The Canine Gourmet Company Produces Delicious Dog Treats
The Canine Gourmet Company Produces Delicious Dog Treats For Canines W
The Canine Gourmet Company produces dog treats with a focus on quality control to ensure consistent product weight. Management has set a target process mean weight of 45 grams per packet. To monitor process stability, an inspector takes samples of 10 packets at regular intervals and records the weights. When the process is in control, the average range of these samples is 6 grams. The task is to design appropriate R and X̄ control charts and analyze recent sample data to determine whether the process is in control.
Paper For Above instruction
The objective of this analysis is to establish control charts suitable for monitoring the weight of dog treat packets produced by The Canine Gourmet Company and to interpret recent sample data to assess process stability.
Designing the R and X̄ Control Charts
The first step involves calculating the control limits for the process. Given the information provided, the key parameters are as follows:
- Target process mean, \(\mu = 45\) grams
- Average range, \(\bar{R} = 6\) grams
- Sample size, \(n = 10\)
Control charts serve to monitor whether the process remains steady over time, detecting any signals indicative of variation caused by assignable causes. The X̄ chart tracks changes in the process average, while the R chart monitors variability within the samples.
Calculating Control Limits
Control limits are derived using standard formulas. The process’s average range (\(\bar{R}\)) is given; hence, the estimate of process variability can be obtained via the constants \(A_2\) and \(D_3, D_4\), which are tabled for specific sample sizes.
For \(n = 10\), the relevant constants are:
- \(A_2 = 0.308\)
- \(D_3 = 0\)
- \(D_4 = 1.72\)
Using these, we calculate the control limits:
- \(\text{X̄} \text{ Chart}\):
\[
\text{Upper Control Limit (UCL)} = \bar{\bar{X}} + A_2 \times \bar{R}
\]
\[
\text{Center Line (CL)} = \bar{\bar{X}} = 45 \text{ grams}
\]
\[
\text{Lower Control Limit (LCL)} = \bar{\bar{X}} - A_2 \times \bar{R}
\]
Substituting the known values:
\[
UCL_{X̄} = 45 + 0.308 \times 6 = 45 + 1.848 = 46.848 \text{ grams}
\]
\[
LCL_{X̄} = 45 - 0.308 \times 6 = 45 - 1.848 = 43.152 \text{ grams}
\]
- R Chart:
\[
UCL_{R} = D_4 \times \bar{R} = 1.72 \times 6 = 10.32 \text{ grams}
\]
\[
LCL_{R} = D_3 \times \bar{R} = 0 \times 6 = 0 \text{ grams}
\]
The control limits are then:
| Chart | Center Line | UCL | LCL (for R chart, LCL ≥ 0) |
|-------|--------------|-------|---------------------------|
| X̄ | 45 grams | 46.848 grams | 43.152 grams |
| R | 6 grams | 10.32 grams | 0 grams |
These control charts will allow the inspector to monitor the process by plotting sample means and ranges over time, flagging any points outside the control limits as signals of potential out-of-control situations.
Analyzing Recent Data
The recent five samples of size 10 are used to evaluate the process control status. Suppose the sample data are as follows:
| Sample | Sample Mean (X̄) | Sample Range (R) |
|---------|------------------|------------------|
| 1 | 44.5 grams | 5 grams |
| 2 | 45.2 grams | 4.8 grams |
| 3 | 45.1 grams | 6.2 grams |
| 4 | 44.8 grams | 5.1 grams |
| 5 | 45.4 grams | 5.7 grams |
Plotting these points against the control limits reveals that all sample means are within the control limits (43.152 to 46.848 grams) and that the ranges are well within the R chart control limits (0 to 10.32 grams). For instance:
- Sample 1: Mean = 44.5 (within limits), Range = 5 (within limit)
- Sample 2: Mean = 45.2 (within limits), Range = 4.8 (within limit)
- Sample 3: Mean = 45.1 (within limits), Range = 6.2 (within limit)
- Sample 4: Mean = 44.8 (within limits), Range = 5.1 (within limit)
- Sample 5: Mean = 45.4 (within limits), Range = 5.7 (within limit)
No points fall outside the control limits, and no discernible patterns, such as trends or cycles, are evident in the control charts. This indicates that the process is stable and in control, with variations likely due to common causes.
Conclusion
Based on the computed control limits and the analysis of the recent samples, it can be concluded that the process producing dog treat packets is in statistical control. This stability ensures consistent product quality, aligning with management’s specifications. Regular monitoring and updating of control charts are essential to sustain process control over time.
References
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