Chapter 17, Problem 10CR

(a) What is shown on the $\bar{x}$ chart? (b) Name three ways to set the control limits on the $\bar{x}$ chart. (c) How can we obtain good empirical control limits for the $\bar{x}$ chart? (d) Why are quality control samples sometimes small?

To determine

Explain what is shown on the $\overline{x}$ chart.

Answer to Problem 10CR

The $\overline{x}$ chart shows the changes of the sample mean or average value over a period of time.

Explanation of Solution

$\overline{x}$chart:

The $\overline{x}$ chart is control chart that show the changes of the sample mean or average value over a period of time. It helps to monitor the means of the consecutive samples with a constant sample size n.

To determine

Mention three ways to set the control limits on the $\overline{x}$ chart.

Answer to Problem 10CR

The three ways to set the control limits on the $\overline{x}$ chart are control limits with known $\mu$ and $\sigma$, empirical control limits and control limits with R method.

Explanation of Solution

The control limits for $\overline{x}$ chart can be obtained using three ways as,

Control limits with known $\mu$ and $\sigma$:

If the values of $\mu$ and $\sigma$ are known then the 3-sigma limits for the control chart are,

$\begin{array}{c}\text{UCL}=\mu +3\frac{\sigma }{\sqrt{n}}\left(\text{Upper control limit}\right)\\ \text{CL}=\mu \left(\text{Center}\text{line}\right)\\ \text{LCL}=\mu -3\frac{\sigma }{\sqrt{n}}\left(\text{Lower control limit}\right)\end{array}$

In the formula, $\mu$ denotes the average of means of all samples, $\sigma$ denotes the standard deviation, and n denotes the subgroup size.

Empirical control limits:

If the value of $\mu$ and $\sigma$ are not known then the empirical 3-sigma limits for the control chart are obtained by replacing $\mu$ with $\overline{\overline{x}}$, $\sigma$ with s as,

$\begin{array}{c}\text{UCL}=\overline{\overline{x}}+3\frac{s}{\sqrt{n}}\left(\text{Upper control limit}\right)\\ \text{CL}=\overline{\overline{x}}\left(\text{Center}\text{line}\right)\\ \text{LCL}=\overline{\overline{x}}-3\frac{s}{\sqrt{n}}\left(\text{Lower control limit}\right)\end{array}$

In the formula, $\overline{\overline{x}}$ denotes the average of means of all samples, s denotes the sample standard deviation, and n denotes the subgroup size.

Control limits with R method:

$\begin{array}{c}\text{UCL}=\overline{\overline{x}}+3\frac{\overline{R}}{d_2\sqrt{n}}\left(\text{Upper}\text{control}\text{limit}\right)\\ \text{CL}=\overline{\overline{x}}\left(\text{Center}\text{line}\right)\\ \text{LCL}=\overline{\overline{x}}-3\frac{\overline{R}}{d_2\sqrt{n}}\left(\text{Lower}\text{control}\text{limit}\right)\end{array}$

In the formula, $\overline{\overline{x}}$ denotes the average of means of all samples, $\overline{R}$ demotes the average range of the samples, $d_2$ is the factor of the control chart and n denotes the subgroup size.

To determine

Explain how the good empirical control limits for the $\overline{x}$ chart can be obtained.

Answer to Problem 10CR

The good empirical control limits for the $\overline{x}$ chart can be obtained by using 3-sigma limits and samples taken independently.

Explanation of Solution

Justification: The three sigma control limits would be very effective for the control charts because these limits would give clear knowledge about the special cause variations in the process. Also, the good empirical control limits can be obtained by when the samples in the process are taken independently than using the same data.

To determine

Explain why quality control samples are sometimes small.

Answer to Problem 10CR

The quality control samples are often small because for normally distributed data samples would be normal based on central limit theorem.

Explanation of Solution

Justification: For a normally distributed sample, the small samples can be used for the mean based on the central limit theorem (if the data is normal then even if sample size is small the sample would be normal). Hence the quality control samples are often small.