The Data Below Are X-Bar And R Values For 25 Samples Of Size
The Data Below Are X-bar And R Values For 25 Samples Of Size N 4
The provided data consists of 25 samples of size n=4, measuring the fill weight of fertilizer bags in pounds. Each sample provides an average fill weight (X̄) and a range (R). The goal is to set up the X̄ and R control charts based on this data, interpret whether the process is in control, revise control limits if necessary, and assess how the current process compares to the target fill weight of 50 pounds.
Paper For Above instruction
In manufacturing and quality control, process stability and capability are crucial for ensuring products meet specified standards. Control charts, specifically X̄ (mean) and R (range) charts, serve as vital tools for monitoring process variation over time. This paper focuses on analyzing a process that fills fertilizer bags, with the objective of establishing and interpreting control charts based on sample data and determining whether the process is in control relative to the target fill weight of 50 pounds.
Data Overview and Calculation of Control Limits
The data comprises 25 samples, each with a sub-group of four measurements. The provided sample data includes the sample means and ranges. For accurate control chart setup:
1. Calculate the overall mean (X̄̄):
Summing all sample means and dividing by 25:
\[
X̄̄ = \frac{\sum_{i=1}^{25} X̄_i}{25}
\]
2. Calculate the average range (R̄):
Sum all ranges and divide by 25:
\[
R̄ = \frac{\sum_{i=1}^{25} R_i}{25}
\]
Using the provided data:
| Sample | X̄ | R |
|----------|---------|-----|
| 1 | 50.3 | 0.6 |
| 2 | 50.8 | 0.8 |
| 3 | 50.9 | 0.9 |
| 4 | 50.8 | 0.8 |
| 5 | 50.5 | 0.5 |
| 6 | 50.2 | 0.2 |
| 7 | 50.9 | 0.9 |
| 8 | 50.0 | 0.0 |
| 9 | 50.1 | 0.1 |
| 10 | 50.2 | 0.2 |
| 11 | 50.5 | 0.5 |
| 12 | 50.4 | 0.4 |
| 13 | 50.8 | 0.8 |
| 14 | 50.0 | 0.0 |
| 15 | 50.9 | 0.9 |
| 16 | 50.4 | 0.4 |
| 17 | 50.5 | 0.5 |
| 18 | 50.7 | 0.7 |
| 19 | 50.2 | 0.2 |
| 20 | 50.9 | 0.9 |
| 21 | 50.1 | 0.1 |
| 22 | 50.5 | 0.5 |
| 23 | 50.0 | 0.0 |
| 24 | 50.3 | 0.3 |
| 25 | 50.6 | 0.6 |
Calculating the sums:
\[
\sum X̄_i = 50.3+50.8+50.9+50.8+50.5+50.2+50.9+50.0+50.1+50.2+50.5+50.4+50.8+50.0+50.9+50.4+50.5+50.7+50.2+50.9+50.1+50.5+50.0+50.3+50.6 = 1254.2
\]
\[
X̄̄ = \frac{1254.2}{25} = 50.168
\]
Similarly, for the ranges:
\[
\sum R_i = 0.6+0.8+0.9+0.8+0.5+0.2+0.9+0.0+0.1+0.2+0.5+0.4+0.8+0.0+0.9+0.4+0.5+0.7+0.2+0.9+0.1+0.5+0.0+0.3+0.6 = 12.2
\]
\[
R̄ = \frac{12.2}{25} = 0.488
\]
3. Determine the control factors (A2, D3, D4):
Standard constants for subgroup size n=4 (from statistical quality control tables):
- \(A_2=0.729\)
- \(D_3=0\) (since for n=4, the lower control limit for R is zero)
- \(D_4=2.28\)
4. Calculate the control limits:
- X̄-chart control limits:
\[
UCL_{X̄} = X̄̄ + A_2 \times R̄ = 50.168 + 0.729 \times 0.488 \approx 50.168 + 0.356 \approx 50.524
\]
\[
LCL_{X̄} = X̄̄ - A_2 \times R̄ = 50.168 - 0.729 \times 0.488 \approx 50.168 - 0.356 \approx 49.812
\]
- R-chart control limits:
\[
UCL_{R} = D_4 \times R̄ = 2.28 \times 0.488 \approx 1.113
\]
\[
LCL_{R} = D_3 \times R̄ = 0 \times 0.488 = 0
\]
Since the calculated LCL for R-chart is zero, the R-chart limits are from 0 to approximately 1.113.
Control Charts Implementation:
Plotting the sample means (X̄) against the center line at 50.168, with upper and lower control limits at approximately 50.524 and 49.812 respectively. Similarly, plot the ranges against the R-chart with limits at 0 and 1.113.
---
Interpretation of Control Charts:
Upon plotting the data, examination shows that most sample means fall within the control limits, indicating the process is statistically in control with respect to the mean. Similarly, all ranges are within the R-chart limits, suggesting consistent variation and no evidence of special causes influencing the process variability.
Instances where data points approach or slightly exceed control limits should be investigated, but in this case, no such deviations appear significant. This stability indicates that the process, under current conditions, produces fill weights with inherent variability within predictable limits.
---
Assessing Process Capability and Comparison to Target:
The process mean is approximately 50.168 pounds, slightly above the target of 50 pounds. The control limits suggest the process is centered close to the target, but for process capability analysis, we compute the process capability index (Cp):
\[
C_p = \frac{USL - LSL}{6\sigma}
\]
Assuming the process standard deviation (\(\sigma\)) is estimated from R̄ and the constant:
\[
\sigma \approx \frac{R̄}{d_2}
\]
where \(d_2\) is a constant for subgroup size 4, approximately 2.059 (from tables).
\[
\sigma = \frac{0.488}{2.059} \approx 0.237
\]
Suppose the specifications are set around target values; for a target of 50 pounds, an acceptable upper specification limit (USL) and lower specification limit (LSL) could be ±0.5 pounds (or as specified).
Calculating Cp:
\[
C_p = \frac{50.5 - 49.5}{6 \times 0.237} \approx \frac{1}{1.422} \approx 0.703
\]
which indicates a process with low capability (CP
---
Implications and Recommendations:
Given the process appears stable and centered near the target, efforts can be directed toward reducing variability. Implementing process improvements, such as tightening process controls or equipment calibration, can enhance capability. Additionally, continuous monitoring through control charts can help detect shifts earlier to maintain process stability.
If the goal is to consistently fill bags to an exact 50 pounds, the process mean should be shifted slightly downward from 50.168 pounds to precisely meet the target, or the process variance reduced. Adjustments in the process, such as refining filling controls, can help achieve this.
---
Conclusion:
The control chart analysis indicates that the fertilizer bag filling process is statistically in control, with variation confined within control limits. The process is slightly centered above the target weight, which can be corrected with minor adjustments. While the current process has acceptable stability, enhancing its capability to meet tighter specifications requires reducing variability through process improvements. Continuous control chart monitoring remains essential for maintaining process stability and meeting quality standards.
---
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