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The Upper Control Limit = 0.151 and Lower Control Limit = 0.001.
Upper Control Limit and Lower Control Limit is defined as control Limits are calculated based on the amount of variation in the process you are measuring. One measure of variation is standard deviation. The mean +/- three standard deviations is an approach that is frequently used to determine control limits.
Given data
Sample number Number detective Size of sample
1 5 50
2 2 50
3 4 50
4 6 50
5 2 50
Number of samples (N) = 5
Sample size (n) = 50
For two sigma limit, Z = 2.
Proportion defective (P-bar) = Number of defects / Total Number of observations
P-bar = Number of defects / (Number of samples x Sample size)
P-bar = (5 + 2 + 4 + 6 + 2) / (5 x 50)
P-bar = 19 / 250
P-bar = 0.076
Standard deviation of defective rate (Sp) = SQRT [P-bar x (1 - P-bar)] / n
Sp = SQRT [0.076 x (1 - 0.076)] / 50
Sp = 0.0375
Calculations for the Upper Control Limit (UCL) and Lower Control Limit (LCL) are as follows:
UCL = P-bar + (Z x Sp)
UCL = 0.076 + (2 x 0.0375)
UCL = 0.076 + 0.075
UCL = 0.151
LCL = P-bar - (Z x Sp)
LCL = 0.076 - (2 x 0.0375)
LCL = 0.076 - 0.075
LCL = 0.001
Therefore the Upper Control Limit = 0.151 and Lower Control Limit = 0.001
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