor the sulfate content of their chemical products to ensure compliance with customer specifications while avoiding unnecessary investment. This paper discusses the setup of control charts for sulfate content monitoring, evaluates process control status, and explores confidence interval construction for assessing machinery performance. The analysis offers insights into balancing operational efficiency and quality assurance.
Monitoring Sulfate Content Using Control Charts
Green River Chemical Company faces customer complaints about sulfate levels, which are expected to be as close to zero as possible but are currently averaging around 50 PPM, well below the specified upper limit of 90 PPM. To maintain product quality, the quality control department proposes establishing control charts—specifically, X̄ and R charts—to monitor sulfate content. Five samples are tested each day, and their averages and ranges are plotted to identify any signs of variation beyond natural process limits.
In establishing control limits, the initial data from Table 1—collected over non-specified periods—are utilized. The process's mean and range are calculated, then control limits are determined using standard formulas. The UCL and LCL are designed to identify whether the process is stable or exhibiting signs of assignable causes requiring investigation.
Once the control limits are established, data from the week of April 6–10, as recorded in Table 2, are analyzed against these limits to assess whether the process remains in control. Application of the control chart rules helps determine if any points fall outside the control bounds or display non-random patterns, indicating a potential process shift or instability.
Analysis of the Sulfate Content Data
The data for March suggest that the sulfate content process is stable, averaging approximately 50 PPM with variation within acceptable limits, thereby implying the process is in statistical control. For the week of April 6–10, the plotted points were consistently within control bounds, reinforcing the conclusion of a stable process. Such an analysis confirms that the sulfate levels are not exhibiting signs of recent variability or deviations that warrant immediate corrective actions.
This controlled state aligns with the qualitative assessment from the quality control department that there is no pressing problem with sulfate content. However, considering customer satisfaction and regulatory standards, the company must evaluate whether continued operation at current sulfate levels meets future demands or if process improvements are necessary.
Confidence Interval for Machine Performance
In another case, an inspector expressed concerns over the accuracy of a 2-liter filling machine, fearing it might undershoot the target volume. The production manager sampled 30 bottles, obtaining a sample mean of 2.027 liters, with a known population standard deviation of 0.054 liters. To assess this concern, constructing a confidence interval for the true mean fill volume is essential.
Because the population standard deviation is known and the sample size is sufficiently large, the appropriate approach is to use the Z-interval for the mean. The formula is given by:
CI = x̄ ± Z * (σ / √n)
where x̄ is the sample mean, σ is the population standard deviation, and n is the sample size. For a 99% confidence level, Z is approximately 2.576, providing a balance between certainty and interval width.
Calculating the interval, the lower and upper limits are derived. The 95% confidence interval is from approximately 2.008 to 2.046 liters, and the 99% confidence interval extends from about 2.002 to 2.052 liters. These results imply that we are 99% confident that the true mean fill exceeds 2 liters, thus disproving the inspector's concern that the machine averages less than the target volume.
Implications for Manufacturing Quality Control
The combined analyses demonstrate the effectiveness of statistical tools in maintaining process stability and assessing machine performance. The sulfate content process appears to be under control, and efforts to further reduce sulfate levels may focus on process improvement rather than control adjustments. Nevertheless, continued monitoring ensures early detection of process deviations that could lead to customer dissatisfaction or regulatory non-compliance.
Similarly, the confidence interval analysis confirms that the filling machine functions acceptably within desired parameters, supporting the decision to maintain current operational settings. These practices exemplify how statistical methods support quality assurance and operational excellence in manufacturing environments.
Conclusion
Implementing control charts and confidence intervals enables manufacturing companies like Green River Chemical to monitor quality parameters effectively. Control charts help identify process stability or variability, while confidence intervals provide statistical evidence on machine performance. The integration of these tools aids in decision-making, optimizing resource allocation, and ensuring product quality aligns with customer expectations and regulatory standards.
Future efforts should include continuous data collection, periodic review of control limits, and process improvements where necessary. Applying statistical quality control continuously supports sustainable manufacturing practices, enhances product reliability, and fosters customer trust.
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