Question

In: Statistics and Probability

Solve step-by-step the following: 1) X-bar and R control Chart Subgroup X1 X2 X3 X4 X-bar...

Solve step-by-step the following:

1) X-bar and R control Chart

Subgroup	X1	X2	X3	X4	X-bar	R
1	1.2	1.1	1.1	2.3
2	3.2	1.3	1.3	3.3
3	1.3	1.4	1.3	1.1
4	3.2	1.1	3.3	2
5	2.1	3.2	2.4	2.3
6	3.1	3.1	3.2	3.3
7	3	1	2.2	2.1
8	2.3	1.3	3.1	2.4
9	4	3.3	4	4.3
10	1.2	3.1	1.3	1.3
11	2.2	1.4	1.1	3
12	3.2	2.3	3.1	1.2
13	1.1	3	1	3.1
14	3.1	1.3	2.2	1.2
15	2.3	2	1	2.3
16	2.2	3.1	3.4	3.2
17	1.2	1.2	2.1	2
18	1.3	3.3	1.2	2.1
19	2.2	2	1.3	3
20	0.7	1.1	1.1	1.2
21	3	3.1	2.2	2.2
22	2.3	2.3	3.1	2.2
23	2.3	1.3	1.2	3.1
24	3.1	3	1.4	1.4
25	3.1	3.2	2.1	2.1

a. Determine the Trial UCL, CL, and LCL for both charts of X-bar and R

b. Determine the Revised UCL, CL, and LCL for both charts of X-bar and R

Expert Solution

a) The control limits for X-bar chart is,

UCL= (X-bar) + A₂ R = 2.400151

CL = X-bar = 2.1701

LCL = (X-bar) - A₂ R = 1.940049

Where A₂ is table value corresponding to n = 25

The control limits for R chart is

UCL = D₃ R = 2.317048

CL = R = 1.5036

LCL = D₄ R = 0.690192

Where D₃ & D₄ is table value corresponding to n = 25.

A₂ = 0.153, D₃ = 0.459, D₄ = 1.541

Sub group	x1	x2	x3	x4	X bar	R
1	1.2	1.1	1.1	2.3	1.425	1.2
2	3.2	1.3	1.3	3.3	2.275	2
3	1.3	1.4	1.3	1.1	1.275	0.3
4	3.2	1.1	3.3	2	2.4	2.2
5	2.1	3.2	2.4	2.3	2.5	1.1
6	3.1	3.1	3.2	3.3	3.175	0.2
7	3	1	2.2	2.1	2.075	2
8	2.3	1.3	3.1	2.4	2.275	1.8
9	4	3.3	4	4.3	3.9	1
10	1.2	3.1	1.3	1.3	1.725	1.9
11	2.2	1.4	1.1	3	1.925	1.9
12	3.2	2.3	3.1	1.2	2.45	2
13	1.1	3	1	3.1	2.05	2.1
14	3.1	1.3	2.2	1.2	1.95	1.9
15	2.3	2	1	2.3	1.9	1.3
16	2.2	3.1	3.4	3.2	2.975	1.2
17	1.2	1.2	2.1	2	1.625	0.9
18	1.3	3.3	1.2	2.1	1.975	2.1
19	2.2	2	1.3	3	2.125	1.7
20	0.7	1.1	1.1	1.2	1.025	0.5
21	3	0.31	2.2	2.2	1.9275	2.69
22	2.3	2.3	3.1	2.2	2.475	0.9
23	2.3	1.3	1.2	3.1	1.975	1.9
24	3.1	3	1.4	1.4	2.225	1.7
25	3.1	3.2	2.1	2.1	2.625	1.1
				Average	2.1701	1.5036

From above graph we see that the process of X-bar chart and R chart is out of control because in both of graph we see that some points goes above UCL line and some points are below LCL line.

b) We see that the process of X-bar and R chart is out of control. Hence we revised control limits of X-bar and R chart.

For revising control limits we delete the points which are above UCL line and below LCL line in both X-bar and R chart and again compute control limits for both of these chart hence we get revised control limits.

Sub group	x1	x2	x3	x4	X bar	R
1	3.2	1.3	1.3	3.3	2.275	2
2	3.2	1.1	3.3	2	2.4	2.2
3	3	1	2.2	2.1	2.075	2
4	2.3	1.3	3.1	2.4	2.275	1.8
5	1.2	3.1	1.3	1.3	1.725	1.9
6	2.2	1.4	1.1	3	1.925	1.9
7	3.2	2.3	3.1	1.2	2.45	2
8	1.1	3	1	3.1	2.05	2.1
9	3.1	1.3	2.2	1.2	1.95	1.9
10	2.3	2	1	2.3	1.9	1.3
11	1.3	3.3	1.2	2.1	1.975	2.1
12	2.2	2	1.3	3	2.125	1.7
13	3	0.31	2.2	2.2	1.9275	2.69
14	2.3	2.3	3.1	2.2	2.475	0.9
15	2.3	1.3	1.2	3.1	1.975	1.9
16	3.1	3	1.4	1.4	2.225	1.7
				Average	2.107969	1.880625

table values corresponding to n = 16 is,

A₂ = 0.212, D₃ = 0.363, D₄ = 1.637

The revised control limits for X-bar chart is,

UCL= (X-bar) + A₂ R = 2.50662

CL = X-bar = 2.107969

LCL = (X-bar) - A₂ R = 1.709277

The revised control limits for R chart is

UCL = D₃ R = 3.078583

CL = R = 1.880625

LCL = D₄ R = 0.870729

orchestra answered 2 years ago

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