91Ó°ÊÓ

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?

Step by step solution

01

Calculate Overall Mean (\(\bar{\bar{X}}\))

To find the overall mean of the process, sum all the means (\(\bar{X}\) values) of the 25 samples and divide by 25. \(\bar{\bar{X}} = \frac{1}{25} \sum_{i=1}^{25} \bar{X}_i\)Performing the calculation:\(\bar{\bar{X}} = \frac{1}{25}(572 + 583.83 + 720.50 + \ldots + 530.67) = \text{value}\)

02

Calculate Average Range (\(\bar{S}\))

Calculate the average of all the standard deviations (\(s\)) across 25 samples:\(\bar{S} = \frac{1}{25} \sum_{i=1}^{25} s_i\)Perform the calculation with the given values.

03

Determine Control Limits for \(\bar{X}\)-Chart

Use the overall mean \(\bar{\bar{X}}\) and average range \(\bar{S}\) to calculate the control limits for the \(\bar{X}\)-chart. Employ the formula:- Lower Control Limit (LCL): \(\bar{\bar{X}} - A_3 \cdot \bar{S}\)- Upper Control Limit (UCL): \(\bar{\bar{X}} + A_3 \cdot \bar{S}\)Where \(A_3\) is a constant based on the sample size (\(n=6\)).Look up the value of \(A_3\) for \(n=6\) in standard statistical tables.

04

Determine Control Limits for S-Chart

For the S-chart, compute the control limits using \(B_3\) and \(B_4\) constants:- Lower Control Limit (LCL): \(B_3 \cdot \bar{S}\)- Upper Control Limit (UCL): \(B_4 \cdot \bar{S}\)Look up the values of \(B_3\) and \(B_4\) for \(n=6\) in standard statistical tables.

05

Analyze Process Control

Examine the calculated control limits against the \(\bar{X}\) and \(s\) values of each sample. If all values fall within the control limits, the process is in control. If any lie outside, it suggests the process is not in control.

06

Calculate Process Capability Indices

Calculate the Process Capability Ratio (PCR) and Process Capability Index (Cpk):- PCR: \(\frac{USL - LSL}{6\sigma}\), where \(USL = 830\) and \(LSL = 330\).- Cpk: \(\min\left(\frac{USL - \bar{\bar{X}}}{3\sigma}, \frac{\bar{\bar{X}} - LSL}{3\sigma}\right)\)Estimate \(\sigma\) using \(\bar{S}\) divided by suggested constant for sample size 6.

07

Determine New Variance for 6-Sigma Process

For PCRk = 2.0, Solve for new \(\sigma^2\) given\(PCRk = \min\left(\frac{USL - \bar{X}_{new}}{6\sigma_{new}}, \frac{\bar{X}_{new} - LSL}{6\sigma_{new}}\right) = 2.0\)Reformulate to find \(\sigma_{new}^2\).

08

Calculate ARL and Shift Detection Probability

Given new mean \(\mu=600\) and original control limits, use normal distribution properties to find probability of detection.Probability: \(P(\bar{X} > LCL)\). ARL: \(1/(1-P(\bar{X} > LCL))\).

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Key Concepts

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

Process Capability

Process capability is a measure that helps to understand how well a process can produce products that meet specifications. It tells us if a process can consistently produce items within given limits. This is crucial in production and quality control.

• **Specifications**: These are the upper and lower limits that a product must fall within to be considered acceptable. In the given exercise, the specification limits for the cement mortar briquettes are set at 580 ± 250, i.e., the range is between 330 and 830.
• **Capability Indices**: These are numerical values that describe how well a process fits within these limits. A key measure is the Process Capability Ratio (PCR). It's calculated as \[PCR = \frac{USL - LSL}{6\sigma}\]where USL and LSL stand for upper and lower specification limits, respectively.
• **Interpreting PCR**: A PCR greater than 1 indicates that the process can meet the specifications most of the time, while a PCR less than 1 suggests the process may not reliably meet the specifications.

This concept ensures that a process is not only effective but also efficient at producing within limits. Understanding process capability helps in minimizing defects and reducing waste.

Six Sigma

Six Sigma is a powerful method used to improve process quality by reducing defects and variability. It's often framed as making a process a "6-sigma process," which aims for very high precision in manufacturing.

• **Six Sigma Goal**: The primary aim is to achieve a process where the variation (standard deviation, \(\sigma\)) is controlled tightly enough so that very few defects occur per million opportunities. In many contexts, this means a defect rate of less than 3.4 per million occurrences.
• **Becoming a 6-Sigma Process**: This transformation requires minimizing variability. In exercises like the original one, this involves calculating new variances. For example, achieving a Process Capability Ratio (PCR) of 2.0, which is a marker of Six Sigma quality.
• **Benefit**: Embracing Six Sigma means less rework and scrap, leading to cost savings and improved customer satisfaction.

Six Sigma isn't just a statistical measure; it's a philosophy of continuous improvement, aiming to streamline processes and boost reliability.

Variance Reduction

Variance reduction is at the heart of quality control and improvement. When we reduce variance, we ensure the product output is consistent and predictable.

• **Importance of Variance**: High variance in a process indicates more unpredictability, leading to more defects and errors. It implies that the process is not stable and can lead to variations in product quality.
• **Techniques for Reduction**: These include statistical tools like control charts, comprehensive training for employees, and implementing standardized processes. By understanding and controlling the variance components using techniques like Analysis of Variance (ANOVA), businesses can identify key areas for improvement.
• **Outcome of Reduction**: The reduction in variance translates into a more consistent product, which aligns with quality standards and customer expectations.

Reducing variance is crucial for processes aiming for higher precision and making strides toward Six Sigma levels of performance.