Question

Checkout time at a supermarket is monitored using a mean and a range chart. Six samples of n = 20 observations have been obtained and the sample means and ranges computed:

Sample	Mean	Range	Sample	Mean	Range
1	3.06	.42	4	3.13	.46
2	3.15	.49	5	3.06	.46
3	3.11	.41	6	3.09	.45

Factors for three-sigma control limits for x¯x¯  and R charts

		FACTORS FOR R CHARTS
Number of Observations in Subgroup, n	Factor for x¯x¯ Chart, A2	Lower Control Limit, D3	Upper Control Limit, D4
2	1.88	0	3.27
3	1.02	0	2.57
4	0.73	0	2.28
5	0.58	0	2.11
6	0.48	0	2.00
7	0.42	0.08	1.92
8	0.37	0.14	1.86
9	0.34	0.18	1.82
10	0.31	0.22	1.78
11	0.29	0.26	1.74
12	0.27	0.28	1.72
13	0.25	0.31	1.69
14	0.24	0.33	1.67
15	0.22	0.35	1.65
16	0.21	0.36	1.64
17	0.20	0.38	1.62
18	0.19	0.39	1.61
19	0.19	0.40	1.60
20	0.18	0.41	1.59

a.	Using the factors in the above table, determine upper and lower limits for mean and range charts. (Round your intermediate calculations and final answers to 4 decimal places.)

Upper limit for mean
Lower limit for mean
Upper limit for range
Lower limit for range

b.	Is the process in control?
	Yes No

Answer 1

Sample	Mean	Range
1	3.06	0.42
2	3.15	0.49
3	3.11	0.41
4	3.13	0.46
5	3.06	0.46
6	3.09	0.45
	3.1000	0.4483
	X-barbar	R bar

Answer a:

D4 (n=6)=			2
D3 (n=6)=			0
control limits for Range,
CL or R-bar= Average of all sample Range			0.4483
UCL=R-bar*D4			0.8967
LCL=R-bar*D3			0.0000

A2 (n=6)			0.48
control limits for X bar,
CL or Xbarbar			3.1000
UCL=Xbarbar+ (A2)*Rbar			3.3152
LCL=Xbarbar- (A2)*Rbar			2.8848

Answer b: Yes, the process is in control because samples of mean and range are in within the control limits. Every sample mean lie between 3.31 and 2.88. and Every range lie betweeen 0.89 and 0.0.

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