Design of Engineering Experiments Chapter 2 Basic Statistical Concepts

Transcription

1 Desig of Egieerig Experimets Chapter 2 Basic tatistical Cocepts imple comparative experimets The hpothesis testig framework The two-sample t-test Checkig assumptios, validit Motgomer_Chap_2

2 Portlad Cemet Formulatio (Table 2-, pp. 22) Observatio (sample), j Modified Mortar (Formulatio ) j Umodified Mortar (Formulatio 2) 2 j Motgomer_Chap_2 2

3 Graphical View of the Data Dot Diagram, Fig. 2-, pp. 22 Dotplots of Form ad Form 2 (meas are idicated b lies) Form Form 2 Motgomer_Chap_2 3

4 Box Plots, Fig. 2-3, pp. 24 Boxplots of Form ad Form 2 (meas are idicated b solid circles) Form Form 2 Motgomer_Chap_2 4

5 The Hpothesis Testig Framework tatistical hpothesis testig is a useful framework for ma experimetal situatios Origis of the methodolog date from the earl 900s We will use a procedure kow as the twosample t-test Motgomer_Chap_2 5

6 The Hpothesis Testig Framework amplig from a ormal distributio tatistical hpotheses: H H : 0 : µ = µ µ µ Motgomer_Chap_2 6

7 Estimatio of Parameters = i= i estimates the populatio mea µ = ( i ) estimates the variace σ i= Motgomer_Chap_2 7

8 ummar tatistics (pg. 35) Formulatio New recipe Formulatio 2 Origial recipe 2 = 6.76 = 0.00 = = = 7.92 = 0.06 = = 0 Motgomer_Chap_2 8

9 How the Two-ample t-test Works: Use the sample meas to draw ifereces about the populatio meas = =.6 Differece i sample meas tadard deviatio of the differece i sample meas 2 2 σ σ = This suggests a statistic: Z 0 = σ σ Motgomer_Chap_2 9

10 How the Two-ample t-test Works: Use ad to estimate σ ad The previous ratio becomes However, we have the case where p = Pool the idividual sample variaces: ( ) + ( ) Motgomer_Chap_2 0 σ σ = σ = σ

11 How the Two-ample t-test Works: The test statistic is t 0 = p + Values of t 0 that are ear zero are cosistet with the ull hpothesis Values of t 0 that are ver differet from zero are cosistet with the alterative hpothesis t 0 is a distace measure-how far apart the averages are expressed i stadard deviatio uits Notice the iterpretatio of t 0 as a sigal-to-oise ratio Motgomer_Chap_2

12 The Two-ample (Pooled) t-test ( ) + ( ) 9(0.00) + 9(0.06) = = = p 2 p = t 0 = = = p The two sample meas are about 9 stadard deviatios apart Is this a "large" differece? Motgomer_Chap_2 2

13 The Two-ample (Pooled) t-test o far, we have t reall doe a statistics We eed a objective basis for decidig how large the test statistic t 0 reall is I 908, W.. Gosset derived the referece distributio for t 0 called the t distributio Tables of the t distributio - text, page 640 Motgomer_Chap_2 3

14 The Two-ample (Pooled) t-test A value of t 0 betwee 2.0 ad 2.0 is cosistet with equalit of meas It is possible for the meas to be equal ad t 0 to exceed either 2.0 or 2.0, but it would be a rare evet leads to the coclusio that the meas are differet Could also use the P-value approach Motgomer_Chap_2 4

15 The Two-ample (Pooled) t-test The P-value is the risk of wrogl rejectig the ull hpothesis of equal meas (it measures rareess of the evet) The P-value i our problem is P = 3.68E-8 Motgomer_Chap_2 5

16 Two-ample t-test Results Two-ample T-Test ad CI: Form, Form 2 Two-sample T for Form vs Form 2 N Mea tdev E Mea Form =.36/3.6 Form = 0.248/3.6 Differece = mu Form - mu Form 2 Estimate for differece: % CI for differece: (-.425, -0.89) T-Test of differece = 0 (vs ot =): T-Value = -9. P-Value = DF = 8 Both use Pooled tdev = Motgomer_Chap_2 6

17 Checkig Assumptios The Normal Probabilit Plot Tesio Bod tregth Data ML Estimates Percet Form Form 2 Goodess of Fit AD* Data Motgomer_Chap_2 7

18 Importace of the t-test Just keep i mid this is for comparig two samples comig from ormal distributios!! Provides a objective framework for simple comparative experimets Could be used to test all relevat hpotheses i a two-level factorial desig, because all of these hpotheses ivolve the mea respose at oe side of the cube versus the mea respose at the opposite side of the cube Motgomer_Chap_2 8

19 Cofidece Itervals (ee pg. 42) Hpothesis testig gives a objective statemet cocerig the differece i meas, but it does t specif how differet the are Geeral form of a cofidece iterval L θ U where P( L θ U) = α The 00(- α )% cofidece iterval o the differece i two meas: t (/ ) + (/ ) µ µ α /2, + 2 p + t (/ ) + (/ ) α /2, + 2 p Motgomer_Chap_2 9