Chapter 10 page 1 One Factor & Randomized Block Experiments ANOVA 1: Completely Randomized Design Suppose that we want to compare the fuel consumption recorded for three different makes of cars over a specified distance. Let’s assume that 18 drivers are chosen: six are randomly assigned to A-cars, six to B-cars, and six to C-cars. H0: The mean fuel consumption of the three makes of cars is the same. H1: At least two car makes have different mean fuel consumptions. Assume that the sample means resulting from our test are as follows: Sample mean for A-cars = 20 mpg Sample means for B-cars = 21 mpg Sample means for C-cars = 22 mpg Are the cars different, with respect to mpg? Test: Chapter 10

Chapter 10 page 3 Let’s investigate setting up a test where our hypotheses are the following: : H0: μ1= μ2= … = μcH1: At least two means are different Notation: c = number of groups or treatments nj= total number of replicates of treatment j (j = 1,..., c), n = total number of replicates for all treatments Assumptions: 1. c independent and random samples. 2. Each population has a Normal probability distribution. 3. The c population variances are equal. Chapter 10 page 4 or among) Consider the breakdown of the variability within our samples: How do we quantify this variability? Terms:

= Sum of Squares for Treatments (variability between or among) Consider the breakdown of the variability within our samples: How do we quantify this variability?

)= Sum of Squares for Error (variability within)

Total Variation = Treatment Variation + Error Variation Total Variation = Between Variation + Within Variation Total Variation = Explained Variation + Unexplained Variation

Chapter 10 page 5 If each of these sums of squares is divided by its associated degrees of freedom, then we have the respective mean squares. That is: 1SSTMST−=n1SSAMSA −=ccn−==SSWMSE)(MSW where: degrees of freedom associated with SST = n – 1 degrees of freedom associated with SSA = c – 1 degrees of freedom associated with SSW = n - cMean Squares are variances. If there are no real differences among the cgroup means, then the three mean squares (above) provide estimates of σ2, the overall measure of the variability inherent in the data. That is, if there are no real differences among the cgroup means: MSW ≈MSA ≈MST ≈σ2Chapter 10 page 6