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Chapter 7: Problem 48

The sample means were calculated for 30 samples of size \(n=10\) for a process that was judged to be in control. The means of the \(30 \bar{x}\) -values and the standard deviation of the combined 300 measurements were \(\overline{\bar{x}}=20.74\) and \(s=.87,\) respectively. a. Use the data to determine the upper and lower control limits for an \(\bar{x}\) chart. b. What is the purpose of an \(\bar{x}\) chart? c. Construct an \(\bar{x}\) chart for the process and explain how it can be used.

Short Answer

Expert verified
Answer: An \(\bar{x}\) chart, also known as a control chart for the process mean, is a type of control chart used to monitor and control the stability of a process over time. The main components of the chart are the individual sample means plotted on the y-axis, the overall mean \(\overline{\bar{x}}\) as a central line, and the Upper and Lower Control Limits (UCL and LCL) as parallel lines. The main purpose of the \(\bar{x}\) chart is to detect small changes in the process mean and distinguish between assignable-cause variation and common-cause variation, thus facilitating continuous process improvement.

Step by step solution

01

Understand the given data

The sample means were calculated for 30 samples of size \(n=10\). The mean of the 30 \(\bar{x}\) values is \(\overline{\bar{x}}=20.74\) and the standard deviation of the combined 300 measurements is \(s=.87\). This information will be used to determine the control limits and construct the control chart.
02

Determine the control limits

To determine the control limits for the \(\bar{x}\) chart, we will use the following formulas: Upper Control Limit (UCL): \(\overline{\bar{x}} + 3\frac{s}{\sqrt{n}}\) Lower Control Limit (LCL): \(\overline{\bar{x}} - 3\frac{s}{\sqrt{n}}\) Given \(\overline{\bar{x}}=20.74\) and \(s=.87\), and \(n=10\), we can calculate the control limits: UCL = \(20.74 + 3\frac{0.87}{\sqrt{10}}\) = \(20.74 + 3(0.275)\) = \(20.74 + 0.825\) = \(21.565\) LCL = \(20.74 - 3\frac{0.87}{\sqrt{10}}\) = \(20.74 - 3(0.275)\) = \(20.74 - 0.825\) = \(19.915\) Thus, the Upper Control Limit (UCL) is \(21.565\) and the Lower Control Limit (LCL) is \(19.915\).
03

Explain the purpose of an \(\bar{x}\) chart

An \(\bar{x}\) chart, also known as the control chart for the process mean, is a type of control chart used to monitor and control the stability of a process over time. It is used to help identify variations in the process so that appropriate corrective actions can be taken, if necessary. The main purpose of the \(\bar{x}\) chart is to detect small changes in the process mean and distinguish between assignable-cause variation and common-cause variation, thus facilitating continuous process improvement.
04

Construct the \(\bar{x}\) chart and usage explanation

An \(\bar{x}\) chart can be constructed by plotting the individual sample means (\(\bar{x}_i\)) on the y-axis and time (or sample number) on the x-axis. The overall mean, \(\overline{\bar{x}}\), is drawn as a central line, and the Upper and Lower Control Limits (UCL and LCL) are drawn as parallel lines. Once the \(\bar{x}\) chart is constructed, it can be used to monitor the process by comparing new data points (i.e., new sample means) to the control limits. A data point or pattern outside the control limits suggests the presence of special (assignable) causes, which warrant investigation and corrective actions. Conversely, if all data points fall within the control limits, the process is considered to be in statistical control, meaning that it is stable and predictable, with only random variations within the acceptable limits. In summary, an \(\bar{x}\) chart can be used to monitor the stability of a process, identify variations, and determine when to take corrective action.

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Key Concepts

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

Statistical Process Control
Statistical Process Control (SPC) is a set of tools used to ensure that a manufacturing or business process is functioning at its best level of efficiency and quality. It uses statistical methods to monitor and control these processes. The goal is to identify and reduce variability in the process, thus maintaining a predictable output. SPC is typically visualized through control charts, like the \(\bar{x}\) chart, which allows for real-time monitoring. By implementing SPC, businesses can effectively reduce waste, increase product quality, and improve overall efficiency.

This methodology relies on the concept of common-cause variation, which is inherent in the process, and special-cause variation, which is due to specific, identifiable factors. By distinguishing between these types of variation, SPC helps determine when a process needs attention or intervention.
Control Limits
Control limits are essential concepts in control charts within Statistical Process Control. They define the boundaries of acceptable variation for a process. There are two primary types of control limits: Upper Control Limit (UCL) and Lower Control Limit (LCL). These are calculated based on a statistical formula that considers the mean of the process and the standard deviation.

In the example provided in the exercise, the UCL is calculated using \( \overline{\bar{x}} + 3\frac{s}{\sqrt{n}} \) and the LCL is \( \overline{\bar{x}} - 3\frac{s}{\sqrt{n}} \). Here, \(\overline{\bar{x}}\) represents the average of the sample means, \(s\) is the standard deviation, and \(n\) is the sample size. Control limits are distinct from specification limits, as they focus purely on process variability and stability, not necessarily on product specifications.
  • UCL and LCL act as visual cues on control charts, signifying the threshold for regular variation.
  • If data falls outside these limits, it indicates an exceptional variation requiring investigation.
Process Variation
Process variation refers to the differences observed in the output of a process. In manufacturing and production, variation is inevitable but must be minimized to ensure consistency and quality. There are two main types of process variation: common-cause and special-cause.

Common-cause variation, also known as natural variation, is inherent to the process itself. It is usually random and predictable within certain limits. On the other hand, special-cause variation arises due to specific factors that are not part of the process's inherent design. These are often irregular and can signify problems.
  • Understanding and managing these variations are essential for maintaining stable and predictable processes.
  • By identifying special-cause variation, a company can implement changes to correct it, thus improving process performance.

Regular monitoring through tools like control charts can help detect these variations early, ensuring that corrective actions are taken promptly.
Mean Control Chart
A Mean Control Chart, often called an \( \bar{x} \) chart, is a specific type of control chart used to monitor the process mean over time. This chart is valuable for tracking trends, shifts, or unusual patterns in the process data. It effectively differentiates between common and special-cause variation.

To set up a Mean Control Chart, the sample means are plotted over time against a central line representing the overall mean (\(\overline{\bar{x}}\)). Additionally, lines for the Upper Control Limit (UCL) and Lower Control Limit (LCL) are drawn to define the boundaries for the process's acceptable range of variation.
  • If a data point falls outside the control limits, it signals a potential special-cause variation that needs investigation.
  • Consistently monitoring the chart helps identify trends that could indicate chronic process issues.
  • The \(\bar{x}\) chart can be paired with other SPC tools for comprehensive process analysis.

A Mean Control Chart is an effective way to ensure a process remains stable over time, allowing for the early detection of any variations that may require corrective interventions.

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Most popular questions from this chapter

A random sample of size \(n=50\) is selected from a binomial distribution with population proportion \(p=.7\) a. What will be the approximate shape of the sampling distribution of \(\hat{p} ?\) b. What will be the mean and standard deviation (or standard error) of the sampling distribution of \(\hat{p} ?\) c. Find the probability that the sample proportion \(\hat{p}\) is less than 8 .

A random sample of size \(n=40\) is selected from a population with mean \(\mu=100\) and standard deviation \(\sigma=20 .\) a. What will be the approximate shape of the sampling distribution of \(\bar{x} ?\) b. What will be the mean and standard deviation of the sampling distribution of \(\bar{x} ?\)

The normal daily human potassium requirement is in the range of 2000 to 6000 milligrams (mg), with larger amounts required during hot summer weather. The amount of potassium in food varies, but bananas are often associated with high potassium, with approximately \(422 \mathrm{mg}\) in a medium sized banana \(^{8}\). Suppose the distribution of potassium in a banana is normally distributed, with mean equal to \(422 \mathrm{mg}\) and standard deviation equal to \(13 \mathrm{mg}\) per banana. You eat \(n=3\) bananas per day, and \(T\) is the total number of milligrams of potassium you receive from them. a. Find the mean and standard deviation of \(T\). b. Find the probability that your total daily intake of potassium from the three bananas will exceed \(1300 \mathrm{mg} .\) (HINT: Note that \(T\) is the sum of three random variables, \(x_{1}, x_{2},\) and \(x_{3},\) where \(x_{1}\) is the amount of potassium in banana number \(1,\) etc. \()\)

The sample means were calculated for 40 samples of size \(n=5\) for a process that was judged to be in control. The means of the 40 values and the standard deviation of the combined 200 measurements were \(\overline{\bar{x}}=155.9\) and \(s=4.3,\) respectively. a. Use the data to determine the upper and lower control limits for an \(\bar{x}\) chart. b. Construct an \(\bar{x}\) chart for the process and explain how it can be used.

Random samples of size \(n=75\) were selected from a binomial population with \(p=.4\). Use the normal distribution to approximate the following probabilities: a. \(P(\hat{p} \leq .43)\) b. \(P(.35 \leq \hat{p} \leq .43)\)
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