1. Slide Slide 1 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Overview and One-Way ANOVA

2. Slide Slide 2 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Overview  Analysis of variance (ANOVA) is a method for testing the hypothesis that three or more population means are equal.  For example: H 0 : µ 1 = µ 2 = µ 3 =... µ k H 1 : At least one mean is different

3. Slide Slide 3 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. F - distribution

4. Slide Slide 4 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. One-Way ANOVA 1.Understand that a small P -value (such as 0.05 or less) leads to rejection of the null hypothesis of equal means. With a large P -value (such as greater than 0.05), fail to reject the null hypothesis of equal means. 2.Develop an understanding of the underlying rationale by studying the examples in this section. An Approach to Understanding ANOVA

5. Slide Slide 5 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. One-Way ANOVA 3.Become acquainted with the nature of the SS (sum of squares) and MS (mean square) values and their role in determining the F test statistic. An Approach to Understanding ANOVA

6. Slide Slide 6 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. One-Way ANOVA Requirements 1. The populations have approximately normal distributions. 2. The populations have the same variance  2 (or standard deviation  ). 3. The samples are simple random samples. 4. The samples are independent of each other.

7. Slide Slide 7 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Procedure for testing H o : µ1 = µ2 = µ3 =... 1. Use Minitab or Excel. 2. Identify the P -value 3. Form a conclusion based on these criteria: If P -value  , reject the null hypothesis of equal means. If P -value > , fail to reject the null hypothesis of equal means.

8. Slide Slide 8 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Weights of Poplar Trees Do the samples come from populations with different means?

9. Slide Slide 9 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Weights of Poplar Trees For a significance level of  = 0.05, use Minitab or Excel, to test the claim that the four samples come from populations with means that are not all the same. H 0 :  1 =  2 =  3 =  4 H 1 : At least one of the means is different from the others. Do the samples come from populations with different means?

10. Slide Slide 10 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Weights of Poplar Trees Do the samples come from populations with different means? Excel

11. Slide Slide 11 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Weights of Poplar Trees H 0 :  1 =  2 =  3 =  4 H 1 : At least one of the means is different from the others. Do the samples come from populations with different means? The P-value of approximately 0.007. Because the P-value is less than the significance level of  = 0.05, we reject the null hypothesis of equal means. There is sufficient evidence to support the claim that the four population means are not all the same. We conclude that those weights come from populations having means that are not all the same.

12. Slide Slide 12 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. An excessively large F test statistic is evidence against equal population means. F = variance between samples variance within samples Test Statistic for One-Way ANOVA ANOVA Fundamental Concepts

13. Slide Slide 13 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Relationships Between the F Test Statistic and P-Value Figure 12-2

14. Slide Slide 14 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Calculations with Equal Sample Sizes where s p = pooled variance (or the mean of the sample variances) 2  Variance within samples = s p 2  Variance between samples = n where = variance of sample means sxsx 2 sxsx 2

15. Slide Slide 15 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Sample Calculations

16. Slide Slide 16 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Key Components of the ANOVA Method SS(total), or total sum of squares, is a measure of the total variation (around x ) in all the sample data combined. SS(total) =  (x – x) 2

17. Slide Slide 17 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Key Components of the ANOVA Method SS (treatment), also referred to as SS(factor) or SS(between groups) or SS(between samples), is a measure of the variation between the sample means. SS(treatment) = n 1 (x 1 – x) 2 + n 2 (x 2 – x) 2 +... n k (x k – x) 2 =  n i (x i - x) 2

18. Slide Slide 18 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. SS(error), (also referred to as SS(within groups) or SS(within samples), is a sum of squares representing the variability that is assumed to be common to all the populations being considered. SS(error) = (n 1 – 1 )s 1 + (n 2 – 1 )s 2 + (n 3 – 1 )s 3... n k (x k – 1 )s i =  (n i – 1 )s i 2 2 2 2 2 Key Components of the ANOVA Method

19. Slide Slide 19 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. SS(total) = SS(treatment) + SS(error) Key Components of the ANOVA Method Given the previous expressions for SS(total), SS(treatment), and SS(error), the following relationship will always hold.

20. Slide Slide 20 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Mean Squares (MS) MS(treatment) is a mean square for treatment, obtained as follows: MS(treatment) = SS (treatment) k – 1 MS(error) is a mean square for error, obtained as follows: MS(error) = SS (error) N – k

21. Slide Slide 21 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Mean Squares (MS) MS(total) is a mean square for the total variation, obtained as follows: MS(total) = SS(total) N – 1

22. Slide Slide 22 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Test Statistic for ANOVA with Unequal Sample Sizes  Numerator df = k – 1  Denominator df = N – k F = MS (treatment) MS (error)

23. Slide Slide 23 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Weights of Poplar Trees Table 12-3 has a format often used in computer displays.

24. Slide Slide 24 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Two-Way ANOVA

25. Slide Slide 25 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Two-Way Analysis of Variance Two-Way ANOVA involves two factors. The data are partitioned into subcategories called cells.

26. Slide Slide 26 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Poplar Tree Weights

27. Slide Slide 27 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. There is an interaction between two factors if the effect of one of the factors changes for different categories of the other factor. Definition

28. Slide Slide 28 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Poplar Tree Weights Exploring Data Calculate the mean for each cell.

29. Slide Slide 29 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Poplar Tree Weights Minitab