Quality and Quality Control

For each of the following cases, fi nd the CL, UCL, and LCL for an x chart based on the given information.

(a) n = 9, x = 26.7, R = 5.3. (b) n = 17, x = 138.6, R = 15.1. (c) n = 4, x = 84.2, R = 9.6. (d) n = 22, x = 8.1, R = 7.4

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Altoona Tire Company sells its ATC-50 tires with a 50,000-mile tread-life warranty. Lorrie Ackerman, a quality control engineer with the company, runs simulated road tests to monitor the life of the output from the ATC-50 production process. From each of the last 12 batches of 1,000 tires, she has tested 5 tires and recorded the following results, with x and R measured in thousands of miles:

Batch 1 2 3 4 5 6 7 8 9 10 11 12 x 50.5 49.7 50.0 50.7 50.7 50.6 49.8 51.1 50.2 50.4 50.6 50.7 R 1.1 1.6 1.8 0.1 0.9 2.1 0.3 0.8 2.3 1.3 2.0 2.1 (a) Use the data above to help Lorrie construct an x chart. (b) Is the production process in-control? Explain.

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For each of the following cases, find the CL, UCL, and LCL for an x chart based on the given information.

(a) n = 12, x = 16.4, x σ = 1.2. (b) n = 12, x = 16.4, R = 7.6. (c) n = 8, x = 4.1, R = 1.3. (d) n =15, x = 141.7, R = 18.6

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The Wilson Piston Company manufactures pistons for LawnGuy mowers, and the diameter of each piston must be carefully monitored. Jeff Wilson, the quality control engineer, has sampled 8 pistons from each of the last 15 batches of 500 pistons and has recorded the following results, with x and R measured in centimeters:

Batch	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
x	15.85	15.95	15.86	15.84	15.91	15.81	15.86	15.84	15.83	15.83	15.72	15.96	15.88	15.84	15.89
R	0.15	0.17	0.18	0.16	0.14	0.21	0.13	0.22	0.19	0.21	0.28	0.12	0.19	0.22	0.24

Use the data above to help Jeff construct an x chart.
1. Is the production process in-control? Explain

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Dick Burney is director of 911 emergency medical services in Ann Arbor, Michigan. He is concerned about response time, the amount of time that elapses between the receipt of a call at the 911 switchboard and the arrival of a municipal rescue squad crew at the calling location. For the last 3 weeks, he has randomly sampled response times for 9 calls each day to get the following results, with x ̅ and R measured in minutes:

Batch	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
x	15.85	15.95	15.86	15.84	15.91	15.81	15.86	15.84	15.83	15.83	15.72	15.96	15.88	15.84	15.89
R	0.15	0.17	0.18	0.16	0.14	0.21	0.13	0.22	0.19	0.21	0.28	0.12	0.19	0.22	0.24

Use the data above to help Jeff construct an x chart.
1. Is the production process in-control? Explain

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Track Bicycle Parts manufactures precision ball bearings for wheel hubs, bottom brackets, head sets, and pedals. Seth Adams is responsible for quality control at Track. He has been checking the output of the 5-mm bearings used in front wheel hubs. For each of the last 18 hours, he has sampled 5 bearings, with the following results:

Batch	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Hour	Bearing Diameters (mm)
1	5.03	5.06	4.86	4.90	4.95
2	4.97	4.94	5.09	4.78	4.88
3	5.02	4.98	4.94	4.95	4.80
4	4.92	4.93	4.90	4.92	4.96
5	5.01	4.99	4.93	5.06	5.01
6	5.00	4.95	5.10	4.85	4.91
7	4.94	4.91	5.05	5.07	4.88
8	5.00	4.98	5.05	4.96	4.97
9	4.99	5.01	4.93	5.10	4.98
10	5.03	4.96	4.92	5.01	4.93
11	5.02	4.88	5.00	4.98	5.09
12	5.09	5.01	5.13	4.89	5.02
13	4.90	4.93	4.97	4.98	5.12
14	5.04	4.96	5.15	5.04	5.02
15	5.09	4.90	5.04	5.19	5.03
16	5.10	5.01	5.04	5.05	5.02
17	4.97	5.10	5.12	4.92	5.04
18	5.01	4.99	5.06	5.04	5.12

(a) Construct an x chart to help Seth determine whether the production of 5-mm bearings is in-control.
1. Should Seth conclude that the process is in-control? Explain.

Solution

Northern White Metals Corp. uses an extrusion process to produce various kinds of aluminum brackets. Raw aluminum ingots are forced under pressure through steel dies to produce long sections of a desired cross-sectional shape. These sections are then fed through an automatic saw, where they are cut into brackets of the desired length. NWMC operates for three shifts of 4 hours each day, and the saw is recalibrated at the beginning of each shift. This week NWMC is producing #409 brackets with a specifi ed cut length of 4 inches. Silvia Serrano, NWMC’s quality specialist, has recorded the lengths of 15 randomly chosen brackets during each half hour of today’s three shifts to get the following data:

Batch	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Hour	Bearing Diameters (mm)
1	5.03	5.06	4.86	4.90	4.95
2	4.97	4.94	5.09	4.78	4.88
3	5.02	4.98	4.94	4.95	4.80
4	4.92	4.93	4.90	4.92	4.96
5	5.01	4.99	4.93	5.06	5.01
6	5.00	4.95	5.10	4.85	4.91
7	4.94	4.91	5.05	5.07	4.88
8	5.00	4.98	5.05	4.96	4.97
9	4.99	5.01	4.93	5.10	4.98
10	5.03	4.96	4.92	5.01	4.93
11	5.02	4.88	5.00	4.98	5.09
12	5.09	5.01	5.13	4.89	5.02
13	4.90	4.93	4.97	4.98	5.12
14	5.04	4.96	5.15	5.04	5.02
15	5.09	4.90	5.04	5.19	5.03
16	5.10	5.01	5.04	5.05	5.02
17	4.97	5.10	5.12	4.92	5.04
18	5.01	4.99	5.06	5.04	5.12

(a) Construct an x chart to help Seth determine whether the production of 5-mm bearings is in-control.
1. Should Seth conclude that the process is in-control? Explain.

Solution

For each of the following cases, fi nd the CL, UCL, and LCL for an R chart based on the given information.

1. n = 9, x = 26.7, R = 5.3.
2. n = 17, x = 138.6, R = 15.1
3. n = 4, x = 84.2, R = 9.6.
4. n = 22, x = 8.1, R = 7.4.

Solution

Construct an R chart for the data given in Exercise SC 10-2. Is the variability in the tread life of the ATC-50 in control? Explain.

Solution

For each of the following cases, find the CL, UCL, and LCL for a p chart based on the given information

Solution