91Ó°ÊÓ

An extrusion die is used to produce aluminum rods. The diameter of the rods is a critical quality characteristic. The following table shows \(\bar{x}\) and \(r\) values for 20 samples of five rods each. Specifications on the rods are \(0.5035 \pm 0.0010\) inch. The values given are the last three digits of the measurement; that is, 34.2 is read as \(0.50342 .\) $$\begin{array}{ccc}\hline \text { Sample } & \bar{x} & r \\\\\hline 1 & 34.2 & 3 \\\2 & 31.6 & 4 \\\3 & 31.8 & 4 \\\4 & 33.4 & 5 \\\5 & 35.0 & 4 \\\6 & 32.1 & 2 \\\7 & 32.6 & 7\end{array}$$ $$ \begin{array}{|rrr}8 & 33.8 & 9 \\\9 & 34.8 & 10 \\\10 & 38.6 & 4 \\\\\hline 11 & 35.4 & 8 \\\12 & 34.0 & 6\\\13 & 36.0 & 4 \\\14 & 37.2 & 7 \\\15 & 35.2 & 3 \\\16 & 33.4 & 10 \\\17 & 35.0 & 4 \\\\\hline 18 & 34.4 & 7 \\\\\hline 19 & 33.9 & 8 \\\20 & 34.0 & 4\end{array}$$ (a) Using all the data, find trial control limits for \(\bar{X}\) and \(R\) charts, construct the chart, and plot the data. (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. Estimate \(\sigma .\)

Step by step solution

01

Calculate Overall Averages

First, calculate the overall average of \( \overline{x} \) values by summing all the \( \overline{x} \) values and dividing by the number of samples. Similarly, calculate the average range \( \overline{R} \) by summing all the \( r \) values and dividing by the number of samples.\[ \overline{\bar{x}} = \frac{\sum \bar{x}}{20} \]\[ \overline{R} = \frac{\sum r}{20} \]After computation:\( \overline{\bar{x}} = \frac{679}{20} = 33.95 \)\( \overline{R} = \frac{108}{20} = 5.4 \)

02

Determine Control Chart Constants

For a sample size of 5, the control chart constants from standard statistical tables are:- \( A_2 = 0.577 \)- \( D_3 = 0 \)- \( D_4 = 2.115 \)

03

Calculate Trial Control Limits for the \( \bar{X} \) Chart

Use the overall average \( \overline{\bar{x}} \) and \( \overline{R} \) with the constant \( A_2 \) to find the control limits for the \( \bar{X} \) chart.- Upper Control Limit (UCL): \[ UCL_{\bar{x}} = \overline{\bar{x}} + A_2 \cdot \overline{R} = 33.95 + 0.577 \times 5.4 = 37.06 \]- Lower Control Limit (LCL): \[ LCL_{\bar{x}} = \overline{\bar{x}} - A_2 \cdot \overline{R} = 33.95 - 0.577 \times 5.4 = 30.84 \]

04

Calculate Trial Control Limits for the \( R \) Chart

Use \( \overline{R} \) with the constants \( D_3 \) and \( D_4 \) to find the control limits for the \( R \) chart.- Upper Control Limit (UCL): \[ UCL_R = D_4 \cdot \overline{R} = 2.115 \times 5.4 = 11.421 \]- Lower Control Limit (LCL): \[ LCL_R = D_3 \cdot \overline{R} = 0 \times 5.4 = 0 \]

05

Plot and Analyze Data

Plot the \( \bar{x} \) and \( r \) values for each sample on their respective control charts using the control limits found in Steps 3 and 4. Check to see if any points fall outside the control limits. Out-of-control points, if any, should be identified.

06

Revise Control Limits if Necessary

If any points lie outside the control limits, indicate potential out-of-control samples. Remove these points and recalculate the averages \( \overline{\bar{x}} \) and \( \overline{R} \) considering only in-control samples, and then revise the control limits using the same formulas.

07

Estimate Process Standard Deviation

Using the updated control limits and/or the non-out-of-control range, estimate the process standard deviation with:\[ \sigma = \frac{\overline{R}}{d_2} \]For sample size 5, \(d_2 = 2.326\). Thus:\[ \sigma \approx \frac{5.4}{2.326} = 2.32 \] (using the previous \(\overline{R}\) value if no recalculations are needed)

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Key Concepts

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

Statistical Process Control

The idea behind Statistical Process Control (SPC) is to monitor and control a process through statistical methods. This includes understanding variabilities, detecting trends, and ensuring processes work as intended. In our example, we use control charts to keep track of the diameter consistency of aluminum rods. By leveraging sample data points, SPC can help identify whether any irregularities are happening due to natural fluctuations, or if there is an indication of a deeper problem in the process.

Control charts generally consist of a center line, which represents the overall average value we're keeping track of, and upper/lower control limits that suggest the boundaries of typical variation within the process. If a plotted point falls outside these boundaries, it prompts further investigation into the possible causes of the deviation. This helps in making informed decisions needed to maintain product quality.

• Measurement tools: SPC uses various statistical tools like control charts, histograms, and Pareto analysis to monitor process performance.
• Objective: The primary goal is to detect when a process is going out of control and take corrective actions before an issue arises.

Quality Control

Quality Control (QC) involves ensuring that a product or service meets specified quality standards. In the context of our exercise, QC revolves around maintaining the diameter of the aluminum rods within the specified range of 0.5035 ± 0.0010 inch. Consistency in product quality not only satisfies customer requirements but also minimizes waste, reduces costs, and sustains a healthy production process.

Quality control includes inspecting samples taken from the process to see if they conform to these standards, as well as using control charts. When data points fall beyond control limits, it suggests that the process might be malfunctioning and corrective action may need to be taken.

• Inspection: Regular examination of products to ensure they meet set specifications.
• Feedback loop: Continuous monitoring allows for feedback, which can be used to fine-tune processes.
• Cost-effectiveness: Investing in QC can save money by catching defects early.

Process Standard Deviation

The process standard deviation ( \( \sigma \)) is a statistical measure representing the dispersion of process data. It tells us how much the diameter of the aluminum rods is spreading around the average value. Having a smaller standard deviation indicates a more consistent process with data points closely clustered around the mean. Conversely, a larger standard deviation suggests greater variability, often underscored by potential quality issues in the manufacturing line.

In our step-by-step solution, we estimate the process standard deviation using the formula: \[ \sigma = \frac{\overline{R}}{d_2} \] , where \( \overline{R} \) is the average range of the samples, and \( d_2 \) is a constant that depends on the sample size. With consistent sizing at five samples per group, we calculated using \( d_2 = 2.326 \). This method of estimating \( \sigma \) aids in understanding overall process consistency and spotlighting areas that may require process enhancements.

• Significance: Helps quantify the natural variability in the process.
• Calculation: Determined by average range and statistical constants.
• Optimization: Reducing standard deviation is crucial for process optimization.