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Chapter 15: Problem 133

Suppose that an \(\bar{X}\) chart is used to control a normally distributed process and that samples of size \(n\) are taken every \(n\) hours and plotted on the chart, which has \(k\) -sigma limits. (a) Find a general expression for the expected number of samples and time that is taken until a false signal is generated. (b) Suppose that the process mean shifts to an out-of-control state, say \(\mu_{1}=\mu_{0}+\delta \sigma\). Find an expression for the expected number of samples that is taken until a false action is generated. (c) Evaluate the in-control ARL for \(k=3 .\) How does this change if \(k=2 ?\) What do you think about the use of 2-sigma limits in practice? (d) Evaluate the out-of-control ARL for a shift of 1 sigma, given that \(n=5\).

Short Answer

Expert verified
For part (c), in-control ARL is about 370 for k=3 and about 22 for k=2. For part (d), out-of-control ARL with a shift of 1 sigma and n=5 is approximately 44.

Step by step solution

01

Understanding False Alarms in Control Charts

In control charts, a false alarm is when the chart signals that the process is out of control when it actually isn't. The Average Run Length (ARL) is the average number of samples taken until a signal is generated. For in-control processes, ARL is computed as the reciprocal of the false alarm probability.
02

Expression for In-Control ARL

The false alarm probability for a control chart with k-sigma limits is related to the probability that a normally distributed observation falls outside those limits. For a standard normal distribution, this is \(P(Z > k) + P(Z < -k)\). Thus, for an in-control ARL, the equation is:\[\text{In-control ARL} = \frac{1}{P(Z > k) + P(Z < -k)}\]
03

Compute Probability of False Alarm for Specific k

Using the standard normal distribution table, find the values: - For \(k=3\), \(P(Z>3) \approx 0.00135\). - For \(k=2\), \(P(Z>2) \approx 0.0228\). Then:\(P(-3 < Z < 3) = 0.9973\) and \(P(-2 < Z < 2) = 0.9544\).Thus, for \(k=3\), \(0.0027\) is the false alarm probability and for \(k=2\), it is \(0.0456\).
04

Calculate In-Control ARL for k=3 and k=2

- For \(k=3\):\[\text{ARL} = \frac{1}{0.0027} \approx 370.37\]- For \(k=2\):\[\text{ARL} = \frac{1}{0.0456} \approx 21.93\]
05

Discuss the Use of 2-Sigma Limits

Using 2-sigma limits significantly reduces the average run length, making false signals more frequent. This may lead to unnecessary interventions, so it is generally considered less practical for detecting true process shifts without increasing false alarms.
06

Out-of-Control ARL for a Shifted Mean

When the process mean shifts to \(\mu_1 = \mu_0 + \delta \sigma\), the probability a sample exceeds control limits changes. This can be calculated using the non-centrality parameter:\[P\left(\left| \bar{X} - \mu_1 \right| > k\frac{\sigma}{\sqrt{n}} \right)\]The numerator shifts further from zero depending on \(\delta\).
07

ARL for Specific Shift \(\delta = 1\) and \(n = 5\)

For \(\delta = 1\), calculate new \(P\left(Z > 3 - \delta \right)\), using \[ Z = \frac{\bar{X} - (\mu_0 + 1*\sigma)}{\sigma/\sqrt{5}}\]This leads to a standard normal with mean \(\delta \sqrt{n}=1*\sqrt{5}=\approx2.24\), and \[\text{Out-of-control ARL} = \frac{1}{P(\text{above } k-\delta\sqrt{n})}\approx 44.11\] after calculating respective probabilities.

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Key Concepts

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

Average Run Length (ARL)
In control charts, the Average Run Length (ARL) is a crucial metric. It tells us the average number of samples or monitoring points needed to trigger an alarm. This can happen when a process deviates from its expected behavior.
ARL is calculated differently depending on whether the process is in control or out of control. For an in-control process, ARL is the inverse of the false alarm probability. This probability represents the likelihood of incorrectly signaling an out-of-control state. Conversely, for an out-of-control process, ARL is calculated by considering the probability of detecting a real process shift.
Understanding ARL enables businesses to set appropriate control limits, thus balancing between reacting to real issues and avoiding unnecessary interventions.
In-Control and Out-of-Control Processes
An in-control process is one that operates within established control limits, indicating it behaves as expected. These control limits are usually set at specific sigma levels, commonly 2 or 3 sigma. The term "sigma" refers to the standard deviation, a measure of variation or spread in the dataset.
On the other hand, an out-of-control process is one that exhibits abnormal variance or drift in its mean. This suggests a deviation from typical operating conditions, possibly requiring correction. Recognizing and distinguishing between these two states is vital for quality control. It ensures that the process runs smoothly and any issues are flagged promptly.
  • In-Control: Falls within control limits, expected process behavior.
  • Out-of-Control: Exceeds control limits, indicating issues.
False Alarm Probability
False alarm probability is the chance of a control chart giving a signal indicating an out-of-control process when it is actually in control. This is a critical concept, as too many false alarms can lead to unnecessary actions, wasting time and resources.
Lowering this probability means reducing the chance of false alarms but might delay detecting an actual issue. The probability depends on the sigma limits set for the control chart. For instance, 3-sigma limits have a lower false alarm probability compared to 2-sigma limits. This trade-off is part of configuring control charts effectively.
  • 3-Sigma: Lower false alarm chance, higher ARL.
  • 2-Sigma: Higher false alarm chance, lower ARL.
Process Mean Shift
A process mean shift refers to a change in the average level of the process output. It signifies an alteration from the norm, which might be due to various factors like machinery wear or changes in input materials. Detecting this shift swiftly is essential for maintaining process quality and efficiency.
In control chart terminology, a shift in the mean can be characterized by the parameter \(\delta\), defined as the shift in terms of the process standard deviation \(\sigma\). When a mean shifts significantly, especially by a full null deviation or more, it signals potential problems requiring immediate investigation and corrective action.
Understanding process mean shifts helps organizations better interpret control charts, thus optimizing process performance.

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Most popular questions from this chapter

An early example of SPC was described in Industrial Quality Control ["The Introduction of Quality Control at Colonial Radio Corporation" (1944, Vol. 1(1), pp. \(4-9\) ) . The following are the fractions defective of shaft and washer assemblies during the month of April in samples of \(n=1500\) each $$\begin{array}{cccc}\hline & \text { Fraction } & & \text { Fraction } \\\\\text { Sample } & \text { Defective } & \text { Sample } & \text { Defective } \\\\\hline 1 & 0.11 & 11 & 0.03 \\\2 & 0.06 & 12 & 0.03 \\\3 & 0.1 & 13 & 0.04 \\\4 & 0.11 & 14 & 0.07 \\\5 & 0.14 & 15 & 0.04 \\\6 & 0.11 & 16 & 0.04 \\\7 & 0.14 & 17 & 0.04 \\\8 & 0.03 & 18 & 0.03 \\\9 & 0.02 & 19 & 0.06 \\\10 & 0.03 & 20 & 0.06\end{array}$$ (a) Set up a \(P\) chart for this process. Is this process in statistical control? (b) Suppose that instead of \(n=1500, n=100\). Use the data given to set up a \(P\) chart for this process. Revise the control limits if necessary. (c) Compare your control limits for the \(P\) charts in parts (a) and (b). Explain why they differ. Also, explain why your assessment about statistical control differs for the two sizes of \(n\).

The purity of a chemical product is measured every two hours. The results of 20 consecutive measurements are as follows: $$\begin{array}{cccc}\text { Sample } & \text { Purity } & \text { Sample } & \text { Purity } \\\\\hline 1 & 89.11 & 11 & 88.55 \\\2 & 90.59 & 12 & 90.43 \\\3 & 91.03 & 13 & 91.04 \\\4 & 89.46 & 14 & 88.17 \\\5 & 89.78 & 15 & 91.23 \\\6 & 90.05 & 16 & 90.92 \\\7 & 90.63 & 17 & 88.86 \\\8 & 90.75 & 18 & 90.87 \\\9 & 89.65 & 19 & 90.73 \\\10 & 90.15 & 20 & 89.78\end{array}$$ (a) Set up a CUSUM control chart for this process. Use \(\sigma=\) 0.8 in setting up the procedure, and assume that the desired process target is \(90 .\) Does the process appear to be in control? (b) Suppose that the next five observations are 90.75,90.00 , \(91.15,90.95,\) and \(90.86 .\) Apply the CUSUM in part (a) to these new observations. Is there any evidence that the process has shifted out of control?

Suppose that the average number of defects in a unit is known to be 8 . If the mean number of defects in a unit shifts to \(16,\) what is the probability that it is detected by a \(U\) chart on the first sample following the shift (a) if the sample size is \(n=4 ?\) (b) if the sample size is \(n=10 ?\) Use a normal approximation for \(U\).

An automatic sensor measures the diameter of holes in consecutive order. The results of measuring 25 holes are in the following table. $$\begin{array}{crcc}\hline \text { Sample } & \text { Diameter } & \text { Sample } & \text { Diameter } \\\\\hline 1 & 9.94 & 14 & 9.99 \\\2 & 9.93 & 15 & 10.12 \\\3 & 10.09 & 16 & 9.81 \\\4 & 9.98 & 17 & 9.73 \\\5 & 10.11 & 18 & 10.14 \\\6 & 9.99 & 19 & 9.96 \\\7 & 10.11 & 20 & 10.06 \\\8 & 9.84 & 21 & 10.11 \\\9 & 9.82 & 22 & 9.95 \\\10 & 10.38 & 23 & 9.92 \\\11 & 9.99 & 24 & 10.09 \\\12 & 10.41 & 25 & 9.85 \\\13 & 10.36 & &\end{array}$$ (a) Using all the data, compute trial control limits for individual observations and moving-range charts. Construct the control chart and plot the data. Determine whether the process is in statistical control. If not, assume that assignable causes can be found to eliminate these samples and revise the control limits. (b) Estimate the process mean and standard deviation for the in-control process.

The following data are the number of spelling errors detected for every 1000 words on a news Web site over 20 weeks. $$\begin{array}{cccc}\hline \text { Week } & \begin{array}{l}\text { No. of Spelling } \\ \text { Errors }\end{array} & \text { Week } & \begin{array}{l}\text { No. of Spelling } \\\\\text { Errors }\end{array} \\\\\hline 1 & 3 & 11 & 1 \\\2 & 6 & 12 & 6 \\\3 & 0 & 13 & 9 \\\4 & 5 & 14 & 8 \\\5 & 9 & 15 & 6 \\\6 & 5 & 16 & 4 \\\7 & 2 & 17 & 13 \\\8 & 2 & 18 & 3 \\\9 & 3 & 19 & 0 \\\10 & 2 & 20 & 7 \\\\\hline\end{array}$$ (a) What control chart is most appropriate for these data? (b) Using all the data, compute trial control limits for the chart in part (a), construct the chart, and plot the data. (c) Determine whether the process is in statistical control. If not, assume that assignable causes can be found and outof-control points eliminated. Revise the control limits.
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