91Ó°ÊÓ

The production manager of a soap manufacturing company wants to monitor the weights of the bars produced on the line. Twenty bars are taken during a stable period of the process. The weights of the bars are shown in the following table. $$\begin{array}{cccc}\hline \text { Bar } & \text { Weight }(\mathrm{g}) & \text { Bar } & \text { Weight }(\mathrm{g}) \\\\\hline 1 & 74 & 11 & 99 \\\2 & 82 & 12 & 75 \\\3 & 97 & 13 & 77 \\\4 & 86 & 14 & 82 \\\5 & 71 & 15 & 93 \\\6 & 68 & 16 & 70 \\\7 & 83 & 17 & 87 \\\8 & 90 & 18 & 76 \\\9 & 88 & 19 & 84 \\\10 & 64 & 20 & 94\end{array}$$ (a) Using all the data, compute trial control limits for individual observations and moving-range charts. 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.

Step by step solution

01

Calculate Moving Range

The moving range is the absolute difference between consecutive sample weights. Compute the moving range for all pairs of weights:\[ \begin{align*}|82 - 74| &= 8, \|97 - 82| &= 15, \|86 - 97| &= 11, \|71 - 86| &= 15, \|68 - 71| &= 3, \|83 - 68| &= 15, \|90 - 83| &= 7, \|88 - 90| &= 2, \|64 - 88| &= 24, \|99 - 64| &= 35, \|75 - 99| &= 24, \|77 - 75| &= 2, \|82 - 77| &= 5, \|93 - 82| &= 11, \|70 - 93| &= 23, \|87 - 70| &= 17, \|76 - 87| &= 11, \|84 - 76| &= 8, \|94 - 84| &= 10.\end{align*} \]

02

Compute Average Moving Range (MRÌ„)

Calculate the average of all moving ranges:\[ \bar{MR} = \frac{8 + 15 + 11 + 15 + 3 + 15 + 7 + 2 + 24 + 35 + 24 + 2 + 5 + 11 + 23 + 17 + 11 + 8 + 10}{19} = \frac{247}{19} \approx 13.00 \]

03

Determine Control Limits for Moving Range Chart

Using known constants for control charts, determine the Upper Control Limit (UCL) for the moving range. Assume \(D_4 = 3.267\) and \(D_3 = 0\):\[ UCL_{MR} = D_4 \times \bar{MR} = 3.267 \times 13.00 \approx 42.47 \]The Lower Control Limit (LCL) is:\[ LCL_{MR} = D_3 \times \bar{MR} = 0 \times 13.00 = 0 \]

04

Calculate Control Limits for Individual Observations

Using the average moving range, compute the standard deviation estimate, \(\hat{\sigma}\):\[ \hat{\sigma} = \frac{\bar{MR}}{d_2} \approx \frac{13.00}{1.128} \approx 11.52 \]Assume \(XÌ„\) is the average weight:\[ \bar{X} = \frac{74 + 82 + 97 + 86 + 71 + 68 + 83 + 90 + 88 + 64 + 99 + 75 + 77 + 82 + 93 + 70 + 87 + 76 + 84 + 94}{20} = \frac{1500}{20} = 75.0 \]Determine the control limits for individual observations using \(\hat{\sigma}\).\[ UCL_{X} = \bar{X} + 3\hat{\sigma} = 75.0 + 3 \times 11.52 \approx 109.56 \]\[ LCL_{X} = \bar{X} - 3\hat{\sigma} = 75.0 - 3 \times 11.52 \approx 40.44 \]

05

Analyze Control Charts for Stability

Compare the moving range and individual weights against the calculated control limits: - If all moving ranges and individual weights fall within the respective UCL and LCL, the process is in control. - If any points fall outside these limits, the process is not in control, and the samples should be reviewed for assignable causes.

06

Revised Control Limits After Removing Outliers

Assuming points outside control limits are due to assignable causes (in the real process, these would be examined and potentially removed), recalculate \(\bar{X}\) and \(\bar{MR}\) without these points and establish new control limits as done in previous steps.

07

Estimate Process Mean and Standard Deviation

Once the process is in control, estimate the process mean and standard deviation from the revised control limits:\[ \text{Process Mean} = \bar{X} \]\[ \text{Process Standard Deviation} = \hat{\sigma} = \frac{\bar{MR}}{d_2} \]

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Key Concepts

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

Control Charts

Control charts are important tools in statistical process control. They help monitor the stability of a process over time. These charts allow us to see when a process is performing normally or when it indicates variation.

• Control charts consist of a central line for the average, an upper control limit (UCL), and a lower control limit (LCL).
• These limits are calculated based on the data collected and help in identifying when the process needs adjustment.
• They make it easy to detect trends and patterns, ensuring consistency in production.

Using control charts, managers can quickly see if a process is likely to produce large discrepancies in the product, like soap weight in our example.

Moving Range

Moving range charts are a specific type of control chart that focuses on variations between consecutive samples. This chart is crucial for tracking the spread of data over time.

• The moving range is the absolute difference between two successive measurements.
• It captures the differences in weights of the soap bars consecutively to see if these are consistent or show unacceptable variation.
• The average moving range helps establish the control limits for the chart.

By analyzing these differences, it's easy to observe patterns that might indicate erratic changes in the production line.

Process Mean

The process mean is the average value that a process aims to achieve. It's a key measure that describes the central tendency of your process data.

• In our scenario, it involves calculating the average weight of all soap bars.
• This value serves as the baseline for the control chart's central line.
• Calculating the process mean helps identify how closely the production aligns with the desired specifications.

When we find the process mean, we understand the typical performance of the process and how much adjustment it might require.

Standard Deviation

Standard deviation measures how spread out data points are from the mean. In the context of statistical process control, it's utilized to understand variation.

• A smaller standard deviation means data points are closely packed around the mean.
• It's used to help compute the control limits for the charts, ensuring they accurately reflect the process.
• In our exercise, we calculate it using the average moving range divided by a constant.

This measure provides insight into the process's consistency—whether producing soap bars or any other product consistently hits the mark or misses often.