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Chapter 15: Problem 44

Suppose that a quality characteristic is normally distributed with specifications at \(150 \pm 20 .\) Natural tolerance limits for the process are \(150 \pm 18\). (a) Calculate the process standard deviation. (b) Calculate \(P C R\) and \(P C R_{k}\) of the process. Calculate the percentage of the specification width used by the process. (c) What fallout level (fraction defective) is produced?

Short Answer

Expert verified
(a) \(\sigma = 6\). (b) \(PCR = 1.11,\ PCR_k = 1.11;\) 90% specification width. (c) Fallout is negligible, near 0%.

Step by step solution

01

Identify Process Limits

First identify the process limits based on the given natural tolerance limits. These are specified as \(150 \pm 18\), which means the process mean \(\mu\) is 150, and the limits are 132 and 168.
02

Calculate Process Standard Deviation (蟽)

The natural tolerance limits are defined as \(\mu \pm 3\sigma\). By setting this equation up based on the limits, we have:\[ 150 + 3\sigma = 168 \]Solving for \(\sigma\) gives us:\[ 3\sigma = 18 \]\[ \sigma = \frac{18}{3} = 6 \]Thus, the process standard deviation \(\sigma\) is 6.
03

Calculate Process Capability Ratio (PCR)

Use the formula:\[PCR = \frac{USL - LSL}{6\sigma}\]where the upper specification limit (USL) is 170 and the lower specification limit (LSL) is 130.Substituting the values, we get:\[PCR = \frac{170 - 130}{6 \, \times \, 6} = \frac{40}{36} \approx 1.11\]
04

Calculate Process Capability Index (PCRk)

To calculate PCRk, use the formula:\[PCR_k = \min \left(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma}\right)\]Substituting the values, we get:\[\frac{170 - 150}{18} = \frac{20}{18} \approx 1.11\]\[\frac{150 - 130}{18} = \frac{20}{18} \approx 1.11\]Therefore, \( PCR_k = \min(1.11, 1.11) = 1.11 \).
05

Calculate Percentage of Specification Width Used by the Process

The process width is given by \(6\sigma = 36\) and the specification width is \(40\).Percentage used by the process is:\[\frac{36}{40} \times 100\% = 90\%\]
06

Determine Fallout Level (Fraction Defective)

The fallout level can be determined using the standard normal distribution. Since the process capability ratios are greater than 1, it's unlikely there is significant fallout. However, integrate the normal distribution over the required limits to find specific probability (standard tables can be used). With around 1.11, probability of defectives approximately corresponds to \(P(z > |1.11|)\). With typical tables, the fallout is very close to 0, indicating almost full capability of the process within specifications.

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Key Concepts

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

Quality Control
Quality control is a crucial aspect of manufacturing and service processes. It ensures that products meet specified standards and function as intended. In this exercise, quality control involves assessing whether a process meets specific tolerance limits. This involves regular monitoring and adjustments, ensuring that products stay within desired specifications.
To achieve effective quality control:
  • Establish clear specifications or limits that products must adhere to.
  • Measure and analyze the process outputs to compare them against these limits.
  • Implement adjustments or improvements to maintain consistency.
Maintaining tight quality control not only enhances product reliability but also enhances customer satisfaction and reduces waste.
Process Standard Deviation
The process standard deviation, denoted by \( \sigma \), is a measure of the variability within a process. It tells us how much the process deviates from its mean, on average. In this exercise, the natural tolerance limits were given as \(150 \pm 18\). This allowed us to calculate the process standard deviation using the relation \( \mu \pm 3\sigma \).

Here's how you can think about process standard deviation:
  • It's calculated as one-third of the difference between the upper and lower natural tolerance limits.
  • A smaller \( \sigma \) implies a more consistent process with less variability.
  • A larger \( \sigma \) indicates more variability and potential issues in meeting specifications.
Understanding and controlling \( \sigma \) is key for any process improvement initiatives.
Capability Indices
Capability indices, such as the Process Capability Ratio (PCR) and the Process Capability Index (PCRk), are statistical measures used in quality control to assess a process's ability to produce outputs that meet specifications. These indices help determine if a process is capable of producing products within the specified limits consistently.

Types of Capability Indices
  • PCR assesses the potential capability of a process by comparing the specification width to the process width, expressed as \( \frac{USL - LSL}{6\sigma} \).
  • PCRk is a more refined index that considers shifts in the process mean, calculated by the minimum of two ratios: \( \frac{USL - \mu}{3\sigma} \) and \( \frac{\mu - LSL}{3\sigma} \).
A higher capability index (usually above 1.0) suggests that the process is capable and likely to produce products within specification limits.
Normal Distribution
The normal distribution is a key concept in statistics, frequently used in quality control to describe the underlying distribution of process variables. It is a symmetric, bell-shaped curve characterized by its mean \( \mu \) and standard deviation \( \sigma \).

Why Normal Distribution?
  • Many processes naturally follow a normal distribution, making it easier to apply statistical methods for analysis and improvement.
  • The properties of the normal distribution allow for predictions about the probability of different outcomes within specified ranges.
  • In quality control, understanding normal distribution is essential for calculating fallout levels, assessing probability of defects, and setting tolerance limits.
A good grasp of normal distribution helps in effectively interpreting process data and making informed decisions to enhance process capability.

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Most popular questions from this chapter

The following data were considered in quality Engineering ["Parabolic Control Limits for The Exponentially Weighted Moving Average Control Charts in Quality Engineering" (1992, Vol. 4(4)]. In a chemical plant, the data for one of the quality characteristics (viscosity) were obtained for each 12 -hour batch's at the batch completion. The results of 15 consecutive measurements are shown in the following table. $$\begin{array}{cccc}\hline \text { Batch } & \text { Viscosity } & \text { Batch } & \text { Viscosity } \\\\\hline 1 & 13.3 & 9 & 14.6 \\\2 & 14.5 & 10 & 14.1 \\\3 & 15.3 & 11 & 14.3 \\\4 & 15.3 & 12 & 16.1 \\\5 & 14.3 & 13 & 13.1 \\\6 & 14.8 & 14 & 15.5 \\\7 & 15.2 & 15 & 12.6 \\\8 & 14.9 & & \\\\\hline\end{array}$$ (a) Set up a CUSUM control chart for this process. Assume that the desired process target is \(14.1 .\) Does the process appear to be in control? (b) Suppose that the next five observations are 14.6,15.3,15.7 , \(16.1,\) and \(16.8 .\) Apply the CUSUM in part (a) to these new observations. Is there any evidence that the process has shifted out of control?

In a semiconductor manufacturing company, samples of 200 wafers are tested for defectives in the lot. See the number of defectives in 20 such samples in the following table. $$\begin{array}{cccc}\hline & \text { No. of } & \text { No. of } \\\\\text { Sample } & \text { Defectives } &\text { Sample } & \text { Defectives } \\\\\hline 1 & 44 & 11 & 52 \\\2 & 63 & 12 & 74 \\\3 & 40 & 13 & 43 \\\4 & 35 & 14 & 50 \\\5 & 29 & 15 & 60 \\\6 & 56 & 16 & 38 \\\7 & 40 & 17 & 36 \\\8 & 38 & 18 & 65 \\\9 & 74 & 19 & 41 \\\10 & 66 & 20 & 95\end{array}$$ (a) Set up a \(P\) chart for this process. Is the process in statistical control? (b) Suppose that instead of samples of size 200 , we have samples of size \(100 .\) Use the data to set up a \(P\) chart for this process. Revise the control limits if necessary. (c) Compare the control limits in parts (a) and (b). Explain why they differ.

Consider an \(\bar{X}\) control chart with k-sigma control limits and subgroup size \(n\). Develop a general expression for the probability that a point plots outside the control limits when the process mean has shifted by \(\delta\) units from the center line.

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 following dataset was considered in Quality Engineering ["Analytic Examination of Variance Components" \((1994-1995,\) Vol. 7(2)\(] .\) A quality characteristic for cement mortar briquettes was monitored. Samples of size \(n=6\) were taken from the process, and 25 samples from the process are shown in the following table: (a) Using all the data, calculate trial control limits for \(\bar{X}\) and \(S\) charts. Is the process in control? $$\begin{array}{ccc}\hline \text { Batch } & \bar{X} & s \\\\\hline 1 & 572.00 & 73.25 \\\2 & 583.83 & 79.30 \\\3 & 720.50 & 86.44 \\\4 & 368.67 & 98.62 \\\5 & 374.00 & 92.36 \\\6 & 580.33 & 93.50 \\\7 & 388.33 & 110.23 \\\8 & 559.33 & 74.79 \\\9 & 562.00 & 76.53 \\\10 & 729.00 & 49.80 \\\11 & 469.00 & 40.52\end{array}$$ $$\begin{array}{lll}12 & 566.67 & 113.82 \\\13 & 578.33 & 58.03 \\\14 & 485.67 & 103.33 \\\15 & 746.33 & 107.88 \\\16 & 436.33 & 98.69 \\\17 & 556.83 & 99.25 \\\18 & 390.33 & 117.35 \\\19 & 562.33 & 75.69 \\\20 & 675.00 & 90.10 \\\21 & 416.50 & 89.27 \\\22 & 568.33 & 61.36 \\\23 & 762.67 & 105.94 \\\24 & 786.17 & 65.05 \\\25 & 530.67 & 99.42\end{array}$$ (b) Suppose that the specifications are at \(580 \pm 250 .\) What statements can you make about process capability? Compute estimates of the appropriate process capability ratios. (c) To make this process a "6-sigma process," the variance \(\sigma^{2}\) would have to be decreased such that \(P C R_{k}=2.0 .\) What should this new variance value be? (d) Suppose the mean shifts to \(600 .\) What is the probability that this shift is detected on the next sample? What is the ARL after the shift?
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