Chapter 10.2, Problem 3E

The thickness, in mm, of metal washers is measured on samples of size 5. The following table presents the means, ranges, and standard deviations for 20 consecutive samples.

Sample	$\bar{X}$	R	s
1	2.49	0.12	0.07
2	2.45	0.17	0.06
3	2.51	0.13	0.06
4	2.53	0.25	0.09
5	2.49	0.11	0.06
6	2.44	0.11	0.06
7	2.44	0.12	0.05
8	2.42	0.18	0.06
9	2.42	0.08	0.05
10	2.47	0.06	0.02
11	2.54	0.19	0.07
12	2.45	0.09	0.04
13	2.54	0.21	0.07
14	2.55	0.10	0.05
15	2.50	0.25	0.08
16	2.53	0.11	0.04
17	2.58	0.16	0.07
18	2.59	0.09	0.03
19	2.60	0.12	0.05
20	2.56	0.14	0.06

The means are $\overline{\stackrel{¯}{X}}\text{ = 2}\text{.505}$, $\bar{R}\text{ = 0}\text{.1395}$, and $\bar{s}\text{ = 0}\text{.057}$.

1. a. Calculate the 3σ control limits for the R chart. Is the variance under control? If not. delete the samples that are out of control and recompute $\overline{\stackrel{¯}{X}}$ and $\bar{R}$.
2. b. Based on the sample range R, calculate the 3σ control limits for the $\bar{X}$ chart. Based on the 3σ limits, is the process mean in control? If not, when is it first detected to be out of control?
3. c. Based on the Western Electric rules, is the process mean in control? If not. when is it first detected to be out of control?