APME_5 - Applied Probability Methods for Engineers Slide...

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Applied Probability Methods for Engineers Slide Set 5
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Chapter 16 Quality Control Methods
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Statistical Process Control A control chart is a useful visual tool for observing an important measurement over time Used to detect when something unusual occurs in a measurable process
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Control Charts Drift in average value: Increase in variation:
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Control Limits How do we know when a process is out of control? Control limits: Upper Control Limit (UCL) and Lower Control Limit (LCL) Hypothesis test: H 0 : process is in control H A : process is out of control Type I error: α associated with hypothesis test (Type I error is probability of observing point outside control limits when it is actually in control) 3 sigma control limits often used α = 1 – 0.9974 = 0.0026
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Note on Six Sigma Six Sigma is actually a set of total quality management (TQM) principles and practices Six Sigma was started at Motorola Processes operating at the six sigma level produce less than 3.4 defects per one million opportunities 3.4 does not correspond to our 0.0026 α value What is going on? Six Sigma permits a 1.5 standard deviation shift in the mean One tail α associated with z = 4.5 equals 0.0000034
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Piston Head Example In control process produces piston heads with radius values with a mean of μ 0 = 30.00 mm and standard deviation σ = 0.05 mm Suppose we take samples of size n and look at the sample mean to determine whether the process is in control Sample mean has expected value μ 0 and standard deviation σ/ n Control chart has center line at μ 0 = 30.00 mm, and LCL and UCL equal to μ 0 - 3σ/ n and μ 0 + 3σ/ n
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Piston Head Example If sample size n = 5, then LCL = 30 – 3 × 0.05/ 5; UCL = 30 + 3 × 0.05/ 5 LCL = 29.933; UCL = 30.067 Probability of Type I error = 1 – P(29.933 ≤ X ≤ 30.067) = 1 – P(-3 ≤ Z ≤ 3) = 0.0026
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Control Charts Even if a process is within the control limits, we can detect problems when a series of consecutive points lies completely above or below the center line
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Control Chart Properties Recall the definition of Type II error:
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