1. Chapter Topics
The Completely Randomized Model: One-Factor Analysis of Variance
F-Test for Difference in c Means
The Tukey-Kramer Procedure
ANOVA Assumptions
Kruksal-Wallis Rank Test for Differences in c
Medians

2. One-Factor Analysis of Variance
Evaluate the Difference Among the Means of 2 or More (c) Populations
e.g., Several Types of Tires, Oven Temperature Settings
Assumptions:
Samples are Randomly and Independently Drawn
(This condition must be met.)
Populations are Normally Distributed
(F test is Robust to moderate departures from normality.)
Populations have Equal Variances

3. One-Factor ANOVA Test Hypothesis
H0: m1 = m2 = m3 = ... = mc
All population means are equal
No treatment effect (NO variation in means among groups)
H1: not all the mk are equal
At least ONE population mean is different
(Others may be the same!)
There is treatment effect
Does NOT mean that all the means are different: m1 ¹ m2 ¹ ... ¹ mc

4. One-Factor ANOVA: No Treatment Effect
H0: m1 = m2 = m3 = ... = mc
H1: not all the mk are equal
The Null Hypothesis is True
m1 = m2 = m3

5. One-Factor ANOVA Partitions of Total Variation
Total Variation SST =
Variation Due to Treatment SSA +
Variation Due to Random Sampling SSW
Commonly referred to as:
Sum of Squares Among, or
Sum of Squares Between, or
Sum of Squares Model, or
Among Groups Variation
Commonly referred to as:
Sum of Squares Within, or
Sum of Squares Error, or
Within Groups Variation

6. Among-Group Variation
nj = the number of observations in group j
c = the number of groups _ Xj
the sample mean of group j _ _ X
the overall or grand mean
mi mj
Variation Due to Differences Among Groups.

7. Within-Group Variation
the ith observation in group j
the sample mean of group j
Summing the variation within each group and then adding over all groups.
m j

8. One-Way ANOVA Summary Table
Source of
Degrees
Sum of
Mean
F Test Statistic
Variation of
Squares
Square
Freedom
(Variance)
MSA =
Among
c - 1
SSA
MSA =
MSW
(Factor)
SSA/(c - 1)
Within
n - c
SSW
MSW =
(Error)
SSW/(n - c)
Total
n - 1
SST =
SSA+SSW

9. One-Factor ANOVA F Test Example
Machine1 Machine2 Machine
As production manager, you want to see if 3 filling machines have different mean filling times. You assign 15 similarly trained & experienced workers, 5 per machine, to the machines. At the .05 level, is there a difference in mean filling times?

10. Summary Table Source of Degrees of Sum of Mean Variation Freedom
Squares
Square
(Variance)
MSA
MSW
Among
3 - 1 = 2
F =
= 25.60
(Machines)
Within
= 12
.9211
(Error)
Total
= 14

11. One-Factor ANOVA Example Solution
H0: m1 = m2 = m3
H1: Not All Equal
a = .05
df1= 2 df2 = 12
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
MSA 23 .
5820 F = = = 25 . 6
MSW .
9211
Reject at a = 0.05
a = 0.05
There is evidence that at least one m i differs from the rest. F
3.89

12. The Tukey-Kramer Procedure
Tells Which Population Means
Are Significantly Different
e.g., m1 = m2 ¹ m3
Post Hoc (a posteriori)
Procedure
Done after rejection of equal means in ANOVA
Ability for Pairwise Comparisons:
Compare absolute mean differences with ‘critical range’
f(X) m m m X = 1 2 3
2 groups whose means may be significantly different.

13. The Tukey-Kramer Procedure: Example
1. Compute absolute mean differences:
Machine1 Machine2 Machine
2. Compute Critical Range:
3. Each of the absolute mean difference is greater. There is a significance difference between each pair of means.
= 1.618

14. Kruskal-Wallis Rank Test for c Medians
Extension of Wilcoxon Rank Sum Test Tests the equality of more than 2 (c) population medians
Distribution-free test procedure
Used to analyze completely randomized experimental designs

15. Kruskal-Wallis Rank Test
Assumptions:
Independent random samples are drawn
Continuous dependent variable
Data may be ranked both within and among samples
Populations have same variability
Populations have same shape
Robust with regard to last 2 conditions
Use F Test in completely randomized designs and when the more stringent assumptions hold.