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Statistics 400 - Lecture 16
 Last Day: Two-Sample T-test (10.2 and 10.3)
 Today: Comparison of Several Treatments (14.1-14.3)
 Last day, we looked at comparing means for two treatments
 When more than two treatments are being compared, we will use a
statistical technique call the Analysis of Variance (ANOVA)
 The same underlying assumptions apply in the ANOVA situation a
the two independent samples case
Example (Pulp Mill)
 An important measure of performance at pulp mills is based on pulp
brightness measured by a reflectance meter
 An investigation was performed (Sheldon, 1960; Industrial and
Engineering Chemistry ) to investigate if there is a difference in
product quality for different mill operators
 Want to see if there are differences in the reflectance for different
operators
 Data:
A
59.8
60.0
60.8
60.8
59.8
Operator
B
C
59.8 60.7
60.2 60.7
60.4 60.5
59.9 60.9
60.0 60.3
D
61.0
60.8
60.6
60.5
60.5
ANOVA Situation
 Situation:
 Have k independent random samples
 Each sample comes from a normal population
 The population standard deviations are equal
 Want to test test a hypothesis about the equality of the population
means
Structure of Data
 Have k independent random samples from k populations…sample
size from each pop. may be different
 Denote jth observation from the ith population as yij
Population 1
y11
y12
.
.
.
y1n1
Population 2
y21
y12
.
.
.
y2n2
…
Population k
yk1
yk2
.
.
.
yk,nk
Estimating

2
 Have assumed that data from each population comes from
independent normal distributions with equal standard deviations
(variances)
 That is, yij has a N (i , ) distribution
 If we wanted to estimate  based on the data from only 1
population, we would use
2
 Combining the data from all of the populations:
Another estimate for

2
 Why are we doing this?
 What is the null hypothesis we have in mind?
 Suppose H0 is true, how could we estimate the mean?
 Variance about true mean:
 When the null hypothesis is true, we expect
 When it is false
 Potential Test Statistic
More Formal Approach
 Model for comparing k treatments:
Yij
  i  eij
for i =1, 2, …, k and j =1, 2, …, ni
 where i is the ith population mean and
 eij had a N (0, ) distribution
 Want to test:
H 0 : 1   2  ...   k
 Sum of Squares for treatment
 Sum of Squares for Error (residual)
 Total Sum of Squares
 Degrees of freedom
 Mean Squares
 Test Statistic
 Hypotheses
 P-value
 ANOVA Table:
Example (Pulp Mill):
 Data:
A
59.8
60.0
60.8
60.8
59.8
Operator
B
C
59.8 60.7
60.2 60.7
60.4 60.5
59.9 60.9
60.0 60.3
D
61.0
60.8
60.6
60.5
60.5
 Summary Statistics:
OPERATOR 1.00
2.00
3.00
4.00
Y
Y
Y
Y
Count
5
5
5
5
Mean Std Deviation
60.26
.50
60.06
.24
60.62
.23
60.68
.22
Plot of Responses By Operator
61.2
61.0
60.8
60.6
60.4
60.2
60.0
Y
59.8
59.6
0.0
1.0
OPERATOR
2.0
3.0
4.0
5.0
ANOVA Table
O
Y
m
e
d
u
F
i
a
g
f
B
9
3
6
1
1
W
0
6
1
T
9
9