Samples Of N4 Items Are Taken From A Process At Regular Inte
Samples Of N 4 Items Are Taken From A Process At Regular Inter
Samples of n = 4 items are taken from a process at regular intervals. A normally distributed quality characteristic is measured, and x-bar and s values are calculated at each sample. After 50 subgroups have been analyzed, we have Σ x̄i = 1,000 and Σ si = 72.
a) Compute the control limit for the x̄ and s charts.
b) Assume that all points on both charts plot within the control limits. What are the natural tolerance limits of the process?
c) If the specification limits are 19 ± 4, what are your conclusions regarding the ability of the process to produce items conforming to specifications?
d) Assuming that if an item exceeds the upper specification limit it can be reworked and if it is below the lower limit it must be scrapped, what percent scrap and rework is the process now producing?
e) If the process were centered at μ = 19.0, what would be the effect on the percent scrap or rework?
Sample Paper For Above instruction
Control charts are essential tools in statistical process control (SPC), enabling manufacturers and quality engineers to monitor process variations and maintain product quality. This paper explores the calculations and implications associated with control charts for a process where samples of size four are taken at regular intervals. Specifically, the focus is on determining control limits for x̄ and s charts, establishing natural tolerance limits, assessing process capability relative to specified limits, and analyzing the impact of process centering on defect rates.
Control Limit Calculations for x̄ and s Charts
Given the data, where 50 subgroups have been analyzed with Σ x̄i = 1,000 and Σ si = 72, the average of the sample means (x̄̄) is calculated as:
x̄̄ = (Σ x̄i) / 50 = 1000 / 50 = 20
The average of the sampling standard deviations (s̄) is:
s̄ = (Σ si) / 50 = 72 / 50 = 1.44
To construct control limits for the x̄ chart, we use the standard formula:
UCL_{x̄} = x̄̄ + A_2 * s̄
LCL_{x̄} = x̄̄ - A_2 * s̄
Where A_2 is a constant depending on the sample size (n=4). From standard SPC tables, A_2 for n=4 is approximately 0.73:
UCL_{x̄} = 20 + 0.73 * 1.44 ≈ 20 + 1.0512 ≈ 21.05
LCL_{x̄} = 20 - 1.0512 ≈ 18.95
Control limits for the s chart are calculated as:
UCL_s = B_4 * s̄
LCL_s = B_3 * s̄
Where B_4 ≈ 2.282 and B_3 ≈ 0 for n=4. Consulting SPC constants, we find:
- B_4 ≈ 2.282
- B_3 ≈ 0
Thus:
UCL_s = 2.282 * 1.44 ≈ 3.29
LCL_s = 0 * 1.44 = 0
In summary, the control limits are:
- x̄ chart: UCL ≈ 21.05, LCL ≈ 18.95
- s chart: UCL ≈ 3.29, LCL = 0
Natural Tolerance Limits of the Process
Assuming the process is stable, the natural tolerance limits are typically defined as:
Mean ± 3 standard deviations
Given the process mean (μ_p) is approximated by x̄̄ = 20, and the process standard deviation (σ) by s̄ = 1.44, the natural limits are:
Lower limit: 20 - 3 * 1.44 ≈ 20 - 4.32 = 15.68
Upper limit: 20 + 4.32 ≈ 24.32
Therefore, the natural tolerance limits are approximately from 15.68 to 24.32, indicating the range within which the process naturally varies when in control.
Process Capability Relative to Specification Limits
The specification limits are set at 19 ± 4, so from 15 to 23. Since the natural tolerance limits extend from approximately 15.68 to 24.32, the process is capable of producing items within the specifications, assuming the process mean remains centered at 19 and is stable. The narrower natural limits compared to the specifications suggest that the process is capable but requires monitoring to prevent drift.
Percent Scrap and Rework Analysis
The specification limits are 15 and 23. With a process mean at 19 and standard deviation estimated at 1.44, the z-scores for the limits are:
- Upper limit: z = (23 - 19) / 1.44 ≈ 2.78
- Lower limit: z = (15 - 19) / 1.44 ≈ -2.78
Using standard normal distribution tables:
- Percent exceeding upper limit: P(z > 2.78) ≈ 0.0027 (0.27%)
- Percent below lower limit: P(z
Total percent of items outside specifications (scrap):
≈ 0.27% + 0.27% = 0.54%
Rework applies to items exceeding the upper limit, which are about 0.27%. Therefore, the process currently produces approximately 0.54% of items requiring rework or scrap, primarily driven by the tails of the process distribution.
Effect of Centering the Process at μ=19.0
Centering the process exactly at μ=19.0 would reduce the proportion of items outside the specification limits, as the process would be ideally aligned with the specifications. The tail probabilities for deviations would decrease slightly, further lowering scrap and rework percentages, approaching the minimal levels dictated by the normal distribution tails.
Conclusion
Control charts serve as vital tools for monitoring process variations and ensuring quality. Accurate calculation of control limits aids in distinguishing special causes from common variation. The analysis demonstrates that with proper control, the process is capable of producing within specifications, and measures such as process centering and stability are crucial for minimizing scrap and rework. Maintaining process stability through regular monitoring of x̄ and s charts, along with understanding natural tolerance limits, is essential for continuous quality improvement.
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