Designing Engineering Experiments. Randomized Single-Factor Experiments

13-1: Designing Engineering Experiments Every experiment involves a sequence of activities: 1. Conjecture – the original hypothesis that motivates the experiment. 2. Experiment – the test performed to investigate the conjecture. 3. Analysis – the statistical analysis of the data from the experiment. 4. Conclusion – what has been learned about the original conjecture from the experiment. Often the experiment will lead to a revised conjecture, and a new experiment, and so forth. 3 13-2: The Completely Randomized Single-Factor Experiment 13-2.1 An Example 4 13-2: The Completely Randomized Single-Factor Experiment 13-2.1 An Example 5 13-2: The Completely Randomized Single-Factor Experiment 13-2.1 An Example • The levels of the factor are sometimes called treatments. • Each treatment has six observations or replicates. • The runs are run in random order. 6 13-2: The Completely Randomized Single-Factor Experiment 13-2.1 An Example Figure 13-1 (a) Box plots of hardwood concentration data. (b) Display of the model in Equation 13-1 for the completely randomized single-factor experiment 7 13-2: The Completely Randomized Single-Factor Experiment 13-2.2 The Analysis of Variance Suppose there are a different levels of a single factor that we wish to compare. The levels are sometimes called treatments. 8 13-2: The Completely Randomized Single-Factor Experiment 13-2.2 The Analysis of Variance We may describe the observations in Table 13-2 by the linear statistical model: The model could be written as 9 13-2: The Completely Randomized Single-Factor Experiment 13-2.2 The Analysis of Variance Fixed-effects Model The treatment effects are usually defined as deviations from the overall mean so that: Also, 10 13-2: The Completely Randomized Single-Factor Experiment 13-2.2 The Analysis of Variance We wish to test the hypotheses: The analysis of variance partitions the total variability into two parts. 11 13-2: The Completely Randomized Single-Factor Experiment 13-2.2 The Analysis of Variance Definition 12 13-2: The Completely Randomized Single-Factor Experiment 13-2.2 The Analysis of Variance The ratio MSTreatments = SSTreatments/(a – 1) is called the mean square for treatments. 13 13-2: The Completely Randomized Single-Factor Experiment 13-2.2 The Analysis of Variance The appropriate test statistic is We would reject H0 if f0 > fα,a-1,a(n-1) 14 13-2: The Completely Randomized Single-Factor Experiment 13-2.2 The Analysis of Variance Definition 15 13-2: The Completely Randomized Single-Factor Experiment 13-2.2 The Analysis of Variance Analysis of Variance Table 16 13-2: The Completely Randomized Single-Factor Experiment Example 13-1 17 13-2: The Completely Randomized Single-Factor Experiment Example 13-1 18 13-2: The Completely Randomized Single-Factor Experiment Example 13-1 19 20 13-2: The Completely Randomized Single-Factor Experiment Definition For 20% hardwood, the resulting confidence interval on the mean is 21 13-2: The Completely Randomized Single-Factor Experiment Definition For the hardwood concentration example, 22 13-2: The Completely Randomized Single-Factor Experiment An Unbalanced Experiment 23 13-2: The Completely Randomized Single-Factor Experiment 13-2.3 Multiple Comparisons Following the ANOVA The least significant difference (LSD) is If the sample sizes are different in each treatment: 24 13-2: The Completely Randomized Single-Factor Experiment Example 13-2 25 13-2: The Completely Randomized Single-Factor Experiment Example 13-2 26 13-2: The Completely Randomized Single-Factor Experiment Example 13-2 Figure 13-2 Results of Fisher’s LSD method in Example 13-2 27 13-2: The Completely Randomized Single-Factor Experiment 13-2.5 Residual Analysis and Model Checking 28 13-2: The Completely Randomized Single-Factor Experiment 13-2.5 Residual Analysis and Model Checking Figure 13-4 Normal probability plot of residuals from the hardwood concentration experiment. 29 13-2: The Completely Randomized Single-Factor Experiment 13-2.5 Residual Analysis and Model Checking Figure 13-5 Plot of residuals versus factor levels (hardwood concentration). 30 13-2: The Completely Randomized Single-Factor Experiment 13-2.5 Residual Analysis and Model Checking Figure 13-6 Plot of residuals versus yi 31