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Chapter 15: Problem 3

Control charts are to be constructed for samples of size \(n=4,\) and \(\bar{x}\) and \(s\) are computed for each of 20 preliminary samples as follows: \(\sum_{i=1}^{20} \bar{x}_{i}=4460 \quad \sum_{i=1}^{20} s_{i}=271.6\) (a) Calculate trial control limits for \(\bar{X}\) and \(S\) charts. (b) Assuming the process is in control, estimate the process mean and standard deviation.

Short Answer

Expert verified
\(\bar{X}\) control limits: [200.88, 245.12]; \(S\) control limits: [0, 28.35]. Process mean: 223; std dev: 14.68.

Step by step solution

01

Calculate the Grand Average for \(\bar{X}\) Chart

The grand average (\(\bar{\bar{x}}\)) for the \(\bar{X}\) chart is calculated by dividing the sum of the individual sample means by the number of samples. Here, \(\bar{\bar{x}} = \frac{\sum \bar{x}_i}{20} = \frac{4460}{20} = 223\).
02

Calculate the Average Standard Deviation

The average standard deviation (\( \bar{s} \)) is obtained by dividing the sum of the standard deviations of the samples by the number of samples. Thus, \( \bar{s} = \frac{\sum s_i}{20} = \frac{271.6}{20} = 13.58\).
03

Determine \(S\) Chart Control Limits

For \(n=4\), the factors are \(B_3 = 0\) and \(B_4 = 2.089\). The control limits for the \(s\) chart are computed as follows:Lower Control Limit (LCL): \(B_3 \times \bar{s} = 0 \times 13.58 = 0\).Upper Control Limit (UCL): \(B_4 \times \bar{s} = 2.089 \times 13.58 \approx 28.35\).
04

Compute \(\bar{X}\) Chart Control Limits

For the \(\bar{X}\) chart, using \(A_3 = 1.628\) we calculate:Lower Control Limit (LCL): \(\bar{\bar{x}} - A_3 \times \bar{s} = 223 - 1.628 \times 13.58 \approx 200.88\).Upper Control Limit (UCL): \(\bar{\bar{x}} + A_3 \times \bar{s} = 223 + 1.628 \times 13.58 \approx 245.12\).
05

Estimate Process Mean and Standard Deviation

Assuming the process is in control, the process mean is the grand average \(\bar{\bar{x}} = 223\). The process standard deviation (\(\sigma\)) is approximately \(\bar{s} / c_4\), where \(c_4\) for \(n=4\) is approximately 0.925. Therefore, \(\sigma \approx \frac{13.58}{0.925} \approx 14.68\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Process Control
Process control is an essential concept in the realm of quality management and engineering. It involves monitoring a production or operational process to ensure it operates within predefined limits. This helps to maintain consistency and quality in the final product. By using data collected from various samples, we can track how a process behaves over time.

Key elements of process control include:
  • Setting up control charts: These charts help visualize variations and identify any trends or abnormalities.
  • Identifying limits: Control limits are calculated to determine the expected range of process performance. Stepping outside these limits indicates potential issues.
  • Sampling: By analyzing samples instead of entire batches, time and resources can be conserved while still ensuring quality.
Process control is essential for optimizing efficiency, reducing waste, and ensuring product quality in any operational system.
Standard Deviation Estimation
Estimating standard deviation is a crucial part of understanding variability within a process. Standard deviation measures how spread out the data points are around the mean. In practical terms, it tells us how much individual measurements vary from the average.

In the context of control charts:
  • We calculate an average standard deviation (\( \bar{s} \)) from multiple samples to understand usual variability.
  • It's used to determine control limits, which demarcate the boundaries within which a process is expected to operate.
The smaller the standard deviation, the more consistent the process is. Larger deviations suggest a process is prone to higher fluctuation, prompting further investigation or tighter controls.
Statistical Process Monitoring
Statistical process monitoring revolves around keeping track of a process through statistical techniques, which frequently employ control charts. The main aim is to identify variations that are statistically significant and require intervention.

This process allows:
  • Early detection of potential issues before they escalate into major problems.
  • Identification of patterns that suggest a shift or drift in the process.
  • Ensuring that the process remains within the set control limits, thereby minimizing defects.
Through methods like control chart analysis, businesses can maintain product quality and operational efficiency with objective and quantitative data.
Mean and Standard Deviation
Mean and standard deviation are fundamental statistical metrics used in process control to measure central tendency and dispersion, respectively. The mean provides a simple average of the data points, offering a central value. The standard deviation indicates how much variation exists from this average.

In process control:
  • The mean helps determine the expected value of a process.
  • The standard deviation shows the expected variability around this mean.
  • Control charts use both the mean and standard deviation to set upper and lower control limits, which define the range of acceptable variation.
Together, these metrics provide a comprehensive overview of a process's performance and its stability over time.

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Most popular questions from this chapter

An article in Quality \& Safety in Health Care ["Statistical Process Control as a Tool for Research and Healthcare Improvement"' \((2003\) Vol. \(12,\) pp. \(458-464)]\) considered a number of control charts in healthcare. An \(X\) chart was constructed for the amount of infectious waste discarded each day (in pounds). The article mentions that improperly classified infectious waste (actually not hazardous) adds substantial costs to hospitals each year. The following tables show approximate data for the average daily waste per month before and after process changes, respectively. The process change included an education campaign to provide an operational definition for infectious waste. Before Process Change $$\begin{array}{lccccccccc}\text { Month } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\\text { Waste } & 6.9 & 6.8 & 6.9 & 6.7 & 6.9 & 7.5 & 7 & 7.4 & 7 \\\\\hline \text { Month } & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 \\\\\text { Waste } & 7.5 & 7.4 & 6.5 & 6.9 & 7.0 & 7.2 & 7.8 & 6.3 & 6.7\end{array}$$ After Process Change $$\begin{array}{lcccccccccccc}\text { Month } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\\\\text { Waste } & 5.0 & 4.8 & 4.4 & 4.3 & 4.6 & 4.3 & 4.5 & 3.5 & 4.0 & 4.1 & 3.8 & 5.0 \\\\\hline \text { Month } & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\\\\text { Waste } & 4.6 & 4.0 & 5.0 & 4.9 & 4.9 & 5.0 & 6.0 & 4.5 & 4.0 & 5.0 & 4.5 & 4.6 \\\\\hline \text { Month } & 25 & 26 & 27 & 28 & 29 & 30 & & & & & & \\\\\text { Waste } & 4.6 & 3.8 & 5.3 & 4.5 & 4.4 & 3.8 & & & & & &\end{array}$$ (a) Handle the data before and after the process change separately and construct individuals and moving-range charts for each set of data. Assume that assignable causes can be found and eliminate suspect observations. If necessary, revise the control limits. (b) Comment on the control of each chart and differences between the charts. Was the process change effective?

The thickness of a metal part is an important quality parameter. Data on thickness (in inches) are given in the following table, for 25 samples of five parts each. $$\begin{array}{cccccc}\hline \begin{array}{l}\text { Sample } \\\\\text { Number }\end{array} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\\\\hline 1 & 0.0629 & 0.0636 & 0.0640 & 0.0635 & 0.0640 \\\2 & 0.0630 & 0.0631 & 0.0622 & 0.0625 & 0.0627 \\\3 & 0.0628 & 0.0631 & 0.0633 & 0.0633 & 0.0630 \\\4 & 0.0634 & 0.0630 & 0.0631 & 0.0632 & 0.0633 \\\5 & 0.0619 & 0.0628 & 0.0630 & 0.0619 & 0.0625 \\\6& 0.0613 & 0.0629 & 0.0634 & 0.0625 & 0.0628 \\\7 & 0.0630 & 0.0639 & 0.0625 & 0.0629 & 0.0627 \\\8 & 0.0628 & 0.0627 & 0.0622 & 0.0625 & 0.0627 \\\9 & 0.0623 & 0.0626 & 0.0633 & 0.0630 & 0.0624 \\\10 & 0.0631 & 0.0631 & 0.0633 & 0.0631 & 0.0630 \\\11 & 0.0635 & 0.0630 & 0.0638 & 0.0635 & 0.0633 \\\12 & 0.0623 & 0.0630 & 0.0630 & 0.0627 & 0.0629 \\\13 & 0.0635 & 0.0631 & 0.0630 & 0.0630 & 0.0630 \\\14 & 0.0645 & 0.0640 & 0.0631 & 0.0640 & 0.0642 \\\15 & 0.0619 & 0.0644 & 0.0632 & 0.0622 & 0.0635 \\\16 & 0.0631 & 0.0627 & 0.0630 & 0.0628 & 0.0629 \\\17 & 0.0616 & 0.0623 & 0.0631 & 0.0620 & 0.0625 \\\18 & 0.0630 & 0.0630 & 0.0626 & 0.0629 & 0.0628 \\\19 & 0.0636 & 0.0631 & 0.0629 & 0.0635 & 0.0634 \\\20 & 0.0640 & 0.0635 & 0.0629 & 0.0635 & 0.0634 \\\21 & 0.0628 & 0.0625 & 0.0616 & 0.0620 & 0.0623 \\\22 & 0.0615 & 0.0625 & 0.0619 & 0.0619 & 0.0622 \\\23 & 0.0630 & 0.0632 & 0.0630 & 0.0631 & 0.0630 \\\24 & 0.0635 & 0.0629 & 0.0635 & 0.0631 & 0.0633 \\\25 & 0.0623 & 0.0629 & 0.0630 & 0.0626 & 0.0628 \\\\\hline\end{array}$$ (a) Using all the data, find trial control limits for \(\bar{X}\) and \(R\) charts, construct the chart, and plot the data. Is the process in statistical control? (b) Use the trial control limits from part (a) to identify outof-control points. If necessary, revise your control limits assuming that any samples that plot outside the control limits can be eliminated. (c) Repeat parts (a) and (b) for \(\bar{X}\) and \(S\) charts.

A article in Graefe's Archive for Clinical and Experimental Ophthalmology ["Statistical Process Control Charts for Ophthalmology," (2011, Vol. 249, pp. \(1103-1105\) ) ] considered the number of cataract surgery cases by month. The data are shown in the following table. (a) What type of control chart is appropriate for these data? Construct this chart. (b) Comment on the control of the process. (c) If necessary, assume that assignable causes can be found, eliminate suspect points, and revise the control limits. (d) In the publication, the data were approximated as normally distributed and an individuals chart was constructed. Construct this chart and compare it to the attribute chart you built in part (a). Why might an individuals chart be reasonable? $$\begin{array}{ccccccc}\hline \text { January } & \text { February } & \text { March } & \text { April } & \text {May } & \text { June } & \text { July } \\\61 & 88 & 80 & 68 & 80 & 70 & 60 \\\\\text { August } & \text { September } & \text { October } & \text { November } & \text { December } & & \\\56 & 72 & 118 & 106 & 60 & & \\\\\hline\end{array}$$

A normally distributed process uses \(66.7 \%\) of the specification band. It is centered at the nominal dimension, located halfway between the upper and lower specification limits. (a) Estimate \(P C R\) and \(P C R_{k}\). Interpret these ratios. (b) What fallout level (fraction defective) is produced?

Suppose that a quality characteristic is normally distributed with specifications from 20 to 32 units. (a) What value is needed for \(\sigma\) to achieve a \(P C R\) of \(1.5 ?\) (b) What value for the process mean minimizes the fraction defective? Does this choice for the mean depend on the value of \(\sigma\) ?
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