1. Chapter 11 – Analysis of Variance
Introduction to Business Statistics, 6e
Kvanli, Pavur, Keeling
Chapter 11 – Analysis of Variance
Slides prepared by Jeff Heyl, Lincoln University
©2003 South-Western/Thomson Learning™

2. Analysis of Variance
Analysis of Variance (ANOVA) determines if a factor has a significant effect on the variable being measured
Examine variation within samples and variation between samples

3. Measuring Variation
SS(factor) measures between-sample variation [SS(between)]
SS(error) measures within-sample variation [SS(within)]
SS(total) measures the total variation in the sample [SS(factor)] [SS(error)]

4. Determining Sum of Squares
SS(factor) = T2 n
T12 n1
T22 n2
SS(total) = ∑x = ∑x2 -
(∑x)2 n T2
SS(error) = ∑x or
T12 n1
T22 n2
SS(error) = SS(total) - SS(factor)

5. ANOVA Test for Ho: µ1 = µ2 Versus Ha: µ1 ≠ µ2
MS(factor) = =
SS(factor)
df for factor 1
MS(error) = =
SS(error)
n1 + n2 - 2
df for error
F = =
estimated population variance based on
the variation among the sample means
the variation within each of the samples
MS(factor)
MS(error)

6. Defining the Rejection Region
Figure 11.1

7. p-Values for Battery Lifetime Example
t curve with 8 df
4.20 t
p-value = 2 (shaded area)
= .0030
F curve with 1 and 8 df
17.64 F
p-value = shaded area
Figure 11.2

8. Dot Array Diagram • • | 25 30 35 40 45 50 B A Number of cartons | 25
Figure 11.3 | 25 30 35 40 45 50 B • A
Number of cartons
Figure 11.4

9. Assumptions
The replicates are obtained independently and randomly from each of the populations
The replicates from each population follow a (approximate) normal distribution
The normal populations all have a common variance

10. Deriving the Sum of Squares
SS(factor) =
T12 n1
T22 n2
Tk2 nk T2 n
SS(total) = ∑x2 - T2 n
SS(error) = ∑x
T12 n1
T22 n2
Tk2 nk
= SS(total) - SS(factor)

11. The ANOVA Table
Source df SS MS F Factor k - 1 SS(factor) MS(factor) MS(factor) Error n - 2 SS(error) MS(error) MS(error) Total n - 1 SS(total)
SS(factor)
k - 1
MS(factor) =
SS(error)
n - k
MS(error) =
MS(factor)
MS(error)
F =

12. Test for Equal Variances
Ho: 12 = 22 = … = k2
Ha: at least 2 variances are unequal
H =
maximum s2
minimum s2
reject Ho if H > HTable A.14

13. Confidence Intervals in One-Factor ANOVA
Xi - t/2,n-ksp to Xi + t/2,n-ksp 1 ni
where
k = number of populations (levels)
ni = number of replicates in the ith sample
n = total number of observations
sp =
MS(error)

14. Confidence Intervals in One-Factor ANOVA
The (1 - ) • 100% confidence interval for µi - µj is
(Xi - Xj) - t/2,n-ksp
to (Xi - Xj) + t/2,n-ksp 1 ni nj

15. Multiple Comparisons Procedure
The multiple comparisons procedure compares all possible pairs of means in such a way that the probability of making one or more Type 1 errors is 
Tukey’s Test
Q =
maximum (Xi) - minimum (Xi)
MS(error)/nr

16. Multiple Comparisons Procedure
1. Find Q,k,v using Table A.16
2. Determine
D = Q,k,v •
MS(error) nr
3. Place the sample means in order, from smallest to largest
4. If two means differ by more than D, the conclusion is that the corresponding population means are unequal

17. Nylon Breaking Strength
Plot of Group Means 1 2 3 4 5
Group
Group Means 26 25 24 23 22 21
Figure 11.5

18. Nylon Breaking Strength
Figure 11.6

19. Nylon Breaking Strength
Figure 11.7

20. Nylon Breaking Strength
Figure 11.7

21. One-Factor ANOVA Procedure
Requirements
The replicates are obtained independently and randomly from each of the populations
The observations from each population follow (approximately) a normal distribution
The populations all have a common variance

22. One-Factor ANOVA Procedure
Ho: µ1 = µ2 = … = µk
Ha: not all µ’s are equal
Hypotheses
Source df SS MS F
Factor k - 1 SS(factor) MS(factor) Error n - 2 SS(error) MS(error) Total n - 1 SS(total)
MS(factor)
MS(error)
reject Ho if F* > F,k-1,n-k

23. Completely Randomized Design
Replicates are obtained in a completely random manner from each population
Null hypothesis is
Ho: µ1 = µ2 = ... = µn

24. Randomized Block Design
The samples are not independent, the data are grouped (blocked) by another variable
The difference between the randomized block design and the completely randomized design is that here we use a blocking strategy rather than independent samples to obtain a more precise test for examining differences in the factor level means

25. Randomized Block Design
SS(factor) = [T12 + T Tk2] - 1 b T2 bk
where
k = number of factor levels in the design
b = number of blocks in the design
n = number of observations = bk
T1, T2, ..., Tk represent the totals for the k factor levels
S1, S2, ..., Sb are the totals for the b blocks
T = T1 + T Tk
= S1 + S Sb = total of all observations

26. Randomized Block Design
SS(blocks) = [S12 + S Sb2] - 1 k T2 bk
SS(total) = ∑x2 - T2 bk
SS(error) + SS(total) - SS(factor) - SS(blocks)
df for factor = k - 1
df for blocks = b - 1
df for error = (k - 1)(b - 1)
df for total = bk - 1

27. Randomized Block Design
Source df SS MS F
Factor k - 1 SS(factor) MS(factor) F1
Blocks b - 1 SS(blocks) MS(blocks) F2
Error (k - 1)(b - 1) SS(error) MS(error)
Total bk - 1 SS(total)
F1 =
MS(factor)
MS(error)
F2 =
MS(blocks)

28. Factor Hypothesis Test
Ho: µ1 = µ2 = … = µk
Ha: not all µ’s are equal
F1 =
MS(factor)
MS(error)
reject Ho if F* > F,k-1,(k-1)(b-1)

29. Block Hypothesis Test Ho: µ1 = µ2 = … = µb Ha: not all µ’s are equal
F2 =
MS(blocks)
MS(error)
reject Ho if F* > F,b-1,(k-1)(b-1)

30. Hardness Test Data Analysis
Figure 11.10

31. Hardness Test Data Analysis
Figure 11.11

32. Confidence Interval Difference Between Two Means
Randomized Block
(1- ) 100% confidence interval
(Xi - Xj) - t/2,df • s •
to (Xi - Xj) + t/2,df • s • 1 b

33. Dental Claim Data Analysis
Figure 11.12

34. Multiple Comparisons Procedure: Randomized Block
|Xi - Xj| > D
D = Q,k,(k-1)(b-1)
MS(error) b

35. Machine Choice Example
Figure 11.13

36. Two-Way Factorial Design
Single Married
Male Low High
Female High Low
Figure 11.14

37. Two-Way Factorial Designb
1 x x x
2 x x x .
a x x x
Factor B
Factor A
Figure 11.15

38. Two-Way Factorial Design
B Totals
1 T1
2 T2 .
a Ta
S1 S2 Sb
Factor A
Factor B
x, x
(total = R11)
(total = R21)
(total = R12)
(total = Ra1)
(total = Ra2)
(total = R22)
(total = R1b)
(total = Rab)
Totals
Figure 11.16

39. Two-Way Factorial Design
Factor A: SSA = [T12 + T Ta2] - 1 br T2
abr r
Interaction: SSAB = [∑R2] - SSA - SSB -
Factor B: SSB = [S12 + S Sa2] - ar
Total: SS(total) = ∑x2 -

40. Two-Way Factorial Design
SS(error) = SS(total) - SSA - SSB - SSAB
MS(error) =
SS(error)
ab(r - 1)
MSA =
SSA
a - 1
MSB =
SSB
b - 1
MSAB =
SSAB
(a - 1)(b - 1)

41. Two-Way Factorial Design
Source df SS MS F
Factor A a - 1 SSA MSA F1
Factor B b - 1 SSB MSB F2
Interaction (a - 1)(b - 1) SSAB MSAB F3
Error ab(r - 1) SS(error) MS(error)
Total abr - 1 SS(total)

42. Hypothesis Test - Factor A
Ho: Factor A is not significant (µM = µF)
Ha: Factor A is significant (µM ≠ µF)
F1 =
MSA
MS(factor)
reject Ho,A if F1 > F,v1,v2

43. Hypothesis Test - Factor B
Ho: Factor B is not significant (µ1 = µ2 = µ3)
Ha: Factor B is significant (not all µ’s are equal)
F2 =
MSB
MS(error)
reject Ho,B if F2 > F,v1,v2

44. Hypothesis Test - Interaction
Ho: Interaction is not significant
Ha: Interaction is significant
F3 =
MSAB
MS(error)
reject Ho,AB if F2 > F,v1,v2

45. Multiple Comparisons Procedure: Two-Way Factorial Design
D = Q,k,v •
MS(error) nr
v = df for error
nr = number of replicates in each sample

46. Annual amount claimed on dental insurance
Interaction Effect –
300 –
250 –
200 –
150 –
100 – |
Category 1
Category 2
Category 3
Category 4
Male
Female
Employee classification
Annual amount claimed on dental insurance A
Figure 11.17

47. Annual amount claimed on dental insurance
Interaction Effect –
300 –
250 –
200 –
150 –
100 – |
Category 1
Category 2
Category 3
Category 4
Male
Female
Employee classification
Annual amount claimed on dental insurance B
Figure 11.17

48. Gender Factor Analysis
Figure 11.18

49. Gender Factor Analysis
Figure 11.19