Question

The time to replace vehicle wiper blades at a service center was 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	.50	5	3.06	.46
3	3.11	.41	6	3.09	.45

Factors for three-sigma control limits for and R charts

		FACTORS FOR R CHARTS
Number of Observations in Subgroup, n	Factor for 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. (Do not round intermediate calculations. Round your mean value to 3 decimal places and range value to 4 decimal places.)

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

Answer 1

Calculate the value of average mean (X) and average range (R) as shown below: Sample Mean Range 3.06 3.15 3.11 3.13 3.06 3.09 0.42 0.5 0.41 0.46 0.46 0.45 AVERAGE(C2:C Average AVERAGE(B2:B7) 1 Sample MeanRange 2 3.06 3.15 3.11 3.13 3.06 3.09 8 Average 3.10 1 0.42 0.50 0.41 0.46 0.46 0.45 0.45 4 H 20, X-3.10 andR-0.45 ence, Sample size (n) - Formula to calculate UCL and LCL for X and R chart is given below: For X chart,

Find out the value of A from the Three-sigma factor table. For a sample size of 20, the value of A2-factor is 0.18. Therefore, UCL 3.10+(0.18 *0.45) = 3.10+0.081 3.181 Similarly, LCL 3.10-(0.18 0.45) 3.10-0.081 = 3.019

R For chart, LCL = D.xR UCL = D.xR R =0.45 Obtain the value of Dj and D4 factor for a sample size of 20 using three-sigma factor table. The values are 0.41 and 1.59. Hence, the value of UCL and LCL for R-bar chart would be calculated as follows LCL 0.45*0.41 0.1845 UCL 0.45 1.59 0.7155 Hence, the following values are obtained: Upper limit for mean: 3.181 awexlt f 3.01 Upper limit for range Lower limit for range: 0.1845