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Example Problems

Example Problems. Hypothesis Testing & Relationship to Continuous Variable Control Charts. Outline. Anthropomorphic Data Hypothesis Test Results Ishikawa’s (Magnificent Seven) Tools Hypothesis Testing to Control Charts X-Bar & R-Charts X-Bar & Sigma-Charts

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Example Problems

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  1. Example Problems Hypothesis Testing & Relationship to Continuous Variable Control Charts

  2. Outline • Anthropomorphic Data Hypothesis Test Results • Ishikawa’s (Magnificent Seven) Tools • Hypothesis Testing to Control Charts • X-Bar & R-Charts • X-Bar & Sigma-Charts • Development of Trial Control Limits • Sensitizing Rules for Control Charts • Practice With Control Charts Continuous Variable Control Charts

  3. R-L Side, Equal Variance Dom. Means Comparison: L = x1 = 129.4, S12 = 2788, n1 = 34 people R = x2 = 104.0, S22 = 1225, n2 = 20 people Sp = 47.1, v = 52 Two-Sided Test at  = .05 HA: There is a difference Test: Is | t0 | > t.025, 52? |1.91| > 2.021 - NO! Keep the Null Hypothesis: There is NOT a difference btwn L & R ! Grip Strength Data Results Continuous Variable Control Charts

  4. R-L Side, No Assumptions Means Comparison: L = x1 = 129.4, S12 = 2788, n1 = 34 people R = x2 = 104.0, S22 = 1225, n2 = 20 people v = 51 Two-Sided Test at  = .05 HA: There is a difference Test: Is | t0 | > t.025, 51? |2.12| > 2.021 - YES! Reject the Null Hypothesis: There IS a difference btwn L & R! Why is this wimpy test significant when the other wasn’t? ANS: Check the equal variance assumption! Grip Strength Data Results Continuous Variable Control Charts

  5. Unknown 0 Variances Comparison: S12 = 2788 n1 = 34, v1 = 33 S22 = 1225 n2 = 20, v2 = 19 Two-Sided Test at  = .10 HA: There is a difference Test: Is F0 > F.05, 33, 19? 2.276 > 2.07 - YES! (Should also checkF1– /2, 33, 19) Reject the Null Hypothesis: There IS a difference in variance! At  = .05, this test is just barely not significant (Should also have checked for Normality with Normal Prob. Plot) Grip Strength Data Results Continuous Variable Control Charts

  6. UCL 0 CL 0 LCL Sample Number 2-Sided Hypothesis Test Sideways Hypothesis Test Shewhart Control Chart  2  2  2  2 Moving from Hypothesis Testing to Control Charts • A control chart is like a sideways hypothesis test • Detects a shift in the process • Heads-off costly errors by detecting trends Continuous Variable Control Charts

  7. Shape, Location, & Spread • Shape is governed by Central Limit Theorem • Normally distributed if sample size is large enough • Usually pick n = 4, 5, or 6 • Location is measured by X-Bar Chart • Measures variation between sample locations • Spread is measured by either R-Chart or -Chart • Measures variation changes within a sample • Requires rational sampling - e.g. avoid a taking a single sample across two shifts - difference in location could show up as change in variation. Continuous Variable Control Charts

  8. X-Bar Control Limits: Approximate 3 limits are found from range & table R-Chart Control Limits: Approximate, asymmetric 3 limits from range & table X-Bar & R-Charts • Most common of all control charts • Can approximate non-Normal processes Continuous Variable Control Charts

  9. X-Bar Control Limits: Approximate 3 limits are found from known 0 & table R-Chart Control Limits: Approximate, asymmetric 3 limits from 0 & table X-Bar & R-Charts • Limits can also be generated from historical data: Continuous Variable Control Charts

  10. The X-Bar Chart checks variability in location between samples The R-Chart checks for changes in sample variation UCL UCL x R LCL LCL Sample Number Sample Number X-Bar (Means) Control Chart R - (Range) Control Chart X-Bar & R-Charts Continuous Variable Control Charts

  11. X-Bar Control Limits: Approximate 3 limits are found from S & table Sigma-Chart Control Limits: Approximate, asymmetric 3 limits from S & table X-Bar & Sigma-Charts • Used when sample size is greater than 10 Continuous Variable Control Charts

  12. X-Bar Control Limits: Approximate 3 limits are found from known 0 & table Sigma-Chart Control Limits: Approximate, asymmetric 3 limits from 0 & table X-Bar & Sigma-Charts • Limits can also be generated from historical data: Continuous Variable Control Charts

  13. Steps for Trial Control Limits • Start with 20 to 25 samples • Use all data to calculate initial control limits • Plot each sample in time-order on chart. • Check for out of control sample points • If one (or more) found, then: • Investigate the process; • Remove the special cause; and • Remove the special cause point and recalculate control limits. • If can’t find special cause - drop point & recalculate anyway Continuous Variable Control Charts

  14. Control Chart Sensitizing Rules • Western Electric Rules: • One point plots outside the three-sigma limits; • Eight consecutive points plot on one side of the center line; • Two out of three consecutive points plot beyond two-sigma warning limits on the same side of the center line; or • Four out of five consecutive points plot beyond one-sigma warning limits on the same side of the center line. • If chart shows lack of control, investigate for special cause Continuous Variable Control Charts

  15. Control Chart Sensitizing Rules • Additional Sensitizing Rules: • One or more points very near a control limit. • Six points in a row steadily increasing or decreasing. • Eight points in a row on both sides of the center line, but none in-between the one-sigma warning limits on both sides of the center line. • Fourteen points in a row alternating above and below the center line. • Fifteen points in a row anywhere between the one-sigma warning limits (including either side of the center line). • Any unusual or non-random pattern to the plotted points. Continuous Variable Control Charts

  16. Practice With Control Charts • X-Bar & R-Chart use for Anthropomorphic Data • Setting Trial Limits • Utilization for Control • Problems from CH 4: • # 5, 9, 11, 13, 23, 24, 32 • Little calculation, more interpretation & thinking Continuous Variable Control Charts

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