This document discusses various statistical tools used in decision making, including regression analysis, confidence intervals, comparison tests, and analysis of variance. It provides examples of how regression analysis can be used to determine correlations and unknown parameters. It also explains how confidence intervals are calculated and used to determine how reliable a sample statistic is in estimating an unknown population parameter. Comparison tests are outlined as a method to determine if one process or supplier is better than another.

1. STATISTICAL INFERENCE

2. STATISTICS AND MAKING CORRECT DECISONS

3. STATISTICAL TOOLS USED TO ASSIST DECISION MAKING Regression Analysis Determining Confidence Interval Comparison Tests Analysis Of Variance Design Of Experiments Linear & Non-linear Programming Queuing Theory

4. Regression Analysis

5. REGRESSION ANALYSIS No Correlation (R = 0) Strong Positive Correlation ( R = .995) Positive Linear correlation (r=0.85) Negative Linear Correlation (r=-0.85)

6. TYPES OF REGRESSION ANALYSIS (AMONG MANY) Exponential Y =AB X Geometric Y = AX B Logarithmic Y = A o + A 1 (logX) + A 2 (logX) 2 Linear Y = A o + A 1 X Linear Regression Is the Most Common

7. THE PURPOSE OF REGRESSION ANALYSIS Correlation r =1 = perfect correlation r = 0 = no correlation Determination Of Unknown Parameters r =  (X i -X) (Y i - Y)  (X i -X) 2 (Y i - Y) 2  1 =  Y i (X i -X) n i =1  (X i -X) 2 n i =1 Y =  0 +  1 X ^ ^ ^  0 = Y - ^  1 X ^

8. Confidence Interval

9. Statistics Usually Do Not Represent Absolute Truth Very Often They Are A Good Guess How Good Of A Guess Is Explained By The Confidence Interval Understanding Confidence Intervals Will Allow You To Better Evaluate Critical Statistics WHY DO WE NEED CONFIDENCE INTERVALS

10. Problem: Commanding General Needs To Know Average Weight of Officers On The Base 1,000 Officers At The Base Six Officers Selected And Weighed Officer # 1: 68 kilos Officer # 2: 57 kilos Officer # 3: 72 kilos Officer # 4: 71 kilos Officer # 5: 100 kilos Officer # 6: 63 kilos Average Weight of Our Sample Is 71.8333 Kilos How Good Is This Statistic? A TYPICAL SITUATION

13. OR OR Single Sided Double-Sided  Known  Unknown CONFIDENCE INTERVAL EQUATIONS   X -  U  n    X +  U  n  X -  U  n     X +  U  n    X - S t  n    X + S t  n  X - S t  n     X + S t  n 

14.  : Population Average  : Population Standard Deviation  : 1 - The Desired Confidence Level S : The Sample Standard Deviation v : Degrees Of Freedom Or n - 1 t : Data Derived From t Distribution U: Data Derived From Normal Distribution n: The Number of Units in The Sample EQUATION HELP Single Sided : Trying to Determine If the Population Average (  ) Is Less Than or Greater Than the Sample Average ( X ) Double Sided : Trying To Determine The Upper& Lower Boundaries of the Population Average (  )

15. SOLUTION TO THE OFFICER WEIGHT PROBLEM X - S t  n     X + S t  n  71.833 - 2.015 (14.878) 6     71.833 + 2.015 (14.878) 6 

16. STANDARD DEVIATION CONFIDENCE INTERVAL Almost The Same But Different See Page 300 Of Implementing Six Sigma We Will Use The Chi Square Distribution (  2 )  2  /2; v  2 (1-  /2; v) ( n - 1) s 2 ( n - 1) s 2    [ ] 1/2 [ ] 1/2

17. 9.999    31.0227 EXAMPLE USING SIX OFFICER WEIGHTS We Are 90% Confident That Standard Deviation Of All Officer Weights Is Between 10 Kilos & 31.0 Kilos  2  /2; v  2 (1-  /2; v) ( n - 1) s 2 ( n - 1) s 2    [ ] 1/2 [ ] 1/2 11.07 1.15 (5) (14.878) 2 (5) (14.878) 2    [ ] 1/2 [ ] 1/2 (99.9796) 1/2    (962.4125) 1/2

18. Comparison Tests

19. Is Process B Better Than Process A? Is Supplier B Better Than Supplier A? These Questions Are Always Being Asked COMPARISON TESTS Comparison Tests Can Give The Right Answers

20. STEPS INVOLVED IN COMPARISON TESTING Define Precisely The Problem Objective Formulate A Null Hypothesis Evaluation By A One Or Two Tail Test Choose A Critical Value Of A Test Statistic Calculate A Test Statistic Make Inference About The Population Communicate The Findings

21. TYPICAL DECISIONS 1. A chemical batch process has yielded average of 802 tons of product for a long period. Production records for last five batches show following results: 803, 786, 806, 791, and 794. Can we predict with 95% confidence that the process is now at a lower average? 2. The average vial height from an injection molding process has been 5.00 inches with a standard deviation of .12”. A vendor claims to have a new material that will reduce the height variation. An experiment, conducted using the new material, yielded the following results: 5.10, 4.90, 4.92, 4.87, 5.09, 4.89, 4.95, 4.88. The average height of the eight vials is 4.95” and the standard deviation is .093”

22. Average vial height from an injection molding process has been 5.00 inches with a standard deviation of .12”. Vendor claims to have a new material that will reduce height variation. An experiment, conducted using new material yielded the following results: 5.10, 4.90, 4.92, 4.87, 5.09, 4.89, 4.95, 4.88. Average height of the eight vials is 4.95” and standard deviation is .093” Is the new material producing shorter vials with the existing molding machine set-up (with 95% confidence)? Is height variation actually less with the new material (with 95% confidence)

23. NULL HYPOTHESIS The Hypothesis To Be Tested A Null Hypothesis Can Only Be Rejected. It Cannot Be Accepted Because of a Lack of Evidence to Reject It Example: If A Claim Is That Process B Is Better Than Process A The Null Hypothesis Is That Process A = Process B H o : A = B

24. Table 19.1(page 322 Implementing Six Sigma ) Most Likely:  1 2   2 2 And  Is Unknown 1. Calculate t 0 = 2. Calculate  = COMPARISON METHODOLOGY  X 1 - X 2  S 1 2 n 1 n 2 S 2 2 +  S 1 2 n 1 ( ) S 2 2 n 2 ( ) + [ ] 2 (S 1 2 / n 1 ) 2 (S 2 2 / n 2 ) 2 n 1 + 1 n 2 + 1 +

25. COMPARISON METHODOLOGY (Continued) 3. Look Up Value Of t  Using Table E Or Table D ( Implementing Six Sigma Pages 697 Or 698) 4. Reject The Null Hypothesis If t 0 Is Greater Than Than t 