1. 1. Complete SRTE on ANGEL 2. Project Due Wed midnight on ANGEL 3. Complete post-test Dec 6, 12pm – Dec 8, 12pm Up to 2% extra credit on your overall grade

2. Today: What if we have more than two samples? Intro to ANOVA and Simple Linear Regression

3. Review example: A team of engineers responsible for the study decides to investigate two levels of hardwood concentration: "low" and "high" What does the engineer interested in: a) Mean strength of product with low concentration vs high concentration b) Variance of strength of product with low concentration vs high concentration A manufacturer of paper used for making grocery bags is interested in improving the tensile strength of the product. Product engineer thinks that tensile strength is dependent on the hardwood concentration in the pulp.

4. A team of engineers responsible for the study decides to investigate four levels of hardwood concentration: 5%, 10%, 15% and 20%. What does the engineer interested in: a) Mean strength of product with 5%, 10%, 15% or 20% concentration b) Mean strength of product with 5 % vs 20% concentration Product engineer thinks that the range of hardwood concentrations of practical interest is between 5% and 20%. What if we have more than two samples

5. Product engineer thinks that the range of hardwood concentrations of practical interest is between 5% and 20%. What if we have more than two samples

6. Terminologies: Factor is an explanatory variable manipulated by the experimenter. The levels of the factor are sometimes called treatments. Variables of interest in an experiment (those that are measured or observed) are called response. The experiment runs are in random order Analysis of Variance

7. Product engineer produce 6 products at each concentration level of 5, 10,15 and 20% and recorded the strength of each products.

8. Effect of any nuisance variable that may influence the observed tensile strength is approximately balanced out. i.e. warm up effect on the machine. changes in experiment environment. Why Randomized Experiment

9. Product engineer produce 6 products at each concentration level of 5, 10,15 and 20% and recorded the strength of each products. What is the response? a) Strength b) Concentration

10. Product engineer produce 6 products at each concentration level of 5, 10,15 and 20% and recorded the strength of each products. What is the factor? a) Strength b) Concentration Each level of concentration is called treatments. Each treatment has 6 replicates

11. Analysis of Variance

12. One-way ANOVA

17. Example

18. Example Con’t

19. Introduction To Empirical Models

22. Based on the scatter diagram, it is probably reasonable to assume that the mean of the random variable Y is related to x by the following straight-line relationship: where the slope and intercept of the line are called regression coefficients. The simple linear regression model is given by where  is the random error term. Introduction To Empirical Models

23. We think of the regression model as an empirical model. Suppose that the mean and variance of  are 0 and  2, respectively, then The variance of Y given x is Introduction To Empirical Models

24. The true regression model is a line of mean values: where  1 can be interpreted as the change in the mean of Y for a unit change in x. Also, the variability of Y at a particular value of x is determined by the error variance,  2. This implies there is a distribution of Y-values at each x and that the variance of this distribution is the same at each x. Introduction To Empirical Models