252anovaex3.doc 3/22/02 (Open this document in 'page layout' view.) Roger Even Bove
Example of 2-way ANOVA with one measurement per cell
The data below represents samples of consumer ratings of three different displays.
a. Assuming that each column represents a random sample from a normal distribution and that variances of
the parent populations are similar, compare the means of the three populations. Column sums are now
given. Each of the three variables represents a sample of ratings of displays.
 x1  225,  x12  10475,  x2  175,  x22  6375,  x3  180,  x32  6850 .
b. Do a confidence interval for the difference between mean ratings of the first and third displays,
assuming that the interval is one of three possible contrasts.
c. Assume that each row represents the opinions of a single individual and find if there are significant
differences between display means and means for individuals.
Solution: New material is added in boldface.
Individual
Display
Display
Display
1
2
3
1
50
45
45
2
45
30
35
3
30
25
20
4
45
35
40
5
55
40
40
Sum
225
175
180
5
5
5
nj
45
x j 
SS
x j 
a)
35
10475
2025
2
6375
1225
36
6850
1296
One-way ANOVA
SST 
 x
2
 nx
2
j .j
6550
4150
1925
4850
6225
23700
2
 xijk
2177.7778
1344.4444
625.0000
1600.0000
2025.0000
7772.2222
x
23700
4546
2
 xijk
 x i 2
 x .2j .
x
2
.j
 nx 2
DF
2
12
14
MS
151.647
80.833
F
1.876
difference between display means.’
b) From the outline, the Scheffe interval is
m  1Fm 1,n  m  s
 45  36   24.75  80 .8333
x
2
i.
 nx 2
 37772 .2222   1538 .6667 2
2,12  4.75 we accept
Since F.05
H 0 which is ‘no
 9  15 .95
46.6667
36.6667
25.0000
40.0000
45.0000
(38.6667)
x
(38.6667)
SSR  C
 303 .2947
SSW  SST  SSB
1  3  x1  x3  
3
3
3
3
3
15
n
 303 .2947
 nx 2
SS
303.2947
970
1293.2947
xi 2
 54546   15 38 .6667 2
 54546   15 38 .6667 2
Source
Between
Within
Total
140
110
75
120
135
580
15
SS
x i 
SSC  R
 1273 .2947
n x
ni
c) Two-way ANOVA
SST  Same as 1-way ANOVA
2
 23700  15 38 .6667 2
SSB 
Sum
1 1
6
6
1
1

n1 n2
 889 .9613
SSW  SST  SSR  SSC
Source
Rows
Columns
Within
Total
SS
889.9613
303.2947
80.0387
1293.2947
DF
4
2
8
14
MS
222.4903
151.6474
10.0048
F
22.24
15.16
4,8  3.84 we reject
Since F.05
H 01 which is ‘no
difference between individual means.’ Since
2,8  4.46
F.05
we reject H 02 which is ‘no
difference between display mea