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Quality: Statistical Control Common Causes x n i 1 n x i σ 2 ( x i x ) n Grams Assignable Causes Average Assignable Causes Average (a) Location Grams (b) Spread Grams Assignable Causes Average Effects

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Quality: Statistical Control Common Causes x n i 1 n x i σ 2 ( x i x ) n Grams Assignable Causes Average Assignable Causes Average (a) Location Grams (b) Spread Grams Assignable Causes Average Effects of Assignable Causes on Control (c) Shape Grams Out of control (assignable causes present) Effects of Assignable Causes on Control Sample Means and the Distribution Mean distribution In Control (no assignable causes) 425 Grams Sample Means and the Distribution Mean Distribution of sample means The Normal Distribution σ Standard deviation distribution 425 Grams Mean 3σ 2σ 1σ +1σ +2σ +3σ 68.26% 95.44% 99.74% Assignable causes likely Samples Using for Improvement Measure the process When changes are indicated, find the assignable cause Eliminate problems, incorporate improvements Repeat the cycle Control Chart Examples Control Chart Examples Control Chart Examples Control Chart Examples Control Chart Examples Control Limits and Errors Type I error: Probability of searching for a cause when none exists Type II error: Probability of concluding that nothing has changed Shift in process average average Two-sigma limits Special Metal Screw Sample Sample Number for Variables for Variables Special Metal Screw R-Charts R R D 4 R R D 3 R Control Chart for Factors Variables Factor for Factor for Factor Size of and for for for Sample x-charts R-Charts R-Charts (n) (A 2 ) (D 3 ) (D 4 ) for Variables Special Metal Screw R-Charts R D D 3 0 R D 4 R R D 3 R R (0.0021) in. R 0 (0.0021) 0 in. Range Chart - Special Metal Screw for Variables Special Metal Screw x-charts R A x x x + A 2 R x x - A 2 R x (0.0021) in. x (0.0021) in. x-chart Special Metal Screw x-chart Special Metal Screw Measure the process Find the assignable cause Eliminate the problem Repeat the cycle for Variables Using σ x x + zσ x x x zσ x σ σ x σ/ n Sunny Dale Bank x 5.0 minutes 1.5 minutes n 6 customers z 1.96 for Variables Using σ x x + zσ x x x zσ x σ σ x σ/ n Sunny Dale Bank x 5.0 minutes 1.5 minutes n 6 customers z 1.96 x (1.5)/ x (1.5)/ min min p p + zσ p p p zσ p σ p p(1 p)/n Sample Number 6 p p + 4zσ p 9 p p -10 zσ p σ p Wrong Account Number p(1 - p)/n Total 147 p n 2500 Total defectives Total observations p p + zσ p p p zσ p σ p p(1 p)/n n 2500 p n 2500 p p (0.0014) p (0.0014) p-chart Wrong Account Numbers σ p p-chart Wrong Account Numbers Measure the process Find the assignable cause Eliminate the problem Repeat the cycle Woodland Paper Company Woodland Paper Company Woodland Paper Company c 20 z 2 c c + z c c c z c Woodland Paper Company Measure the process Find the assignable cause Incorporate the problem Repeat the cycle Lower value distribution Upper Hours (a) is capable value distribution value Six sigma Four sigma Lower Upper Lower Two sigma Upper (b) is not capable Hours Mean Upper 1200 hours Lower 800 hours Average life 900 hours σ 48 hours Upper 1200 hours Lower 800 hours Average life 900 hours σ 48 hours Cp Upper - Lower 6σ Ratio Upper 1200 hours Cp 1.39 Lower 800 hours Average life 900 hours σ 48 hours Upper 1200 hours Cp 1.39 Lower 800 hours Average life 900 hours σ 48 hours Cpk Minimum of Capability Index x Lower, 3σ Upper x 3σ Upper 1200 hours Lower 800 hours Average life 900 hours σ 48 hours Cpk 0.69 Capability Index Cp 1.39 Capability Ratio

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