TWO ANOVAs
1) An experiment was performed to determine the effect of four different chemicals on the
strength of a fabric. These chemicals are used as part of the permanent press finishing process.
Five fabric samples were selected, and a randomized complete block design was run by testing
each chemical type once in random order on each fabric. The data are shown below.
Test for differences in means using ANOVA with α=0.01.
Fabric Sample 1 2 3 4 5 Chemical Type 1 1.3 1.6 0.5 1.2 1.1 2 2.2 2.4 0.4 2.0 1.8 3 1.8 1.7 0.6
1.5 1.3 4 3.9 4.4 2.0 4.1 3.4
DATA
Fabric
Sample
Chemical
Type
1
1.3
2.2
1.8
3.9
1
2
3
4
2
1.6
2.4
1.7
4.4
3
0.5
0.4
0.6
2
4
1.2
2
1.5
4.1
5
1.1
1.8
1.3
3.4
EXCEL OUTPUT
Anova: Two-Factor Without Replication
SUMMARY
Row 1
Row 2
Row 3
Row 4
Count Sum
5 5.7
5 8.8
5 6.9
5 17.8
Average
1.14
1.76
1.38
3.56
Variance
0.163
0.628
0.227
0.893
4 9.2
4 10.1
4 3.5
4 8.8
4 7.6
2.3
2.525
0.875
2.2
1.9
1.273333
1.689167
0.569167
1.713333
1.086667
Column 1
Column 2
Column 3
Column 4
Column 5
ANOVA
Source of
Variation
SS
df
MS
F
P-value
F crit
Rows
Columns
Error
Total
18.044
3 6.014667 75.89485
6.693
0.951
4
12
25.688
19
1.67325 21.11356
0.07925
4.52E08 3.490295
2.32E05 3.259167
Interpretations
The rows represent chemical types. The F statistic for testing differences in rows(chemical types)
is highly significant. (p-value is 4.52E-08). So the means of chemical types are not all equal.
The columns represent fabric samples. The F statistic for testing differences in columns(fabric
samples) is highly significant. (p-value is 3.25E-05). So the means of chemical types are not all
equal.
2) An article in the Journal of Agricultural Engineering Research (Vol. 52, 1992, pp. 53-76)
described an experiment to investigate the effect of drying temperature of wheat grain on the
baking quality of bread. Three temperature levels were used, and the response variable measured
was the volume of the loaf of bread produced. The data are show below. Test for the differences
in means using ANOVA with α=0.01. Temperature (C°) Volume (CC) 70 1245 1235 1285 1245
1235 75 1235 1240 1200 1220 1210 80 1225 1200 1170 1155 1095 USE EXCEL or TI 83/84 to
theses questions.
DATA
TEMPERATURE
VOLUME
70
1245
1235
1285
1245
1235
75
1235
1240
1200
1220
1210
80
1225
1200
1170
1155
1095
EXCEL OUTPUT
SUMMARY
Groups
Column 1
Column 2
Column 3
ANOVA
Count
5
5
5
Sum
Average Variance
6245
1249
430
6105
1221
280
5845
1169
2442.5
Source of
Variation
Between Groups
Within Groups
SS
16480
12610
Total
29090
df
MS
F
P-value
F crit
2
8240 7.841396 0.006635 3.885294
12 1050.833
14
The p-value of the F test is 0.006635 < 0.05. The F is significant at 0.05 level.
Conclusion: There is significant differences between the mean volumes of the different temperature
levels.