Cairo 02 Stat Inference

STATISTICS AND MAKING CORRECT DECISONS

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

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

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

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 ^

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

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

We Can Be 90% Confident That The Average Weight Is Between 59.6 Kilos And 84.1 Kilos We Can Be 95% Confident That The Average Weight Is Between 56.2 Kilos And 87.5 Kilos We Can Be 99% Confident That The Average Weight Is Between 47.3 Kilos And 96.3 Kilos HOW GOOD IS OUR SAMPLE? 59.6 84.1 87.5 52.6 47.3 96.3 90% 95% 99% Average Weight of All Officers In Kilos

HOW DID WE GET OUR CONFIDENCE INTERVAL? Use Table 17.1 (Page 297 of Implementing Six Sigma ) Depends On Is  Known Or Not Known Sample Variation Sample Size Level Of Desired Confidence

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 

 : 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 (  )

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 

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

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

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

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

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”

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)

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

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 +

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 

Cairo 02 Stat Inference

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Cairo 02 Stat Inference