Homework SourseResistors for electronic circuits are manufactured on a high
Resistors for electronic circuits are manufactured on a high-speed automated machine. The machine is set up to produce a large run of resistors of 1,000 ohms each. Use Exhibit 13.7.
To set up the machine and to create a control chart to be used throughout the run, 15 samples were taken with four resistors in each sample. The complete list of samples and their measured values are as follows: Use three-sigma control limits.
Calculate the mean and range for the above samples. (Round \"Mean\" to 2 decimal places and \"Range\" to the nearest whole number.)
Determine the UCL and LCL for R-chart. (Leave no cells blank - be certain to enter \"0\" wherever required. Round your answers to 3 decimal places.)
| Resistors for electronic circuits are manufactured on a high-speed automated machine. The machine is set up to produce a large run of resistors of 1,000 ohms each. Use Exhibit 13.7. |
| To set up the machine and to create a control chart to be used throughout the run, 15 samples were taken with four resistors in each sample. The complete list of samples and their measured values are as follows: Use three-sigma control limits. |
Solution
a.
Calculate the mean and range for the above samples. (Round \"Mean\" to 2 decimal places and \"Range\" to the nearest whole number.)
Ans:
There are four registors in each sample. If they are r1, r2, r3 and r4
Mean = (r1 +r 2+ r3+ r4) /4
For sample 1, Mean = (1014 +1019 +976+1018)/4 = 1006.75
Range is the difference between maximum value and minimum value within each sample.
For sample 1, Range = (1019 – 976) = 43
SAMPLE NUMBER
Mean
Range
1
1006.75
43.00
2
994.25
25.00
3
1008.25
22.00
4
1011.75
42.00
5
990.50
30.00
6
1005.00
39.00
7
991.50
39.00
8
993.75
44.00
9
998.75
57.00
10
997.75
22.00
11
999.75
22.00
12
983.25
24.00
13
996.50
56.00
14
995.25
28.00
15
1001.00
37.00
b.
X double bar = average of all mean values
= (1006.75 + 994.25 + 1008.25 + ……… + 995.25 + 1001.00 )/ 15 = 998.27
R = average of all range values
= (43.00+ 25.00+22.00+ … + 28.00+37.00)/15 = 35.33
c.
The X chart examines the variation from day to day, while the R chart examines the variation within a day.
Control Limits for Xbar chart are given by following formula,
UCL = Xbar + A2*Rbar
LCL = Xbar - A2*Rbar
Wherein A2 is a control chart constant. Value of this constant depends on sub-group size. In the given case, subgroup size is 4 as four registors are measured in each sample.
Given below is the table of constants used in control charts based on 3 sigma level
Subgroup
Size (n)
A2
D3
D4
d2
2
1.880
0
3.267
1.128
3
1.023
0
2.574
1.693
4
0.729
0
2.282
2.059
5
0.577
0
2.114
2.326
6
0.483
0
2.004
2.534
7
0.419
0.076
1.924
2.704
8
0.373
0.136
1.864
2.847
9
0.337
0.184
1.816
2.970
10
0.308
0.223
1.777
3.078
11
0.285
0.256
1.774
3.173
12
0.266
0.284
1.716
3.258
13
0.249
0.308
1.692
3.336
14
0.235
0.329
1.671
3.407
15
0.223
0.348
1.652
3.472
16
0.212
0.364
1.636
3.532
17
0.203
0.379
1.621
3.588
18
0.194
0.392
1.608
3.640
19
0.187
0.404
1.596
3.689
20
0.180
0.414
1.586
3.735
21
0.173
0.425
1.575
3.778
22
0.167
0.434
1.566
3.819
23
0.162
0.443
1.557
3.858
24
0.157
0.452
1.548
3.895
25
0.153
0.459
1.541
3.931
From the table we see, value of A2 for sample size 4 is 0.729
Therefore,
UCL = 998.27 + (0.729*35.33) = 1024.025
LCL = 998.27 - (0.729*35.33) = 972.509
d.
Control Limits for Rbar chart are given by following formula,
UCL = D4*Rbar
LCL = D3*Rbar
Wherein D3 and D4 are constants. Value of these constants depend on sample size. From the table in previous answer
D4=2.282
D3= 0
UCL = 2.282 * 35.33 = 80.631
LCL = 0* 35.33 = 0
e.
What comments can you make about the process?
Xbar-R chart is used in statistical process control when sample size is less than 8. In our case, as sample size is 4, so it is justified to used Xbar-R chart for examining process variation.
In order to determine whether the process is stable or not, we need to check first whether R values are within UCL and LCL of Rbar i.e. within 80.631 and 0.
As we see Max 57.00 and Min 22.00 are well within UCL and LCL. Therefore, we interpret that process is stable, so we can go ahead with examining Xbar chart.
From the table of Xbar, we see,
Maximum value of Xbar is 1011.75 and
Minimum value of Xbar is 983.25
These values are within UCL and LCL of Xbar. i.e. 1024.025 and 972.509
Therefore, we conclude that “The process is in statistical control”.
| SAMPLE NUMBER | Mean | Range |
| 1 | 1006.75 | 43.00 |
| 2 | 994.25 | 25.00 |
| 3 | 1008.25 | 22.00 |
| 4 | 1011.75 | 42.00 |
| 5 | 990.50 | 30.00 |
| 6 | 1005.00 | 39.00 |
| 7 | 991.50 | 39.00 |
| 8 | 993.75 | 44.00 |
| 9 | 998.75 | 57.00 |
| 10 | 997.75 | 22.00 |
| 11 | 999.75 | 22.00 |
| 12 | 983.25 | 24.00 |
| 13 | 996.50 | 56.00 |
| 14 | 995.25 | 28.00 |
| 15 | 1001.00 | 37.00 |
