Question

A manufacturing process is designed to produce an electronic component for use in small portable television sets. The components are all of standard size and need not conform to any measurable characteristic, but are sometimes inoperable when emerging from the manufacturing process. Fifteen samples were selected from the process at times when the process was known to be in statistical control. Fifty components were observed within each sample, and the number of inoperable components was recorded. $$6,7,3,5,6,8,4,5,7,3,1,6,5,4,5$$ Construct a \(p\) chart to monitor the manufacturing process.

Short answer

Answer: The four steps to construct a p-chart are: 1. Calculate the proportion of inoperable components (p) for each sample. 2. Calculate the average proportion of inoperable components (p-bar) across all samples. 3. Calculate the upper control limit (UCL) and lower control limit (LCL) using the average proportion and sample size. 4. Plot the p-chart.

Step-by-step solution

Calculate the proportion of inoperable components (p) for each sample

Divide the number of inoperable components in each sample by the total number of components (50) in that sample to find the proportion of inoperable components: $$p_i = \frac{\text{number of inoperable components in sample i}}{\text{total number of components in sample i}}$$ Calculate \(p_i\) for each sample.

Calculate the average proportion of inoperable components (p-bar) across all samples

Add up all the proportions calculated in step 1, then divide the sum by the number of samples (15) to find p-bar: $$\bar{p} = \frac{1}{n}\sum_{i=1}^n p_i$$

Calculate the upper control limit (UCL) and lower control limit (LCL)

Using the average proportion and the sample size, calculate the UCL and LCL as follows: $$\text{UCL} = \bar{p} + 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}}$$ $$\text{LCL} = \bar{p} - 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}}$$ Where \(n_i\) is the number of components in a sample (50 in this case).

Plot the p-chart

On a graph, plot the proportion of inoperable components (\(p_i\)) for each sample along the vertical axis and the sample number along the horizontal axis. Draw horizontal lines at UCL and LCL to show the control limits. If all data points fall within the control limits, it indicates that the process is stable and in control. If any data points fall outside the control limits, it signals that the process may be out of control and requires further investigation.

Key concepts

Understanding Control Limits in a p-chart

Control limits are vital in statistical quality control, especially when using a p-chart. They help us determine if a process is stable or if adjustments are needed. In a p-chart, we deal with proportions of non-conforming items in samples, such as inoperable components.

Here is how control limits work:

• The **Upper Control Limit (UCL)** indicates the upper threshold of acceptable variation. If data points exceed this limit, the process may have special cause variations.
• The **Lower Control Limit (LCL)** sets the lower boundary. If data points fall below this level, it indicates a possible issue that needs checking.

To calculate these limits, we use the formula:
\[\text{UCL} = \bar{p} + 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}} \]\[\text{LCL} = \bar{p} - 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}} \]
where \(\bar{p}\) is the average proportion of defectives, and \(n_i\) is the sample size. Control limits here are essential for identifying unusual patterns that could lead to improvements.

Average Proportion in p-chart Analysis

The average proportion is a critical component when constructing a p-chart. It is the baseline measure that shows the typical rate of defects or non-conformity in sampled units.

Here's why understanding the average proportion matters:

• It reflects the overall quality level of the production process.
• It serves as the central line on a p-chart, against which variations are measured.

To find the average proportion \(\bar{p}\), sum up all individual sample proportions and divide by the total number of samples:
\[\bar{p} = \frac{1}{n}\sum_{i=1}^n p_i\]
Calculating the average proportion allows us to understand the process's expected behavior under normal conditions. It’s also a foundation for setting control limits, making it a cornerstone of effective quality monitoring.

Recognizing Statistical Control Using a p-chart

Statistical control is achieved when a process operates consistently within defined control limits. The goal of creating a p-chart is to visualize whether a process is in statistical control, meaning it varies naturally due to random causes.

Here's what this implies:

• All plotted points fall within the UCL and LCL, demonstrating a reliable and stable process.
• Control limits guide us in ensuring that any variations are due to normal fluctuations rather than systemic errors.

When a process is in statistical control:

• It suggests that the process's resources and procedures are functioning as expected.
• It minimizes the risk of product waste or quality issues, ultimately optimizing production efficiency.

Monitoring a process through a p-chart helps identify when factors like machine wear or changes in materials could cause deviations, enabling proactive management of potential quality issues.