Experimental Designs Examples

Latin Square Problem
Problem:
A manufacturing firm wants to investigate the effects of 5 color additives on the setting time of a new concrete mix. Variations in the setting time can be expected from day-to-day changes in temperature and humidity and also from different workers who prepare the test molds. To eliminate this extraneous sources of variation, a 5x5 Latin square design was used in which the Letters A-E represent 5 additives. The setting times, in hours, for the 25 molds are shown in the table:
Table here:
WORKER
1
2
3
4
5
1
D 10.7
E 10.3
B 11.2
A 10.9
C 10.5
2
E 11.0
C 10.5
D 12.0
B 11.5
A 10.3
3
A 11.8
B 10.9
C 10.5
D 11.3
E 7.5
4
B 14.1
A 11.6
E 11.0
C 11.7
D 11.5
5
C 14.5
D 11.5
A 11.5
E 12.7
B 10.0

Step 1: State the hypothesis
Ho : μ..1 = μ..2 = μ..ρ
Ha: at least two means are not equal
Ho: τk = 0 for all k
Ha: τk ≠ 0 for at least one k

Step 2: Compute the necessary values
Rows and Colums
WORKER
DAYS
Row total

Row mean

1
2
3
4
5

1
10.7 (D)
10.3 (E)
11.2 (B)
10.9 (A)
10.5 (C)
53.6
10.72
2
11 (E)
10.5 (C)
12 (D)
11.5 (B)
10.3 (A)
55.3
11.06
3
11.8 (A)
10.9 (B)
10.5 (C)
11.3 (C)
7.5 (E)
52
10.4
4
14.1 (B)
11.6 (A)
11 (E)
11.7 (D)
11.5 (D)
59.9
11.98
5
14.5 (C)
11.5 (D)
11.5 (A)
12.7 (E)
10 (B)
60.2
12.04
Column total

62.1
54.8
56.2
58.1
49.8

Column mean

12.42
10.96
11.24
11.62
9.96

Treatments
ADDITIVES(Treatments)
A
B
C
D
E
Totals
56.1
57.7
57.7
57
52.5
Means
11.22
11.54
11.54
11.4
10.5

Step 3: Compute sum of squares
Compute Correction Factor

Compute Total Sum of Squares

Compute Sum of Squares Rows

Compute Sum of Squares Columns

Compute Sum of Squares Treatments

Compute Sum of Squares Error

Step 4: Fill the ANOVA Table
Sources of Variation df Sum of squares
Mean square
Fc
F

0.05
0.01