Question

During long production runs of canned tomatoes, the average weights (in ounces) of samples of five cans of standard-grade tomatoes in pureed form were taken at 30 control points during an 11 -day period. These results are shown in the table. \({ }^{20}\) When the machine is performing normally, the average weight per can is 21 ounces with a standard deviation of 1.20 ounces. a. Compute the upper and lower control limits and the centerline for the \(\bar{x}\) chart. b. Plot the sample data on the \(\bar{x}\) chart and determine whether the performance of the machine is in control.

Short answer

Based on the given data and the control limits calculated in Step 2, determine if the plotted points on the \(\bar{x}\) chart fall within the upper and lower control limits. If there are no discernible patterns and most of the points are within the limits, the machine is working normally and is considered to be in control. If there are points outside the control limits or any patterns that show instability, the machine's performance is not in control and may require further investigation or adjustments.

Step-by-step solution

Calculate the centerline

The centerline for the \(\bar{x}\) chart is the average weight per can when the machine is performing normally. It's given in the problem as 21 ounces.

Compute the upper and lower control limits

To compute the upper and lower control limits, we will use the formula: Control limit = Centerline \(\pm\) 3 × (Standard deviation / \(\sqrt{n}\)) where n is the sample size. In our case: Centerline = 21 ounces Standard deviation = 1.20 ounces Sample size (n) = 5 cans Upper control limit = 21 + 3 × (1.20 / \(\sqrt{5}\)) = 21 + 1.61 ≈ 22.61 ounces Lower control limit = 21 - 3 × (1.20 / \(\sqrt{5}\)) = 21 - 1.61 ≈ 19.39 ounces

Plot the sample data on the \(\bar{x}\) chart

The next step is to plot the sample data from the 30 control points on the \(\bar{x}\) chart. Along the horizontal axis, we have the control points, and on the vertical axis, we have the average weight per can in ounces. At each control point, plot a point representing the average weight of the five cans. Draw lines to connect these points, creating the chart. Then, draw horizontal lines at the centerline (21 ounces), upper control limit (22.61 ounces), and lower control limit (19.39 ounces).

Analyze the chart

Analyze the \(\bar{x}\) chart by observing the plotted points and their relationship to the centerline and control limits. If most of the points are within the control limits and there is no discernible pattern (such as continuous increase or decrease in average weight), the machine's performance is considered to be in control. However, if there are points outside the control limits or any patterns indicating instability, the machine's performance is not in control, and further investigation or adjustments may be necessary. Based on the chart you've created, determine whether the machine's performance is in control.

Key concepts

Statistical Process Control

Statistical Process Control (SPC) is a method used by industries to maintain and improve quality by monitoring the manufacturing process through statistical means. It involves collecting data from production activities and plotting them on control charts to identify any variations from the standard process. The aim is to ensure the process stays within defined control limits and operates predictably. If an anomaly is detected, adjustments can be made to bring the process back in line before major quality issues arise.
Using SPC offers several benefits:

• It helps in reducing variability in production.
• It can identify potential quality problems before they become defects.
• It improves overall efficiency and consistency in manufacturing.

By employing SPC, manufacturers not only meet quality standards but also extend the lifespan of machinery by keeping operations within specified parameters. It is a proactive approach, focusing on understanding and controlling the process, which ultimately safeguards product quality.

Sampling Methods

Sampling in quality control is a strategic way to gather insights about a production process without inspecting every single item. Methods such as random sampling, systematic sampling, and stratified sampling are common. In random sampling, each item has an equal chance of being chosen. This method minimizes bias and provides a representative sample of the entire batch. Systematic sampling involves selecting every nth item, offering simplicity and ease of implementation during production.
Stratified sampling divides the production lot into distinct 'strata,' and random samples are drawn from each stratum. This ensures that all parts of the lot are proportionally represented in the samples.
Effective sampling:

• Requires determination of the appropriate sample size, which is crucial for reliable results.
• Incorporates randomness to minimize selection bias.
• Provides valuable information quickly, aiding timely quality decisions.

Whether choosing random, systematic, or stratified sampling, the intent is always to maintain control over manufacturing quality with minimal effort while maximizing accuracy.

Quality Control in Manufacturing

Quality Control (QC) is a critical aspect of manufacturing that ensures products meet specified standards and customer satisfaction. QC involves a series of checks and measures throughout the production cycle to identify and correct defects. It often uses tools like control charts, as seen in Statistical Process Control, to maintain consistent quality.
The process of QC in manufacturing includes:

• Setting quality benchmarks for raw materials and finished products.
• Conducting tests and inspections at various stages of production.
• Implementing feedback from these inspections to correct processes and improve standards.

QC doesn't just focus on the end product; it encompasses the entire manufacturing process. By employing thorough QC measures, manufacturers can reduce waste, avoid costly recalls, and maintain a competitive edge. The key to successful QC is finding a balance between rigorous standards and practical application, ensuring efficiency without compromising quality.