1. Determine Critical X’s Statistical Tests for a Continuous Single Variable Deliverable 10A

2. Define Module Roadmap Define 1D – Define VOC, VOB, and CTQ’s 2D – Define Project Boundaries 3D – Quantify Project Value 4D – Develop Project Mgmt. Plan Measure 5M – Document Process 6M – Prioritize List of X’s 7M – Create Data Collection Plan 8M – Validate Measurement System 9M – Establish Baseline Process Cap. Analyze 10A – Determine Critical X’s Improve 12I – Prioritized List of Solutions 13I – Pilot Best Solution Control 14C – Create Control System 15C – Finalize Project Documentation Green 11G – Identify Root Cause Relationships Queue 1 Queue 2

6. Statistical Tests Continuous Y Discreet Y Discreet X 2 Sample t Test Test for Equal Variance One-Way ANOVA (Tukeys) Moods Median Paired t Test Two Way ANOVA GLM CHI Square TOA Two Proportion Continuous X Correlation Simple Linear Regression Multiple Linear Regression These tools are not taught as part of Black Belt training Vs. Target Normality 1 Sample T One Sample Sign CHI Square GOF One Proportion

7. Hypothesis Test Categories Continuous Y, Continuous X(s) Tests Continuous Y, Discrete X(s) Tests Discrete Y, Continuous X(s) Tests Discrete Y, Discrete X(s) Tests Continuous Y? Y N Continuous X(s)? Continuous X(s)? Y N N Y Start

8. Continuous Y, Discrete X(s) Test for Normality ( Shape = normal) Residuals Normal? ResidualsEqual Variance? Residuals Stable? Y Y See MBB N N N Y Done Testing vs. a Target Value(s)? Y N 1 Sample t  1 Sample Sign (m = #) Data Symmetric? 1 Sample Wilcoxon (m = #) Y N Not Normal Normal Done No of X’s? 1 > 2 2 Sample t (Assume equal variance) (  No of levels? 2 Data Paired? N Y > 3 Paired t (  See MBB 1 Way ANOVA (  General Linear Model (  Go to “B” Perform Box-Cox Transform and Reanalyze Data already Transformed? N Y

9. Testing for Normality One Variable, Continuous Data H o : The data is normally distributed H a : The data is not normally distributed

11. Descriptive Statistics Graphical Output Is the data normal?

12. Normality Test Graphical Output

14. 1 Sample t Test Normally Distributed Continuous Data vs. Target H o :  a = Target H a :  a ≠ Target

16. Chlorine Residuals Graphical Summary Data is normal!

18. 1 Sample t Test Check ‘histogram of the data’ Make the test one-tailed by choosing ‘greater than’ The test mean is 0.03 Summarized data would go here

19. 1 Sample t Test One-Sample T: PPM Chlorine Test of mu = 0.3 vs > 0.3 95% Lower Variable N Mean StDev SE Mean Bound T P PPM Chlorine 55 0.335889 0.045808 0.006177 0.325552 5.81 0.000 P value < 0.05, reject the null and conclude the mean is greater than 0.3 Target Value (H o ) Confidence interval for the lower bound H o :  =0.3 H a :  >0.3

21. One Sample Sign Test Non-Normally Distributed Continuous Data vs. a Target H o : Median = Target H a : Median ≠ Target

25. Vacation Days Example First, determine if the data is normal… Did this catch you? Don’t forget to check for normality. Technically, we should use a 1 sample t test of the mean for this data. However, let’s proceed with a 1- sample sign test for the median for illustration.

27. Vacation Days Example Sign Test for Median: Vacation Days Sign test of median = 15.00 versus > 15.00 N Below Equal Above P Median Vacation Days 78 30 5 43 0.0801 18.00 Note the lack of power in a nonparametric statistical test. 43 of 78 data points are above 15 yet there is still insufficient evidence to prove the population is above 15. What if we had used the 1-sample t after all? The number of observations below the test median. The number of observations above the test median. The number of observations equal to the test median.

28. Vacation Days Revisited One-Sample T: Vacation Days Test of mu = 15 vs > 15 95% Lower Variable N Mean StDev SE Mean Bound T P Vacation Days 78 16.6154 7.6145 0.8622 15.1800 1.87 0.032 There is enough data to confirm the mean is >15!