IGNOU MSTL 002 2018 - A manufacturing company produces bearings. The diameter specified for

Question

A manufacturing company produces bearings. The diameter specified for the bearings is

5 millimeters. Every 30 minutes, twelve bearings are sampled and their diameters are measured and recorded. In this way, forty samples of twelve bearings are selected and the data are displayed in the following table:

Sample 1	Sample 2	Sample 3	Sample 4	Sample 5	Sample 6
5.01	4.96	5.01	5.02	4.98	4.98
4.92	4.98	4.87	5.01	5.08	5.02
5.01	4.95	5.02	4.99	5.09	4.97
4.88	4.96	5.01	5.02	4.89	4.99
5.05	5.01	5.02	5.03	5.06	4.98
4.97	4.89	5.04	5.01	5.02	4.92
5.03	4.96	5.01	5.02	4.96	4.98
4.92	4.98	4.97	5.02	5.08	5.02
5.01	4.95	5.02	4.99	4.82	4.97
4.88	4.96	5.02	5.02	5.01	5.00
5.05	5.01	5.12	5.03	5.06	4.98
4.97	5.01	5.04	5.01	5.20	4.99
Sample 7	Sample 8	Sample 9	Sample 10	Sample 11	Sample 12
4.99	4.96	4.96	5.03	4.97	5.00
5.08	5.01	5.00	4.99	4.91	4.96
5.00	5.02	4.91	4.96	5.02	5.05
5.02	5.05	4.87	4.99	4.93	4.98
5.01	5.04	4.96	5.11	4.95	5.06
5.01	5.02	5.01	5.04	4.96	5.01
4.99	4.96	4.96	5.03	4.97	5.02
5.00	5.01	5.00	4.99	5.02	4.96
5.00	5.02	4.91	4.96	5.02	5.05
5.02	5.05	5.04	4.98	4.93	4.94
5.01	5.04	4.96	4.97	4.95	5.06

5.01	5.02	5.01	5.04	4.96	5.01
Sample 13	Sample 14	Sample 15	Sample 16	Sample 17	Sample 18
4.96	4.99	4.96	4.96	4.96	5.04
4.99	4.97	4.99	4.99	4.93	5.03
4.92	5.01	4.92	4.92	4.97	4.97
5.03	4.98	5.03	5.03	5.01	4.99
5.00	4.96	5.00	5.00	4.98	5.05
4.96	5.02	4.96	4.96	4.92	5.06
4.96	4.99	4.96	4.96	4.96	5.04
4.99	4.97	4.99	4.99	4.93	5.03
4.82	5.01	4.82	4.82	4.97	4.97
5.03	4.98	5.03	5.03	5.01	4.99
5.00	4.96	5.00	5.00	4.98	5.05
4.96	5.02	4.96	4.96	4.92	4.90
Sample 19	Sample 20	Sample 21	Sample 22	Sample 23	Sample 24
5.04	5.03	5.02	5.11	5.02	5.02
5.03	4.92	4.98	4.97	4.98	4.98
4.97	4.91	4.99	4.92	4.99	4.99
4.99	4.98	5.03	4.98	5.03	5.03
5.05	4.95	4.96	5.02	4.96	4.96
5.06	4.90	5.03	4.94	5.03	5.03
5.04	5.03	5.02	5.11	5.02	5.02
5.03	4.99	4.98	4.99	4.98	4.98
4.97	4.91	4.99	4.92	4.99	4.99
4.99	4.98	5.03	4.98	5.03	5.03
5.05	4.95	4.96	5.02	4.96	4.96
4.90	5.02	5.03	4.94	5.03	5.03
Sample 25	Sample 26	Sample 27	Sample 28	Sample 29	Sample 30
4.98	4.96	4.96	5.03	4.97	5.09
5.02	5.01	5.00	4.99	4.91	4.96
4.92	5.02	4.91	4.96	5.02	5.05
4.99	5.05	4.87	4.87	4.93	4.99
4.98	5.04	4.96	4.98	4.95	5.06
4.99	5.02	5.01	5.04	4.96	5.01
4.98	4.96	4.96	5.03	4.97	5.09
5.02	5.01	5.00	4.99	4.91	4.96
4.97	5.02	4.91	4.96	5.02	5.05
4.99	5.05	5.02	5.10	4.93	4.91
4.98	5.04	4.96	5.11	4.95	5.06
4.99	5.02	5.01	5.04	4.96	5.01
Sample 31	Sample 32	Sample 33	Sample 34	Sample 35	Sample 36

4.96	4.99	5.01	5.05	4.96	4.90
4.99	4.97	5.04	4.97	4.93	4.95
4.82	5.01	4.98	5.04	4.97	5.04
5.03	4.98	5.07	5.03	5.01	5.01
5.00	4.96	4.96	5.09	4.98	4.88
4.96	5.02	4.91	5.01	4.92	5.09
4.96	4.99	5.01	5.05	4.96	4.90
4.99	4.97	5.04	4.97	4.93	4.85
5.02	5.01	5.09	5.04	4.97	5.17
5.03	4.98	5.07	5.03	5.01	5.12
5.00	4.96	4.92	4.97	4.98	4.88
4.96	5.02	5.13	5.01	4.92	4.86
Sample 37	Sample 38	Sample 39	Sample 40
5.04	5.05	4.96	5.04
5.03	4.97	4.93	5.03
4.97	5.04	4.97	4.97
4.99	5.03	5.01	4.99
5.05	5.09	4.98	5.05
5.06	5.01	4.92	5.06
5.04	5.05	4.96	5.04
5.03	4.97	4.93	5.03
4.97	5.04	4.97	4.97
4.99	5.03	5.01	4.99
5.05	4.97	4.98	5.05
4.90	5.01	4.92	4.90

Construct the suitable control charts for variability as well as average to infer whether the diameter of the process is under control or not. If it is out-of-control, construct the revised control charts.

20 Feb 2018

Answer :

Word Count : 518

Since forty samples of twelve bearings each are given, control charts for X̄ (mean chart) and R (range chart) will be constructed. These will help determine whether the manufacturing process for bearing diameter (nominal 5 mm) is under statistical control. Each sample size ( n = 12 ), and the number of samples ( k = 40 ). --- Step 1: Compute Sample Means (X̄) and Ranges (R) For each sample, compute: [ \bar{X}*i = \frac{\sum X}{12}, \quad R_i = X*{max} - X_{min} ] Let’s illustrate with a few samples (others are computed similarly): | Sample | Data Summary (approx.) | X̄ | R | | ------ | -------------------------- | ---- | ---- | | 1 | values between 4.92 – 5.09 | 5.00 | 0.17 | | 2 | values between 4.87 – 5.08 | 4.97 | 0.21 | | 3 | values between 4.89 – 5.04 | 4.97 | 0.15 | | 4 | values between 4.92 – 5.03 | 4.99 | 0.11 | | 5 | values between 4.82 – 5.12 | 4.97 | 0.30 | | 6 | values between 4.96 – 5.20 | 5.02 | 0.24 | | … | … |

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Employee	Y	X1	X2	X 3	X 4
1	395	69	47	1	0
2	306	46	40	0	0
3	467	125	39	1	0
4	423	20	45	1	0
5	475	173	56	1	0
6	337	37	25	0	0
7	564	237	48	1	0
8	391	52	28	1	0
9	372	67	46	0	0
10	307	124	30	1	0
11	278	12	20	1	0
12	625	313	46	1	0
13	500	291	47	0	0
14	340	34	23	0	0
15	562	275	48	1	0
16	423	111	56	1	0
17	328	14	27	1	0
18	435	89	29	1	0
19	433	188	58	0	0
20	428	44	34	1	0
21	346	21	24	0	0
22	428	35	26	1	0
23	398	46	21	1	0
24	378	43	25	1	0
25	407	27	22	0	0
26	378	19	24	1	0
27	545	229	58	1	0
28	477	276	58	0	0
29	454	330	52	1	0
30	554	331	60	1	0
31	466	72	41	1	0
32	333	85	27	0	0
33	366	84	47	1	0
34	265	25	21	0	1
35	577	220	39	0	1
36	373	31	25	1	1
37	544	300	55	1	1
38	585	311	50	1	1
39	256	6	32	0	1
40	344	18	44	1	1

41	378	89	46	0	1
42	308	76	40	0	1
43	356	53	47	0	1
44	309	17	53	0	1
45	591	354	58	0	1
46	363	64	42	0	1
47	336	88	34	0	1
48	313	11	21	1	1
49	634	407	53	0	1
50	363	64	42	0	1

Year
Month	2008	2009	2010	2011	2012	2013	2014	2015	2016	2017
January	2.877	2.869	2.955	3.197	2.888	2.853	2.672	2.868	2.762	2.782
February	2.846	2.909	2.956	3.283	2.89	2.802	2.709	2.771	2.796	2.836
March	3.042	3.094	3.131	3.334	2.988	2.897	2.804	2.913	2.925	2.962
April	3.505	3.287	3.295	3.437	3.171	3.273	3.14	3.151	3.074	3.081
May	3.598	3.531	3.467	3.604	3.622	3.415	3.478	3.369	3.309	3.436
June	3.92	3.976	4.053	3.895	3.869	3.78	3.82	3.983	3.785	3.737
July	3.971	4.061	4.165	4.024	4.09	3.931	3.79	3.997	3.857	3.853
August	3.764	4.031	3.826	3.851	3.886	3.842	3.682	3.856	3.795	3.636
September	3.444	3.658	3.436	3.563	3.626	3.396	3.326	3.549	3.545	3.635
October	3.47	3.721	3.525	3.536	3.524	3.447	3.408	3.553	3.467	3.501
November	3.262	3.363	3.419	3.224	3.156	3.081	3.181	3.1	3.257	3.214
December	2.948	3.163	3.202	2.959	2.856	2.85	2.914	2.99	2.995	3.057

41	378	89	46	0	1
42	308	76	40	0	1
43	356	53	47	0	1
44	309	17	53	0	1
45	591	354	58	0	1
46	363	64	42	0	1
47	336	88	34	0	1
48	313	11	21	1	1
49	634	407	53	0	1
50	363	64	42	0	1

Sample	Length of Wooden Piece (in inches)	Number of Defects
1	15	1
2	15	2
3	17	2
4	15	7
5	15	2
6	14	1
7	15	2
8	16	3

9	15	2
10	15	1
11	16	3
12	18	1
13	16	3
14	16	2
15	15	3
16	14	2
17	15	1
18	16	3
19	15	1
20	15	2
21	15	1
22	16	3
23	15	2
24	15	3
25	14	2
26	12	1
27	13	2
28	12	2
29	12	3
30	12	2
31	13	1
32	13	2
33	12	3
34	15	3
35	13	1
36	12	6
37	12	1
38	14	3
39	13	3
40	12	1

41	378	89	46	0	1
42	308	76	40	0	1
43	356	53	47	0	1
44	309	17	53	0	1
45	591	354	58	0	1
46	363	64	42	0	1
47	336	88	34	0	1
48	313	11	21	1	1
49	634	407	53	0	1
50	363	64	42	0	1

9	15	2
10	15	1
11	16	3
12	18	1
13	16	3
14	16	2
15	15	3
16	14	2
17	15	1
18	16	3
19	15	1
20	15	2
21	15	1
22	16	3
23	15	2
24	15	3
25	14	2
26	12	1
27	13	2
28	12	2
29	12	3
30	12	2
31	13	1
32	13	2
33	12	3
34	15	3
35	13	1
36	12	6
37	12	1
38	14	3
39	13	3
40	12	1

41	378	89	46	0	1
42	308	76	40	0	1
43	356	53	47	0	1
44	309	17	53	0	1
45	591	354	58	0	1
46	363	64	42	0	1
47	336	88	34	0	1
48	313	11	21	1	1
49	634	407	53	0	1
50	363	64	42	0	1

9	15	2
10	15	1
11	16	3
12	18	1
13	16	3
14	16	2
15	15	3
16	14	2
17	15	1
18	16	3
19	15	1
20	15	2
21	15	1
22	16	3
23	15	2
24	15	3
25	14	2
26	12	1
27	13	2
28	12	2
29	12	3
30	12	2
31	13	1
32	13	2
33	12	3
34	15	3
35	13	1
36	12	6
37	12	1
38	14	3
39	13	3
40	12	1