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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: The purpose of an \(\bar{x}\) chart is to monitor the process performance over time by plotting the means of samples taken from the process. It is used to detect shifts in the process mean and determine whether the process is in control (stable) or out of control (unstable). To use an \(\bar{x}\) chart, one must plot the sample means against their respective sample numbers, add the upper and lower control limits on the chart, and analyze the chart by looking for points falling outside the control limits or unusual patterns. If all points are within the control limits and no unusual patterns are found, the process is considered in control. If any points are outside the limits or systematic patterns are observed, the process is considered out of control, requiring investigation to determine the cause of variation.

Step by step solution

01

Calculate the average of the sample means

To calculate the average of the sample means, we have been given the following information: \(\overline{\bar{x}}=20.74\)
02

Calculate the standard deviation of the samples

The given standard deviation of the combined 300 measurements is: \(s = 0.87\)
03

Calculate the standard error of the sample means

The standard error of the sample means (or the standard deviation of the distribution of sample means) can be calculated using the following formula: \(SE = \frac{s}{\sqrt{n}}\) where \(SE\) is the standard error, \(s\) is the standard deviation, and \(n\) is the sample size. In this case, the sample size \(n = 10\). Therefore, \(SE = \frac{0.87}{\sqrt{10}} = 0.275\)
04

Calculate the upper and lower control limits

To calculate the upper and lower control limits, we will use the following formulas: \(UCL = \overline{\bar{x}} + 3SE\) \(LCL = \overline{\bar{x}} - 3SE\) Plugging in the values, we get: \(UCL = 20.74 + 3(0.275) = 20.74 + 0.825 = 21.565\) \(LCL = 20.74 - 3(0.275) = 20.74 - 0.825 = 19.915\) So, the upper control limit is 21.565 and the lower control limit is 19.915. #b. Explain the purpose of an \(\bar{x}\) chart#
05

Definition and purpose of an \(\bar{x}\) chart

An \(\bar{x}\) chart (also known as an average chart or a means chart) is a type of control chart that is used to monitor the process performance over time by plotting the means of samples taken from the process. The purpose of an \(\bar{x}\) chart is to detect shifts in the process mean and to determine whether the process is in control (stable) or out of control (unstable). #c. Construct an \(\bar{x}\) chart for the process and explain how it can be used#
06

Plot the sample means

To construct an \(\bar{x}\) chart, we would first plot the 30 sample means on the vertical axis against their respective sample numbers on the horizontal axis.
07

Plot the upper and lower control limits

Next, we would plot the upper control limit (UCL =21.565) and lower control limit (LCL = 19.915) as horizontal lines on the chart.
08

Analyze the chart

Now, we will analyze the chart and specifically look for points that fall outside of the control limits or patterns that suggest that the process might be out of control. If all points fall within the control limits and there are no unusual patterns, the process is considered to be in control. If there are one or more points outside the control limits or any systematic patterns, the process is considered to be out of control, and an investigation should be carried out to determine the cause of the observed variation. In summary, an \(\bar{x}\) chart can be used to monitor the process performance over time by detecting shifts in the process mean and identifying whether the process is in control or out of control.

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