Petroleum Is Cracked To Make Gasoline And Other Products
Petroleum Is Cracked To Make Gasoline And Other Products Such As Lubri
Petroleum is cracked to make gasoline and other products such as lubricating oil and kerosene. The process is too slow to allow sample sizes greater than one. The following are the octane ratings of the output from the process recorded at intervals of thirty minutes each. Obs. Number Octane rating 1 86. Compute the control limits of appropriate control charts, which can be used to monitor the process. Use alpha = 0.0026.
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Monitoring manufacturing processes is essential to ensure that products meet quality standards and that processes operate within desirable variability limits. Control charts are statistical tools used to observe process stability over time. In this context, we are analyzing the octane ratings obtained from the petroleum cracking process to determine whether the process remains in control or exhibits variability that warrants intervention. Given the data and a significance level (alpha) of 0.0026, the task involves calculating the appropriate control chart limits.
Data and Context
The process produces octane ratings, which are critical parameters influencing fuel quality. Data is collected at thirty-minute intervals, with each observation representing a single sample. The provided data indicate one initial observation of an octane rating of 86. To construct control charts, especially the X̄ and R charts, multiple observations are typically required, but given the constraints and the data, we may proceed under assumptions or with a simplified approach, such as calculating control limits based on estimated process standard deviation.
Understanding Control Charts
Control charts are categorized mainly into two types:
- The X̄ chart, which monitors the process mean.
- The R chart (range chart), which monitors the process variability.
Given the small sample size (single observations per interval), an individual moving range (I-MR) chart is appropriate, as it combines individual measurements with their ranges (between successive measurements). This choice is suitable because the process is slow, and only one data point per interval is recorded.
Calculating Control Limits
The significance level alpha (0.0026) corresponds to the probability of a Type I error, i.e., the false detection of a process out-of-control condition. This alpha value relates to the control limits because control limits are set such that the probability of a point falling outside (due to random variation) is alpha.
For individual control charts (I-MR):
- The control limits are calculated based on the average moving range (MR̄) and an estimated process standard deviation (σ̂).
- The constant d2, used for estimating σ̂ from the average range, is approximately 1.128 for a sample size of 2 (since the range is between two observations).
Step 1: Compute the average of the observed octane ratings
Since only one observation is provided (86), we cannot compute an average across multiple points. To proceed, we need more data points. Let's assume, for illustration purposes, that additional sample data at various intervals are as follows:
| Observation Number | Octane Rating |
|----------------------|--------------|
| 1 | 86 |
| 2 | 85 |
| 3 | 87 |
| 4 | 86 |
| 5 | 88 |
| 6 | 85 |
| 7 | 86 |
| 8 | 87 |
| 9 | 86 |
| 10 | 85 |
This hypothetical data will allow us to compute the necessary statistics and control limits.
Step 2: Calculate the individual differences (ranges)
Calculate the differences between successive observations:
| Difference | Value |
|--------------|--------|
| |85 - 86| = 1 |
| |87 - 85| = 2 |
| |86 - 87| = 1 |
| |88 - 86| = 2 |
| |85 - 88| = 3 |
| |86 - 85| = 1 |
| |87 - 86| = 1 |
| |86 - 87| = 1 |
| |85 - 86| = 1 |
Average of these ranges, MR̄:
MR̄ = (1 + 2 + 1 + 2 + 3 + 1 + 1 + 1 + 1) / 9 = 14 / 9 ≈ 1.56
Step 3: Estimate process standard deviation (σ̂)
σ̂ = MR̄ / d2 ≈ 1.56 / 1.128 ≈ 1.38
Step 4: Determine the control limits
Using the standard normal distribution corresponding to alpha = 0.0026, find the z-value:
Since alpha = 0.0026 (two-sided), each tail is alpha/2 ≈ 0.0013. Looking up in the standard normal table:
z ≈ 3.00 (approximate from standard z-tables for 0.0013 in each tail).
Control limits for individual observations are:
- Upper Control Limit (UCL) = mean + z * σ̂
- Lower Control Limit (LCL) = mean - z * σ̂
Calculate the process mean:
X̄ = (86 + 85 + 87 + 86 + 88 + 85 + 86 + 87 + 86 + 85) / 10 = 860 / 10 = 86.0
Finally:
UCL = 86.0 + 3.00 * 1.38 ≈ 86.0 + 4.14 ≈ 90.14
LCL = 86.0 - 3.00 * 1.38 ≈ 86.0 - 4.14 ≈ 81.86
Conclusion
The control chart limits operative for monitoring the octane ratings are approximately:
- UCL ≈ 90.14
- LCL ≈ 81.86
Any subsequent measurement falling outside these bounds would signal a potential out-of-control process, prompting investigation to maintain process stability and product quality. These limits are based on assumed data; in practice, actual observations are necessary for precise calculations.
References
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