Question

Sampling 4 pieces of​ precision-cut wire​ (to be used in computer​ assembly) every hour for the past 24 hours has produced the following​ results:

Hour   Xbar   R   Hour   Xbar   R   Hour   Xbar   R   Hour   Xbar   R
1 3.15   0.76   7   3.15   0.53   13   3.11   0.90   19   3.31   1.66
2 3.10   1.13   8   2.65   1.08   14   2.73   1.31   20   2.89   1.14
3 3.12   1.48   9 3.12 0.71 15   3.02   1.11   21   2.65   1.08
4 3.49   1.31   10   2.95 1.38 16   2.84   0.45   22   3.18   0.46
5 3.17   1.22   11   2.73 1.17   17   2.76   1.38 23   2.84   1.53
6 2.86   0.37   12 2.87 0.35   18   2.74   1.24   24   2.74 1.02

Based on the sampling​ done, the control limits for ​3-sigma x overbarx chart are ​(round all intermediate calculations to three decimal places before proceeding with further​calculations)​:
Upper Control Limit ​(UCL Subscript x overbarUCLx​) ​= ?
Lower Control Limit ​(LCL Subscript x overbarLCLx​) ​= ?
Based on the x overbarx​-chart, the wire cutting process has been
IN CONTROL or OUT OF CONTROL?

The control limits for the ​3-sigma​ R-chart are ​(round all intermediate calculations to three decimal places before proceeding with further ​calculations)​:
Upper Control Limit ​(UCL Subscript Upper RUCLR​) ​= ?
Lower Control Limit ​(LCL Subscript Upper RLCLR​) ​= ?
Based on the​ R-chart, the wire cutting process has been
IN CONTROL or OUT OF CONTROL ?

Answer 1

Answer:

Based on the data available “X double overbar” and “R overbar” is to be calculated:

Sample	Average (X overbar)	Range (R )
1	3.15	0.76
2	3.1	1.13
3	3.12	1.48
4	3.49	1.31
5	3.17	1.22
6	2.86	0.37
7	3.15	0.53
8	2.65	1.08
9	3.12	0.71
10	2.95	1.38
11	2.73	1.17
12	2.87	0.35
13	3.11	0.9
14	2.73	1.31
15	3.02	1.11
16	2.84	0.45
17	2.76	1.38
18	2.74	1.24
19	3.31	1.66
20	2.89	1.14
21	2.65	1.08
22	3.18	0.46
23	2.84	1.53
24	2.74	1.02
	2.965	1.032
	X double overbar	R overbar

X double overbar = Average of Mean of sample 1 to 15 = 2.965

R overbar = Average of Range of sample 1 to 15 = 1.032

For sample size of 4, following will be the control constants:

Sample size	4
A2	0.729
D3	0
D4	2.282

X bar control chart:

UCL = X double overbar + A2*R bar = 2.965+(0.729*1.032) = 3.718

LCL = X double overbar - A2*R bar = 2.965-(0.729*1.032) = 2.213

Therefore,

Upper Control Limit ​(UCL Subscript x overbarUCLx​) ​= 3.718

Lower Control Limit ​(LCL Subscript x overbarLCLx​) ​= 2.213

Populate the data as under:

X bar chart

Based on the x overbar x​-chart, the wire cutting process has been IN CONTROL (as all X over bar values all within UCL & LCL).

R over bar chart:

UCL = D4*R bar = 2.282*1.032 = 2.355

LCL = D3*R bar = 0*1.032 = 0

Therefore,

Upper Control Limit ​(UCL Subscript Upper RUCLR​) ​= 2.355
Lower Control Limit ​(LCL Subscript Upper RLCLR​) ​= 0

Populate the data as under:

R overbar chart:

Based on the​ R-chart, the wire cutting process has been
IN CONTROL or OUT OF CONTROL ?

Based on the R-chart, the wire cutting process has been IN CONTROL (as all X over bar values all within UCL & LCL).