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Stat 151
Ch.22 -1/5
Comparing Several Means (ANOVA)
(Chapt.22)
Heart-rate (beats/min.) while performing a stressful task in the presence of a
friend, pet and in a control situation.
Control
80.37
87.45
90.02
99.05
75.48
87.23
91.75
87.79
77.80
62.65
84.74
84.88
73.28
84.52
70.88
Friend
99.69
83.40
102.15
80.28
88.02
86.99
92.49
91.35
100.88
101.06
97.05
81.60
89.82
98.20
76.91
Pet
69.17
70.17
75.99
86.45
68.86
64.17
97.54
85.00
65.45
72.26
58.69
79.66
69.23
69.54
70.08
Is there any difference in mean stress level (i.e.,
heart-rate) of the three groups?
Stat 151
Ch.22 -2/5
Recall that we used a two sample t-test to test for a
difference between two population means. That is,
we wished to test:
H 0 : m1 = m 2
We can extend this procedure to test whether any
differences exist between more than two means.
That is, we wish to test:
H 0 : m1 = m 2 = m 3 = L = m k
vs
Ha : not all m1 , m 2 , K, m k are equal
where k is the number of groups (populations)
involved in the study.
This test is called the Analysis of Variance test, or
ANOVA.
Stat 151
Ch.22 -3/5
ANOVA (F-test)
Some definitions we will need:
The Mean Squared Error:
( n1 - 1) s12 + ( n2 - 1) s22 + L + ( nk - 1) sk2
MSE =
N -k
where
N is the total number of observations in the sample
ni is the number of observations in the ith group, and
si is the sample standard deviation of the ith group.
Note:
MSE is the pooled sample standard
deviation, and is an estimate of the standard
deviation s, that is assumed to be the same for all
groups.
The Mean Squared Group:
2
2
2
n1 ( x1 - x ) + n2 ( x2 - x ) + L + nk ( xk - x )
MSG =
k -1
where x is the overall sample mean and and xi is the
sample mean of the ith group.
Stat 151
Ch.22 -4/5
The test statistic we use to test
H 0 : m1 = m 2 = m 3 = L = m k
vs
Ha : not all m1 , m 2 , K, m k are equal
is
MSG
F=
MSE
where under H0, F has a F distribution on k - 1
and N - k degrees of freedom.
Anova: Single Factor
SUMMARY
Groups
Pet
Friend
Control
Count
15
15
15
Sum Average Variance
1237.86
82.52
85.41
1369.88
91.33
69.57
1102.25
73.48
99.40
ANOVA
Source of Variation
SS
Between Groups
2387.69
Within Groups
3561.30
df
2
42
Total
44
5948.99
MS
1193.84
84.79
F
P-value
14.08
0.00
Stat 151
Ch.22 -5/5
ANOVA Assumptions
· The k samples, one from each of the k populations,
are independent.
· Each of the k samples is drawn from a population
that has a Normal distribution.
· All of the k populations have the same standard
deviation, s.
Confidence Intervals from ANOVA Output
We can obtain confidence intervals for the ith
population mean, m i from the usual form using the
pooled sample standard deviation as the estimate of
s. That is, a confidence interval for m i is of the form
xi ± t * SE ( xi )
sp
MSE
=
.
where SE ( xi ) =
ni
ni