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Chapter 7: Problem 49

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.

Short Answer

Expert verified
Answer: The calculated upper control limit (UCL) is approximately 157.01, and the lower control limit (LCL) is approximately 154.79. By plotting the sample means on the \(\bar{x}\) chart and comparing them to the UCL and LCL, we can monitor the process performance over time. If most sample means fall within the control limits, the process is considered to be in control. Points falling outside these limits or any patterns and trends in the plotted points may indicate potential problems or assignable causes that should be investigated.

Step by step solution

01

Determine the value of A2 and calculate standard error

The first step is to determine the value of A2 for n = 5. You can find this in a statistical control chart constants table or use online resources. Generally, for n = 5, A2 鈮� 0.58. Next, we will calculate the standard error (SE) of the sample mean using the given standard deviation (s = 4.3). The SE is given by the formula: \(SE = \frac{s}{\sqrt{n}}\) Plugging in our values, \(SE = \frac{4.3}{\sqrt{5}} \approx 1.92\)
02

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

Now that we have the value of A2 and the standard error, we can calculate the upper and lower control limits using the formulas: \(UCL = \overline{\bar{x}} + A_2 \frac{s}{\sqrt{n}}\) \(LCL = \overline{\bar{x}} - A_2 \frac{s}{\sqrt{n}}\) Plugging in the given values, \(UCL = 155.9 + 0.58(1.92) \approx 157.01\) \(LCL = 155.9 - 0.58(1.92) \approx 154.79\) The upper control limit (UCL) is approximately 157.01, and the lower control limit (LCL) is approximately 154.79.
03

Construct the \(\bar{x}\) chart

Now that we have calculated the UCL and LCL, we can construct the \(\bar{x}\) chart. For this, plot the points of the control chart as follows: 1. On the y-axis, indicate the mean of the sample means (\(\overline{\bar{x}} = 155.9\)). 2. Draw a horizontal line at the upper control limit (UCL = 157.01). 3. Draw a horizontal line at the lower control limit (LCL = 154.79). 4. Plot the 40 sample means on the control chart.
04

Interpret the \(\bar{x}\) chart

Once the control chart has been constructed, you can use it to monitor the process performance over time. If most of the plotted sample means fall within the control limits (between UCL and LCL), the process is considered to be in control. If any points fall outside these control limits, they indicate potential problems or assignable causes that should be investigated. You can also look for patterns or trends in the plotted points, such as runs above or below the centerline of the chart or continuous trends in one direction, which may indicate a systematic problem in the process.

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