91Ó°ÊÓ

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.

Step by step solution

01

Identify the Appropriate Control Chart

Since the data represents the number of spelling errors per week and can be categorized as count data, a c-chart is most suitable. A c-chart is used for monitoring the count of defects or errors in a constant-size sample.

02

Calculate Trial Control Limits

First, calculate the average number of spelling errors, denoted as \( \bar{c} \). Add up all the errors and divide by the number of weeks: \( \bar{c} = \frac{3 + 6 + 0 + 5 + 9 + 5 + 2 + 2 + 3 + 2 + 1 + 6 + 9 + 8 + 6 + 4 + 13 + 3 + 0 + 7}{20} = 4.5 \). Control limits are then calculated using the formulas: \[ UCL = \bar{c} + 3\sqrt{\bar{c}} \] \[ LCL = \bar{c} - 3\sqrt{\bar{c}} \] Calculate \( UCL = 4.5 + 3\sqrt{4.5} \approx 10.02 \) and \( LCL = 4.5 - 3\sqrt{4.5} \approx -1.02 \), but LCL cannot be negative, hence LCL = 0.

03

Plot the Control Chart

Create a graph with weeks on the x-axis and number of errors on the y-axis. Plot the control limits, with UCL at approximately 10.02, center line at \( \bar{c} = 4.5 \), and LCL at 0. Plot each week's spelling error count on the chart, marking any points outside of control limits.

04

Analyze Process for Statistical Control

Examine the plotted data to see if any points fall outside of the control limits. Points beyond the control limits suggest the process is out of control. Analyze these points to check for assignable causes.

05

Revise Control Limits (if necessary)

If out-of-control points are identified, assume assignable causes are found and remove those data points from the calculation. Recompute the average \( \bar{c} \) using only the points within control limits and adjust UCL and LCL using the revised \( \bar{c} \). Re-plot the revised chart and check for control again.

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Key Concepts

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

c-chart

A c-chart is a type of control chart used in Statistical Process Control to monitor the count of defects or occurrences within a fixed-size sample. In our exercise, we're dealing with spell-check errors over several weeks, which makes the c-chart an ideal tool.
The purpose of the c-chart is to visually track the number of occurrences, such as spelling errors, and determine if they deviate from what is expected under normal conditions.

• The y-axis on the c-chart represents the number of errors or defects.
• The x-axis typically denotes time, such as weeks in our case.
• The center line reflects the average number of defects, and control limits set thresholds for acceptable variation.

Using a c-chart helps you quickly see if there is a week with unusually high or low errors, which could signal a problem.

control limits

Control limits on a c-chart dictate the boundaries within which the process should operate under normal circumstances. These limits are not set arbitrarily; they are statistically calculated to represent the expected variation for a stable process.
For our exercise:

• The Center Line (\( \bar{c} \)) is the average number of errors over the observed periods.
• The Upper Control Limit (UCL) and Lower Control Limit (LCL) are calculated using the average and its standard deviation.

The formulas involve adding and subtracting three times the square root of the average to the average itself. This gives us:\[ UCL = \bar{c} + 3\sqrt{\bar{c}} \]\[ LCL = \bar{c} - 3\sqrt{\bar{c}} \]These boundaries help identify when the process might be experiencing something unusual, challenging you to investigate further.

statistical control

A process is considered in statistical control if all the data points lie within the calculated control limits, and there are no patterns or trends, like continuous upward or downward slopes.
To determine if our spelling error data is in statistical control, we plot the points on the c-chart in relation to the control limits.

• Points within the limits indicate the process is generally stable.
• Points outside suggest potential issues needing attention, such as investigating unusual events.

Continuously assessing statistical control ensures processes remain consistent, making it easier to predict outcomes and improve quality over time.

assignable causes

Assignable causes are significant sources of variation that can affect a process, causing it to go out of control. In the context of our spelling errors, an assignable cause could be a faulty spell-check algorithm for a particular week.
When points lie outside the control limits:

• Analyze them to search for any known or identifiable causes.
• Remove points attributed to known assignable causes to adjust the control limits better reflecting true process behavior.

Once the assignable causes are found and addressed, the remaining data will be used to recalculate control limits, providing a more accurate representation of the ongoing process stability. By doing this, the quality of output is improved, ensuring that systematic errors are reduced or eliminated efficiently.