ASQ/Penn State Univ. Great Valley Statistics Symposium 10/3/09

1. Proportion Testing Chris Connors, Ph.D. Jay Armstrong, MSc., M.C.E. October 2, 2009 Statistics Symposium

2. Applied Statistics in Business, Healthcare, Pharmaceuticals, Education,and Industry

4. Hypothesis Tests: What we are covering? Continuous Data Attribute Data 1 sample t-test : Δ mean from known test mean 2 sample t-test: Δ mean between 2 independent sample means Paired t-test: Δ mean between 2 dependent sample means One Way ANOVA : At least 1 sample mean Δ between 3 or more samples Kruskal Wallis & Mood’s Median: At least 1 sample median Δ between 3 or more samples F-test, Levene’s test, & Bartlett’s test: At least 1 sample standard deviation Δ between 3 or more samples Correlation/Regression/DOE: 2 or more factors are correlated/ Predictor affects the sampled process 1 proportion test: A sample proportion Δ against a known value 2 proportion test: Proportions from the two samples are different Chi Square test: At least one sample proportion Δ from others:

8. 6S Black Belt Level of Cognition for Hypothesis Testing *1= learned, 2= know, 3 = used, 4 = taught Topic Level of Cognition My Development Introduction to Statistical Comparisons 2* Normality and Transformation 2 Correlation Analysis 3 Regression Analysis 3 Introduction to Multiple Linear Regression 1 t-tests 3 ANOVA 3 1 and 2 proportion test 3 Chi-Square Analysis 3 Binary Logistic Regression 1

9. 6S BB Level of Cognition for Hypothesis Testing *1= learned, 2= know, 3 = used, 4 = taught Topic Level of Cognition My Development Introduction Experimental Design 2* Background on Experimental design 2 DOE Designs and terminology 2 Full Factorial design 2 Half factorial designs 2 Robust Designs 2 Checklists for designing and conducting DOE BBC exercise: DOE Simulation 2 Results of DOE Simulation 2

11. Selecting the Right Statistical Tool Discrete Discrete Continuous Proportion Tests Logistic Regression t test ANOVA DOE Correlation Regression X Y Continuous

12. Determine if a statistically significant difference of proportion exists between: - A sample and a target - Two independent samples - Two samples or less Tests of Proportion Use samples to make inferences about population proportions 1 Proportion Test 1 Sample Comparing Proportions 2 Proportion Test Chi-Square Test More Than 2 Samples 2 Samples

14. One Tail or Two Tails: Placing the Alpha Risk

16. Binomial - key facts Useful fact: has approximately a normal distribution when n is large (more than 25 or 30) and np and n( 1 -p) are not too small (say >5).

17. Binomial - Normal Approximation

18. Histogram: n=20

19. Histogram: n=100

23. Hypothesis tests are either one tailed or two tail tests Fail to Reject H 0 Reject H 0 1% or 5% significance level Fail to Reject H 0 Reject H 0 One tail test - Answers only ONE question - is the test statistic less than or greater than the known distribution Fail to Reject H 0 Reject H 0 Reject H 0 Two tailed test – Only asks if the test statistic is different from the known distribution – H A usually has “not equal to” in the wording 2.5% significance level 2.5% significance level

27. Regulatory Compliance Documentation Sample Size: Minitab

28. The Business Approach 1-Proportion Test Statistical Problem Statistical Problem Business Problem Business Solution Decision Statistical Solution Potential Root Causes Identified Root Causes Verified

32. Compliance Documentation Example – Minitab Results What’s our interpretation? Test and CI for One Proportion Test of p = 0.5 vs p > 0.5 95% Lower Exact Sample X N Sample p Bound P-Value 1 74 130 0.569231 0.493309 0.068

34. The Business Approach 2-Proportion Test Statistical Problem Statistical Problem Business Problem Business Solution Decision Statistical Solution Potential Root Causes Identified Root Causes Verified

35. Analysis of Proportions for Workload Balance Jack Lairdieson, MBB, Vanguard Interpret as an Interval Plot for Multiple Proportions Total Region 5 Region 6 Region 3 Region 1 Region 3 Region 2

36. The Workload Balance (WLB) metrics were being discussed at a regional meeting. The Region 1 representative scoffed at the Region 2 representative that the Region 2’s “In-range” WLB performance metrics were at the “bottom of the barrel”. The Region 2 representative quickly responded, “Really, Region 1 is no better than Region 2.” Once back to the office the concerned Region 1 representative gave the following Workload Balance data to a Black Belt. WLB Stats In-Range Staff Region 1 663 1411 Region 2 141 353 Should Region 1 be concerned about his conclusion? What is the null hypothesis? Workload Balance Example

39. Workload Balance Example – Minitab Results Session Window Output What’s our interpretation? What Hypothesis did we choose to test? Is the sample size a concern? Test and CI for Two Proportions Sample X N Sample p 1 663 1411 0.469880 2 141 353 0.399433 Difference = p (1) - p (2) Estimate for difference: 0.0704461 95% lower bound for difference: 0.0223190 Test for difference = 0 (vs > 0): Z = 2.41 P-Value = 0.008

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