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Chapter 2 –Basic Statistical Methods • Describing sample data – – – – – Random samples Sample mean, variance, standard deviation Populations versus samples Population mean, variance, standard deviation Estimating parameters • Simple comparative experiments – The hypothesis testing framework – The two-sample t-test – Checking assumptions, validity Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 1 Portland Cement Formulation (page 24) Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 2 Graphical View of the Data Dot Diagram, Fig. 2.1, pp. 24 Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 3 If you have a large sample, a histogram may be useful Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 4 Box Plots, Fig. 2.3, pp. 26 Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 5 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 Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 6 The Hypothesis Testing Framework • Sampling from a normal distribution • Statistical hypotheses: H : 0 1 2 H1 : 1 2 Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 7 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 Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 8 Summary Statistics (pg. 36) Formulation 1 Formulation 2 “New recipe” “Original recipe” y1 16.76 y2 17.04 S 0.100 S22 0.061 S1 0.316 S2 0.248 n1 10 n2 10 2 1 Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 9 How the Two-Sample t-Test Works: Use the sample means to draw inferences about the population means y1 y2 16.76 17.04 0.28 Difference in sample means Standard deviation of the difference in sample means 2 y 2 n This suggests a statistic: Z0 y1 y2 12 n1 Chapter 2 22 n2 Design & Analysis of Experiments 7E 2009 Montgomery 10 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 Chapter 2 2 1 2 2 Design & Analysis of Experiments 7E 2009 Montgomery 11 How the Two-Sample t-Test Works: The test statistic is y1 y2 t0 1 1 Sp n1 n2 • 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 • t0 is a “distance” measure-how far apart the averages are expressed in standard deviation units • Notice the interpretation of t0 as a signal-to-noise ratio Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 12 The Two-Sample (Pooled) t-Test (n1 1) S12 (n2 1) S22 9(0.100) 9(0.061) S 0.081 n1 n2 2 10 10 2 2 p S p 0.284 t0 y1 y2 16.76 17.04 2.20 1 1 1 1 Sp 0.284 n1 n2 10 10 The two sample means are a little over two standard deviations apart Is this a "large" difference? Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 13 William Sealy Gosset (1876, 1937) Gosset's interest in barley cultivation led him to speculate that design of experiments should aim, not only at improving the average yield, but also at breeding varieties whose yield was insensitive (robust) to variation in soil and climate. Developed the t-test (1908) Gosset was a friend of both Karl Pearson and R.A. Fisher, an achievement, for each had a monumental ego and a loathing for the other. Gosset was a modest man who cut short an admirer with the comment that “Fisher would have discovered it all anyway.” Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 14 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 – see textbook appendix Chapter 2 t0 = -2.20 Design & Analysis of Experiments 7E 2009 Montgomery 15 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 Chapter 2 t0 = -2.20 Design & Analysis of Experiments 7E 2009 Montgomery 16 The Two-Sample (Pooled) t-Test t0 = -2.20 • • • • The P-value is the area (probability) in the tails of the t-distribution beyond -2.20 + the probability beyond +2.20 (it’s a two-sided test) The P-value is a measure of how unusual the value of the test statistic is given that the null hypothesis is true The P-value 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 = 0.042 Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 17 Computer Two-Sample t-Test Results Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 18 Checking Assumptions – The Normal Probability Plot Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 19 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 Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 20 Confidence Intervals (See pg. 44) • 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 • The 100(1- α)% confidence interval on the difference in two means: y1 y2 t / 2,n1 n2 2 S p (1/ n1 ) (1/ n2 ) 1 2 y1 y2 t / 2,n1 n2 2 S p (1/ n1 ) (1/ n2 ) Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 21 Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 22 Other Chapter Topics • Hypothesis testing when the variances are known • One sample inference • Hypothesis tests on variances • Paired experiments Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery 23