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Design of Engineering Experiments Part 2 – Basic Statistical Concepts • Simple comparative experiments – The hypothesis testing framework – The two-sample t-test – Checking assumptions, validity • Comparing more than two factor levels…the analysis of variance – – – – ANOVA decomposition of total variability Statistical testing & analysis Checking assumptions, model validity Post-ANOVA testing of means • Sample size determination 1 Portland Cement Formulation (Table 2-1, pp. 24) Observation (sample), j Modified Mortar (Formulation 1) 1 16.85 16.62 2 16.40 16.75 3 17.21 17.37 4 16.35 17.12 5 16.52 16.98 6 17.04 16.87 7 16.96 17.34 8 17.15 17.02 9 16.59 17.08 10 16.57 17.27 y1 j Unmodified Mortar (Formulation 2) y2 j 2 Graphical View of the Data Dot Diagram, Fig. 2-1, pp. 24 3 Box Plots, Fig. 2-3, pp. 26 4 The Hypothesis Testing Framework • Statistical hypothesis testing is a useful framework for many experimental situations • Origins of the methodology date from the early 1900s • We will use a procedure known as the twosample t-test 5 The Hypothesis Testing Framework • Sampling from a normal distribution • Statistical hypotheses: H : 0 1 2 H1 : 1 2 6 Estimation of Parameters 1 n y yi estimates the population mean n i 1 n 1 2 2 S ( yi y ) estimates the variance n 1 i 1 2 7 Summary Statistics (pg. 36) Formulation 1 Formulation 2 “New recipe” “Original recipe” y1 16.76 y2 17.04 S12 0.100 S 22 0.061 S1 0.316 S 2 0.248 n2 10 n2 10 8 How the Two-Sample t-Test Works: Use the sample means to draw inferences about the population means Difference in sample means is y1 y2 16.76 17.04 0.28 The inference is drawn by comparing the difference with standard deviation of the difference in sample means: Difference in sample means ──────────────────────────────── Standard deviation of the difference in sample means This suggests a statistic: Zo y1 y2 12 n1 22 n2 9 How the Two-Sample t-Test Works Use S and S to estimate and 2 1 2 2 2 1 The previous ratio becomes 2 2 y1 y2 2 1 2 2 S S n1 n2 However, we have the case where 2 1 2 2 2 Pool the individual sample variances: (n1 1) S (n2 1) S S n1 n2 2 2 p 2 1 2 2 10 How the Two-Sample t-Test Works: The test statistic is y1 y2 t0 1 1 Sp n1 n2 • t0 is a “distance” measure-how far apart the averages are expressed in standard deviation units • Values of t0 that are near zero are consistent with the null hypothesis • Values of t0 that are very different from zero are consistent with the alternative hypothesis • Notice the interpretation of t0 as a signal-to-noise ratio 11 The Two-Sample (Pooled) t-Test (n1 1) S12 (n2 1) S 22 9(0.100) 9(0.061) S 0.081 n1 n2 1 10 10 2 2 p S p 0.284 y1 y2 16.76 17.04 to 2.20 1 1 1 1 Sp 0.284 n1 n2 10 10 The two sample means are about 2 standard deviations apart. Is this a “large” difference? 12 The Two-Sample (Pooled) t-Test • So far, we haven’t really done any “statistics” • We need an objective basis for deciding how large the test statistic t0 really is • In 1908, W. S. Gosset derived the reference distribution for t0 … called the t distribution • Tables of the t distribution - text, page 606 13 The Two-Sample (Pooled) t-Test • A value of t0 between –2.101 and 2.101 is consistent with equality of means • It is possible for the means to be equal and t0 to exceed either 2.101 or –2.101, but it would be a “rare event” … leads to the conclusion that the means are different • Could also use the P-value approach 14 The Two-Sample (Pooled) t-Test • The P-value is the risk of wrongly rejecting the null hypothesis of equal means (it measures rareness of the event) • The P-value in our problem is P = 0.0411 • The null hypothesis Ho: 1 = 2 would be rejected at any level of significance a ≥ 0.0411. 15 Minitab Two-Sample t-Test Results Two-Sample T-Test and CI: Form 1, Form 2 Two-sample T for Form 1 vs Form 2 N Mean StDev SE Mean Form 1 10 16.764 0.316 0.10 Form 2 10 17.042 0.248 0.078 Difference = mu Form 1 - mu Form 2 Estimate for difference: -0.278000 95% CI for difference: (-0.545073, -0.010927) T-Test of difference = 0 (vs not =): T-Value = -2.19 P-Value = 0.042 DF = 18 Both use Pooled StDev = 0.2843 16 Checking Assumptions – The Normal Probability Plot 17 Importance of the t-Test • Provides an objective framework for simple comparative experiments • Could be used to test all relevant hypotheses in a two-level factorial design, because all of these hypotheses involve the mean response at one “side” of the cube versus the mean response at the opposite “side” of the cube 18 Confidence Intervals (See pg. 43) • Hypothesis testing gives an objective statement concerning the difference in means, but it doesn’t specify “how different” they are • General form of a confidence interval L U where P( L U ) 1 a a • The 100(1- )% confidence interval on the difference in two means: y1 y2 ta / 2,n1 n2 2 S p (1/ n1 ) (1/ n2 ) 1 2 y1 y2 ta / 2,n1 n2 2 S p (1/ n1 ) (1/ n2 ) 19