Chapter 15 – Statistical Methods for Quality Control
POINTS:
1
59. The following data represent the filling weights based on samples of 14.5 ounce cans of whole peeled tomatoes. Ten
samples of size 5 were taken. Use Excel to develop an x-bar chart.
Sample
Observ. 1
Observ. 2
Observ. 3
Observ. 4
Observ. 5
1
14.34988
13.86116
14.62213
15.13824
15.09918
2
14.15490
13.65478
13.57654
14.01119
14.11325
3
14.33650
14.31488
15.17132
14.45736
14.40692
4
15.33073
13.69380
14.76947
14.95110
15.45946
5
13.77791
14.07638
13.73921
14.31856
14.48376
6
13.21121
15.22384
13.86012
14.17321
14.87886
7
14.84700
14.66132
14.03008
14.37953
14.56577
8
14.53612
14.91492
14.93100
14.18173
14.03840
9
15.60284
15.22188
15.15195
14.55648
14.50098
10
14.72211
14.80895
14.60674
13.98653
15.11910
60. The following data represent the filling weights based on samples of 350-gram containers. Ten samples of size 5 were
taken. Use Excel to develop an R chart.
Sample
Observ. 1
Observ. 2
Observ. 3
Observ. 4
Observ. 5
1
333.6226
339.3906
361.9761
339.1192
346.4578
2
365.5820
347.4967
349.5748
352.6524
363.7096
3
363.8708
367.4003
335.0422
328.8487
355.8509
4
338.4916
338.6541
346.3491
366.9538
343.1767
5
355.2305
345.7635
356.5218
347.2718
334.5434
6
345.6990
326.0756
328.9903
362.4881
352.8718
7
334.7083
359.4960
333.1609
352.2697
360.8256
8
341.2400
356.8819
369.7263
336.0729
361.5562
9
356.7090
343.1499
373.2071
352.1363
353.2949
10
351.4613
338.4823
366.3254
346.1882
343.1589
1
61. The following data represent the filling weights based on samples of 350-gram containers. Ten samples of size 5 were
taken. Use Excel to develop an x-bar chart.
Sample
Observ. 1
Observ. 2
Observ. 3
Observ. 4
Observ. 5
1
333.6226
339.3906
361.9761
339.1192
346.4578
2
365.5820
347.4967
349.5748
352.6524
363.7096
3
363.8708
367.4003
335.0422
328.8487
355.8509
4
338.4916
338.6541
346.3491
366.9538
343.1767
5
355.2305
345.7635
356.5218
347.2718
334.5434
6
345.6990
326.0756
328.9903
362.4881
352.8718
7
334.7083
359.4960
333.1609
352.2697
360.8256
8
341.2400
356.8819
369.7263
336.0729
361.5562
9
356.7090
343.1499
373.2071
352.1363
353.2949
10
351.4613
338.4823
366.3254
346.1882
343.1589
1
62. A production process is considered in control if 6% of the items produced are defective. Samples of size 300 are used
for the inspection process.
a.
Determine the standard error of the proportion.
b.
Determine the upper and the lower control limits for the p chart.
a.
b.
63. A production process is considered in control if 4% of the items produced are defective. Samples of size 100 are used
for the inspection process.
a.
Determine the standard error of the proportion.
b.
Determine the upper and the lower control limits for the p chart.
a.
0.0196 (rounded)
b.
Note: Since the lower control limit is negative, it is set equal to zero.
64. Brakes Shop, Inc., is a franchise that specializes in repairing brake systems of automobiles. The company purchases
brake shoes from a national supplier. Currently, lots of 1,000 brake shoes are purchased, and each shoe is inspected before
being installed on an automobile. The company has decided, instead of 100% inspection, to adopt an acceptance sampling
plan.
a.
Explain what is meant by the acceptance sampling plan.
b.
If the company decides to adopt an acceptance sampling plan, what kinds of risks are there?
c.
The quality control department of the company has decided to select a sample of 10 shoes and
inspect them for defects. Furthermore, it has been decided that if the sample contains no
defective parts, the entire lot will be accepted. If there are 50 defective shoes in a shipment,
what is the probability that the entire lot will be accepted?
d.
What is the probability of accepting the lot if there are 100 defective units in the lot?
defective units that can be found in the sample and still consider the lot as acceptable.
65. The quality control department of a company has decided to select a sample of 20 items from each shipment of goods
it receives and inspect them for defects. It has been decided that if the sample contains no defective parts, the entire lot
will be accepted. Each shipment contains 1,000 items.
a.
What is the probability of accepting a lot that contains 10% defective items?
b.
What is the probability of accepting a lot that contains 5% defective items?
c.
What is the probability of rejecting a lot that contains 15% defective items?
a.
0.1216
b.
0.3585
c.
0.9612
66. An acceptance sampling plan uses a sample of 18 with an acceptance criterion of zero. Determine the probability of
accepting shipments that contain 5, 10, 15, 20, 25, 30, 35, 40, and 45% defective units.
67. The quality control department of a company has decided to select a sample of 10 items from the shipments received;
and if the sample contains no defective parts, the entire shipment will be accepted.
a.
If there are 40 defective items in a shipment, what is the probability that the entire lot will be
accepted?
b.
Use the binomial table and read the probability of accepting lots that contain 5, 10, 15, 20, 25,
30, 35, 40, 45, and 50% defective units.
a.
0.9044
b.
0.5987
10
0.3487
15
0.1969
20
0.1074
25
0.0563
30
0.0282
35
0.0135
40
0.0060
45
0.0025
50
0.0010
c.
0.5987
d.
.3487
68. The weight of bags of cement filled by Granite Rock Company’s packaging process is normally distributed with a
mean of 50 pounds and a standard deviation of 1.5 pounds when the process is in control. What should the control limits
be for a sample mean, , chart if 9 bags are sampled at a time?
69. Snipper, Inc. manufactures lawnmowers that require minor, final assembly by the customer. A sealed plastic bag
containing the hardware (nuts, bolts, washers, and so on) needed for final assembly is included with each lawnmower
shipped. During a week of normal, in-control operation, twenty samples of 200 bags of hardware were examined for
content (hardware type and count) accuracy. A total of 104 bags of the 4000 examined failed to have the correct contents.
a. Compute the upper limit, center line, and lower limit for a p chart.
b. Compute the upper limit, center line, and lower limit for an np chart.
70. A U.S. manufacturer of video cassette recorders purchases a circuit board from a Taiwanese firm. The circuit boards
are shipped in lots of 2000. The acceptance sampling procedure uses 12 randomly selected circuit boards. The acceptance
number is 1. If p0 is .03 and p1 is .20, what are the producer’s and consumer’s risks for this plan?
71. To inspect incoming shipments of components, a manufacturer is considering samples of sizes 12, 15, and 18. Use
binomial probabilities to select a sampling plan that provides a producer’s risk of α = .12 when p0 is .04 and a consumer’s
risk of β = .08 when p1 is .25.
72. A process sampled 30 times with a sample of size nine resulted in = 12.7 and = 0.8. Compute the upper and lower
control limits for the and charts for this process.
73. A process that is in control has a mean of μ = 56.5 and a standard deviation of σ = 3.4. What should the control limits
be for a sample mean chart if samples of size 8 are taken?
74. An acceptance sampling plan with n = 20 and c = 1 has been designed with a producer’s risk of .12.
a. Was the value of p0 equal to .02, .03, .04, or .05?
b. What is the consumer’s risk associated with this plan if p1 is .08?
c. Assume the consumer’s risk found in (b) is unacceptably high. Which modification of the sampling plan will result in
the greater reduction of the consumer’s risk, increasing n to 30 or decreasing c to 0?
75. Every check cashed or deposited at Lincoln Bank must be encoded with the amount of the check before it can begin
the Federal Reserve clearing process. If there is any discrepancy between the amount a check is made out for and the
encoded amount, the check is defective. The manager of the check encoding department knows from past experience that
when the encoding process is in control, an average of 2.5% of the encoded checks are defective. She wants to construct a
p chart, assuming the sample size will be 200 checks. Determine the center line and the 3− control limits for the p
chart.
76. Ledd Electronics has received a large shipment of power supply units for the desktop computers being assembled. The
units are coming from a new supplier and Ledd is not sure what the actual defect rate will be for this component. Ledd is
considering an acceptance sampling plan with n = 30 and c = 1.
a. Find the probability of accepting a lot when the defect rate is 2%, 4%, and 6%.
b. What happens to the producer’s risk as the defect rate increases?
c. What happens to the consumer’s risk as the defect rate increases?
77. Harry Coates wants to construct and R charts at the bag-filling operation for Meow Chow cat food. He knows that
when the filling operation is functioning correctly, bags of cat food should average 50.00 pounds and 5-bag samples
should have an average range of .330 pounds. Harry had twenty 5-bag samples taken at 2-hour intervals and the sample
means and ranges are shown below. Determine the center lines and upper and lower control limits for the and R charts.
Sample
Number
Sample
Mean ( )
Sample
Range (R)
Sample
Number
Sample
Mean ( )
Sample
Range (R)
1
50.018
0.43
11
49.958
0.32
2
50.132
0.19
12
50.060
0.27
3
50.060
0.41
13
49.978
0.39
4
50.112
0.33
14
49.986
0.40
5
49.998
0.29
15
50.036
0.52
6
49.892
0.29
16
50.066
0.27
7
49.944
0.30
17
49.946
0.33
8
49.922
0.30
18
49.980
0.24
9
50.004
0.25
19
49.972
0.49
10
50.118
0.14
20
49.966
0.28
Chapter 15 – Statistical Methods for Quality Control
78. Jane Hochkiss, director of production at the center, has decided to record the number of defective labels in random
daily samples on control charts. Jane estimates that 1.5 percent loose labels is typical when the labeling process is in
control. Twelve daily samples, each consisting of 200 pairs of jeans, were selected and examined. The number of
defective labels found in each sample is shown below.
Sample
Number
Number of
Defectives
Sample
Number
Number of
Defectives
1
2
7
3
2
3
8
0
3
5
9
5
4
2
10
3
5
7
11
9
6
1
12
2
a. Determine the center line and the 3-sigma control limits for the p chart.
b. Decide if the labeling operation is in control.
79. Five samples were taken, with five observations each. The sample findings are listed below. The sample values
represent service times in minutes.
Observation Number
Sample Number
1
2
3
4
5
1
10.1
10.6
9.8
9.9
10.9
2
9.7
9.5
10.3
9.9
10.5
3
10.1
10.7
9.2
10.0
10.1
4
9.9
9.8
10.5
10.4
10.1
5
10.4
10.1
10.9
9.9
10.3
a. Determine the control limits for the R-chart.
Chapter 15 – Statistical Methods for Quality Control
b. Would you conclude the R-chart is in statistical control and proceed to develop the chart?