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Tata McGraw
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Chapter 9AProcess Capability and SPC
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Process VariationProcess Capability Process Control ProceduresVariable dataAttribute dataAcceptance SamplingOperating Characteristic CurveOBJECTIVES 9A-*
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Basic Forms of VariationAssignable variation is caused by factors that can be clearly identified and possibly managedCommon variation is inherent in the production process Example: A poorly trained employee that creates variation in finished product output.Example: A molding process that always leaves burrs or flaws on a molded item.9A-*
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Taguchis View of VariationTraditional view is that quality within the LS and US is good and that the cost of quality outside this range is constant, where Taguchi views costs as increasing as variability increases, so seek to achieve zero defects and that will truly minimize quality costs.9A-*
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Process Capability Index, CpkShifts in Process MeanCapability Index shows how well parts being produced fit into design limit specifications.As a production process produces items small shifts in equipment or systems can cause differences in production performance from differing samples.9A-*
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A simple ratio: Specification Width_________________________________________________________ Actual Process Width
Generally, the bigger the better.Process Capability A Standard Measure of How Good a Process Is.9A-*
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Process CapabilityThis is a one-sided Capability IndexConcentration on the side which is closest to the specification - closest to being bad9A-*
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The Cereal Box ExampleWe are the maker of this cereal. Consumer reports has just published an article that shows that we frequently have less than 16 ounces of cereal in a box.Lets assume that the government says that we must be within 5 percent of the weight advertised on the box.Upper Tolerance Limit = 16 + .05(16) = 16.8 ouncesLower Tolerance Limit = 16 .05(16) = 15.2 ouncesWe go out and buy 1,000 boxes of cereal and find that they weight an average of 15.875 ounces with a standard deviation of .529 ounces.
9A-*
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Cereal Box Process CapabilitySpecification or Tolerance LimitsUpper Spec = 16.8 ozLower Spec = 15.2 ozObserved WeightMean = 15.875 ozStd Dev = .529 oz9A-*
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What does a Cpk of .4253 mean?An index that shows how well the units being produced fit within the specification limits. This is a process that will produce a relatively high number of defects.Many companies look for a Cpk of 1.3 or better 6-Sigma company wants 2.0!
9A-*
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Types of Statistical SamplingAttribute (Go or no-go information)Defectives refers to the acceptability of product across a range of characteristics.Defects refers to the number of defects per unit which may be higher than the number of defectives.p-chart application Variable (Continuous)Usually measured by the mean and the standard deviation.X-bar and R chart applications9A-*
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Statistical Process Control (SPC) ChartsUCLLCLSamples over time 1 2 3 4 5 6UCLLCLSamples over time 1 2 3 4 5 6UCLLCLSamples over time 1 2 3 4 5 6Normal BehaviorPossible problem, investigatePossible problem, investigate9A-*
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Control Limits are based on the Normal Curvex0123-3-2-1zmStandard deviation units or z units. 9A-*
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Control LimitsWe establish the Upper Control Limits (UCL) and the Lower Control Limits (LCL) with plus or minus 3 standard deviations from some x-bar or mean value. Based on this we can expect 99.7% of our sample observations to fall within these limits. LCLUCL99.7%9A-*
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Example of Constructing a p-Chart: Required DataSample No.No. ofSamplesNumber of defects found in each sample9A-*
Sheet1
SampleObs 1Obs 2Obs 3Obs 4Obs 5AvgRangenA2D3D4
110.68210.68910.77610.79810.71410.7320.11621.8803.27
210.78710.8610.60110.74610.77910.7550.25931.0202.57
310.7810.66710.83810.78510.72310.7590.17140.7302.28
410.59110.72710.81210.77510.7310.7270.22150.5802.11
510.69310.70810.7910.75810.67110.7240.11960.4802
610.74910.71410.73810.71910.60610.7050.14370.420.081.92
710.79110.71310.68910.87710.60310.7350.27480.370.141.86
810.74410.77910.6610.73710.82210.7480.16290.340.181.82
910.76910.77310.64110.64410.72510.7100.132100.310.221.78
1010.71810.67110.70810.8510.71210.7320.179110.290.261.74
1110.78710.82110.76410.65810.70810.7480.163
1210.62210.80210.81810.87210.72710.7680.250
1310.65710.82210.89310.54410.7510.7330.349
1410.80610.74910.85910.80110.70110.7830.158
1510.6610.68110.64410.74710.72810.6920.103
Averages10.7370.187
A2
SamplenDefectives
11004
21002
31005
41003
51006
61004
71003
81007
91001
101002
111003
121002
131002
141008
151003
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Sheet1
10.7318
10.7546
10.7586
10.727
10.724
10.7052
10.7346
10.7484
10.7104
10.7318
10.7476
10.7682
10.7332
10.7832
10.692
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Avg
Sheet2
0.116
0.259
0.171
0.221
0.119
0.143
0.274
0.162
0.132
0.179
0.163
0.25
0.349
0.158
0.103
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Range
Sheet3
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Sheet4
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Sheet5
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Sheet6
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Sheet7
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Sheet9
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Sheet11
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Sheet16
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Statistical Process Control Formulas:Attribute Measurements (p-Chart)Given:Compute control limits: 9A-*
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1. Calculate the sample proportions, p (these are what can be plotted on the p-chart) for each sampleExample of Constructing a p-chart: Step 19A-*
Chart1
0.040.10140180630
0.040.10140180630
0.050.10140180630
0.030.10140180630
0.080.10140180630
0.040.10140180630
0.030.10140180630
0.140.10140180630
0.010.10140180630
0.020.10140180630
0.030.10140180630
0.020.10140180630
0.020.10140180630
0.080.10140180630
0.030.10140180630
Observation
p
Sheet1
SampleObs 1Obs 2Obs 3Obs 4Obs 5AvgRangenA2D3D4
110.68210.68910.77610.79810.71410.7320.11621.8803.27
210.78710.8610.60110.74610.77910.7550.25931.0202.57
310.7810.66710.83810.78510.72310.7590.17140.7302.28
410.59110.72710.81210.77510.7310.7270.22150.5802.11
510.69310.70810.7910.75810.67110.7240.11960.4802
610.74910.71410.73810.71910.60610.7050.14370.420.081.92
710.79110.71310.68910.87710.60310.7350.27480.370.141.86
810.74410.77910.6610.73710.82210.7480.16290.340.181.82
910.76910.77310.64110.64410.72510.7100.132100.310.221.78
1010.71810.67110.70810.8510.71210.7320.179110.290.261.74
1110.78710.82110.76410.65810.70810.7480.163
1210.62210.80210.81810.87210.72710.7680.250
1310.65710.82210.89310.54410.7510.7330.349
1410.80610.74910.85910.80110.70110.7830.158
1510.6610.68110.64410.74710.72810.6920.103
Averages10.7370.187
A2
SamplenDefectivespUCLLCL
110040.040.09304928810
210020.020.09304928810
310050.050.09304928810
410030.030.09304928810
510060.060.09304928810
610040.040.09304928810
710030.030.09304928810
810070.070.09304928810
910010.010.09304928810
1010020.020.09304928810
1110030.030.09304928810
1210020.020.09304928810
1310020.020.09304928810
1410080.080.09304928810
1510030.030.09304928810
n-barsum defectsp-bar
100550.0366666667
sp
15000.0187942071
UCL0.0930492881
LCL-0.0197159548
0
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Sheet1
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Avg
Sheet2
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Range
Sheet3
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Sheet6
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Sheet7
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Sheet9
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2. Calculate the average of the sample proportions3. Calculate the standard deviation of the sample proportion Example of Constructing a p-chart: Steps 2&39A-*
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4. Calculate the control limitsUCL = 0.0924LCL = -0.0204 (or 0)Example of Constructing a p-chart: Step 49A-*
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Example of Constructing a p-Chart: Step 55. Plot the individual sample proportions, the average of the proportions, and the control limits 9A-*
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Example of x-bar and R Charts: Required Data9A-*
Sheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Sheet: Sheet4
Sheet: Sheet5
Sheet: Sheet6
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
Sample
Obs 1
Obs 2
Obs 3
Obs 4
Obs 5
Avg
Range
n
A2
D3
D4
1.0
10.682
10.689
10.776
10.798
10.714
10.731800000000002
0.11599999999999966
2.0
10.787
10.86
10.601
10.746
10.779
10.7546
0.25899999999999856
10.667
10.838
10.785
10.723
10.758599999999998
0.17099999999999937
4.0
10.591
10.727
10.812
10.726999999999999
0.22100000000000009
5.0
10.693
10.708
10.79
10.758
10.671
10.724
0.11899999999999977
6.0
10.749
10.714
10.738
10.719
10.606
10.705200000000001
0.14300000000000068
7.0
10.791
10.713
10.689
10.877
10.603
10.7346
0.2740000000000009
8.0
10.744
10.779
10.11
10.737
10.75
10.623999999999999
0.6690000000000005
9.0
10.769
10.773
10.641
10.644
10.725
10.7104
0.13199999999999967
10.0
10.718
10.671
10.708
10.85
10.712
10.731800000000002
0.17900000000000027
11.0
10.787
10.821
10.764
10.658
10.708
10.7476
0.16300000000000026
12.0
10.622
10.802
10.818
10.872
10.727
10.768199999999998
0.25
13.0
10.657
10.822
10.893
10.544
10.75
10.7332
0.3490000000000002
14.0
10.806
10.749
10.859
10.801
10.701
10.7832
0.15799999999999947
10.681
10.644
10.747
10.728
10.692
0.10299999999999976
A2
0.58
X-Bar
UCL
LCL
D3
0.0
10.856245333333334
10.600581333333334
Averages
10.728413333333334
0.22039999999999996
D4
2.11
Mean
10.728413333333334
R-Bar
0.4650439999999999
0.0
Range
0.22039999999999996
A2
Sample
n
Defectives
p
0.04
0.04
0.05
0.03
0.08
0.04
0.03
0.16
0.01
0.02
0.03
0.02
0.02
0.08
0.03
1375.0
56.0
0.04072727272727273
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
10.731800000000002
10.7546
10.758599999999998
10.726999999999999
10.724
10.705200000000001
10.7346
10.7484
10.7104
10.731800000000002
10.7476
10.768199999999998
10.7332
10.7832
10.692
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
0.11599999999999966
0.25899999999999856
0.17099999999999937
0.22100000000000009
0.11899999999999977
0.14300000000000068
0.2740000000000009
0.16199999999999903
0.13199999999999967
0.17900000000000027
0.16300000000000026
0.25
0.3490000000000002
0.15799999999999947
0.10299999999999976
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Example of x-bar and R charts: Step 1. Calculate sample means, sample ranges, mean of means, and mean of ranges.9A-*
Chart1
Chart2
10.731810.856245333310.6005813333
10.754610.856245333310.6005813333
10.758610.856245333310.6005813333
10.72710.856245333310.6005813333
10.72410.856245333310.6005813333
10.705210.856245333310.6005813333
10.734610.856245333310.6005813333
10.62410.856245333310.6005813333
10.710410.856245333310.6005813333
10.731810.856245333310.6005813333
10.747610.856245333310.6005813333
10.768210.856245333310.6005813333
10.733210.856245333310.6005813333
10.783210.856245333310.6005813333
10.69210.856245333310.6005813333
Sample
Means
Chart3
0.1160.4650440
0.2590.4650440
0.1710.4650440
0.2210.4650440
0.1190.4650440
0.1430.4650440
0.2740.4650440
0.6690.4650440
0.1320.4650440
0.1790.4650440
0.1630.4650440
0.250.4650440
0.3490.4650440
0.1580.4650440
0.1030.4650440
Sample
Ranges
Sheet1
x-barRange
SampleObs 1Obs 2Obs 3Obs 4Obs 5AvgRangeUCLLCLUCLLCLnA2D3D4
110.68210.68910.77610.79810.71410.7320.11610.856245333310.60058133330.465044021.8803.27
210.78710.8610.60110.74610.77910.7550.25910.856245333310.60058133330.465044031.0202.57
310.7810.66710.83810.78510.72310.7590.17110.856245333310.60058133330.465044040.7302.28
410.59110.72710.81210.77510.7310.7270.22110.856245333310.60058133330.465044050.5802.11
510.69310.70810.7910.75810.67110.7240.11910.856245333310.60058133330.465044060.4802
610.74910.71410.73810.71910.60610.7050.14310.856245333310.60058133330.465044070.420.081.92
710.79110.71310.68910.87710.60310.7350.27410.856245333310.60058133330.465044080.370.141.86
810.74410.77910.1110.73710.7510.6240.66910.856245333310.60058133330.465044090.340.181.82
910.76910.77310.64110.64410.72510.7100.13210.856245333310.60058133330.4650440100.310.221.78
1010.71810.67110.70810.8510.71210.7320.17910.856245333310.60058133330.4650440110.290.261.74
1110.78710.82110.76410.65810.70810.7480.16310.856245333310.60058133330.4650440
1210.62210.80210.81810.87210.72710.7680.25010.856245333310.60058133330.4650440
1310.65710.82210.89310.54410.7510.7330.34910.856245333310.60058133330.4650440
1410.80610.74910.85910.80110.70110.7830.15810.856245333310.60058133330.4650440
1510.6610.68110.64410.74710.72810.6920.10310.856245333310.60058133330.4650440
Averages10.7280.220400
A20.58X-BarUCLLCL
D3010.856245333310.6005813333
D42.11
Mean10.728R-Bar0.4650440
Range0.220
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Example of x-bar and R charts: Step 2. Determine Control Limit Formulas and Necessary Tabled ValuesFrom Exhibit TN8.79A-*
Sheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Sheet: Sheet4
Sheet: Sheet5
Sheet: Sheet6
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
Sample
Obs 1
Obs 2
Obs 3
Obs 4
Obs 5
Avg
Range
n
A2
D3
D4
1.0
10.682
10.689
10.776
10.798
10.714
10.731800000000002
0.11599999999999966
2.0
10.787
10.86
10.601
10.746
10.779
10.7546
0.25899999999999856
10.667
10.838
10.785
10.723
10.758599999999998
0.17099999999999937
4.0
10.591
10.727
10.812
10.726999999999999
0.22100000000000009
5.0
10.693
10.708
10.79
10.758
10.671
10.724
0.11899999999999977
6.0
10.749
10.714
10.738
10.719
10.606
10.705200000000001
0.14300000000000068
7.0
10.791
10.713
10.689
10.877
10.603
10.7346
0.2740000000000009
8.0
10.744
10.779
10.66
10.737
10.822
10.7484
0.16199999999999903
9.0
10.769
10.773
10.641
10.644
10.725
10.7104
0.13199999999999967
10.0
10.718
10.671
10.708
10.85
10.712
10.731800000000002
0.17900000000000027
11.0
10.787
10.821
10.764
10.658
10.708
10.7476
0.16300000000000026
12.0
10.622
10.802
10.818
10.872
10.727
10.768199999999998
0.25
13.0
10.657
10.822
10.893
10.544
10.75
10.7332
0.3490000000000002
14.0
10.806
10.749
10.859
10.801
10.701
10.7832
0.15799999999999947
10.681
10.644
10.747
10.728
10.692
0.10299999999999976
Averages
10.736706666666668
0.18659999999999985
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
10.731800000000002
10.7546
10.758599999999998
10.726999999999999
10.724
10.705200000000001
10.7346
10.7484
10.7104
10.731800000000002
10.7476
10.768199999999998
10.7332
10.7832
10.692
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
0.11599999999999966
0.25899999999999856
0.17099999999999937
0.22100000000000009
0.11899999999999977
0.14300000000000068
0.2740000000000009
0.16199999999999903
0.13199999999999967
0.17900000000000027
0.16300000000000026
0.25
0.3490000000000002
0.15799999999999947
0.10299999999999976
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Example of x-bar and R charts: Steps 3&4. Calculate x-bar Chart and Plot Values9A-*
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Example of x-bar and R charts: Steps 5&6. Calculate R-chart and Plot ValuesUCLLCL9A-*
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Basic Forms of Statistical Sampling for Quality ControlAcceptance Sampling is sampling to accept or reject the immediate lot of product at handStatistical Process Control is sampling to determine if the process is within acceptable limits9A-*
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Acceptance SamplingPurposesDetermine quality levelEnsure quality is within predetermined levelAdvantagesEconomyLess handling damageFewer inspectorsUpgrading of the inspection jobApplicability to destructive testingEntire lot rejection (motivation for improvement) 9A-*
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Acceptance Sampling (Continued)DisadvantagesRisks of accepting bad lots and rejecting good lotsAdded planning and documentationSample provides less information than 100-percent inspection 9A-*
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Acceptance Sampling: Single Sampling PlanA simple goal Determine (1) how many units, n, to sample from a lot, and (2) the maximum number of defective items, c, that can be found in the sample before the lot is rejected9A-*
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RiskAcceptable Quality Level (AQL)Max. acceptable percentage of defectives defined by producerThe a (Producers risk)The probability of rejecting a good lotLot Tolerance Percent Defective (LTPD)Percentage of defectives that defines consumers rejection pointThe (Consumers risk)The probability of accepting a bad lot9A-*
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Operating Characteristic CurveThe OCC brings the concepts of producers risk, consumers risk, sample size, and maximum defects allowed togetherThe shape or slope of the curve is dependent on a particular combination of the four parameters9A-*
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Example: Acceptance Sampling ProblemZypercom, a manufacturer of video interfaces, purchases printed wiring boards from an outside vender, Procard. Procard has set an acceptable quality level of 1% and accepts a 5% risk of rejecting lots at or below this level. Zypercom considers lots with 3% defectives to be unacceptable and will assume a 10% risk of accepting a defective lot.
Develop a sampling plan for Zypercom and determine a rule to be followed by the receiving inspection personnel. 9A-*
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Example: Step 1. What is given and what is not? In this problem, AQL is given to be 0.01 and LTDP is given to be 0.03. We are also given an alpha of 0.05 and a beta of 0.10.What you need to determine is your sampling plan is c and n.9A-*
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Example: Step 2. Determine cFirst divide LTPD by AQL.Then find the value for c by selecting the value in the TN7.10 n(AQL)column that is equal to or just greater than the ratio above. So, c = 6.9A-*
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Question BowlA methodology that is used to show how well parts being produced fit into a range specified by design limits is which of the following?Capability indexProducers riskConsumers riskAQLNone of the aboveAnswer: a. Capability index9A-*
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Question BowlOn a quality control chart if one of the values plotted falls outside a boundary it should signal to the production manager to do which of the following?System is out of control, should be stopped and fixedSystem is out of control, but can still be operated without any concernSystem is only out of control if the number of observations falling outside the boundary exceeds statistical expectationsSystem is OK as isNone of the aboveAnswer: c. System is only out of control if the number of observations falling outside the boundary exceeds statistical expectations9A-*
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Question BowlYou want to prepare a p chart and you observe 200 samples with 10 in each, and find 5 defective units. What is the resulting fraction defective?252.50.00250.00025Can not be computed on data aboveAnswer: c. 0.0025 (5/(2000x10)=0.0025)9A-*
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Question BowlYou want to prepare an x-bar chart. If the number of observations in a subgroup is 10, what is the appropriate factor used in the computation of the UCL and LCL?1.880.310.221.78None of the aboveAnswer: b. 0.319A-*
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Question BowlYou want to prepare an R chart. If the number of observations in a subgroup is 5, what is the appropriate factor used in the computation of the LCL?00.881.882.11None of the aboveAnswer: a. 09A-*
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Question BowlYou want to prepare an R chart. If the number of observations in a subgroup is 3, what is the appropriate factor used in the computation of the UCL?0.871.001.882.11None of the aboveAnswer: e. None of the above9A-*
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Question BowlThe maximum number of defectives that can be found in a sample before the lot is rejected is denoted in acceptance sampling as which of the following?AlphaBetaAQL cNone of the aboveAnswer: d. c9A-*
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End of Chapter 9A9A-*
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