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Used
to evaluate means from two or more subgroups. A
statistically significant ANOVA indicates there is more variation between subgroups than
would be expected by chance. It does not
identify which subgroup pairs are significantly different from each other. Used to evaluate multiple means of one independent
variable to avoid conducting multiple t-tests.
Problem:
You obtained the number of years of education from one random sample of 38 police officers
from City A, the number of years of education from a second random sample of 30 police
officers from City B, and the number of years of education from a third random sample of
45 police officers from City C. The average years of education for the sample from City A
is 15 years with a standard deviation of 2 years. The average years of education for the
sample from City B is 14 years with a standard deviation of 2.5 years. The average years
of education for the sample from City C is 16 years with a standard deviation of 1.2
years. Is there a statistically significant difference between the education levels of
police officers in City A, City B, and City C?
|
City
A |
City
B |
City
C |
Mean
(years) |
15 |
14 |
16 |
S
(Standard Deviation) |
2 |
2.5 |
1.2 |
S2
(Variance) |
4 |
6.25 |
1.44 |
N
(number of cases) |
38 |
30 |
45 |
Sum
of Squares |
152 |
187.5 |
64.8 |
Sum
of Scores (mean*n) |
570 |
420 |
720 |
Assumptions
Independent
Random sampling
Interval/ratio
level data
Population
variances equal
Groups
are normally distributed
State
the Hypothesis
Ho:
There is no statistically significant difference among the three cities in the mean years
of education for police officers. 
Ha:
There is a statistically significant difference among the three cities in the mean years
of education for police officers. 
Set
the Rejection Criteria
Determine
the degrees of freedom for the F Distribution
Numerator
Degrees of Freedom
df=k-1
where k=3 (number of independent samples/groups)
df=2
Denominator
Degrees of Freedom
df=n-k
where n=113 (sum of all independent samples) df=110
Establish
Critical Value
Determine
the level of confidence -- alpha
At
alpha.05,
df=(2,110)
Consult
f-distribution,
Fcv = 3.072
Compute
the Test Statistic
where
 = each group mean
 = grand mean for
all the groups = ( sum of all scores)/N
 = number in each group
Note: F=
Mean
Squares between groups
Mean Squares within groups
Estimate
Grand Mean
   
Estimate
F Statistic
 
Decide
Results of Null Hypothesis
Compare
the F statistic to the F critical value. If the F statistic equals or exceeds the Fcv, the
null hypothesis is rejected. This suggests that the population means of the groups sampled
are not equal -- there is a difference between the group means.
Since
the F-statistic (9.931) exceeds the F critical value (3.072), we reject the null
hypothesis and conclude there is a statistically significant difference between the three
cities in the mean years of education for police officers.
Software Output Example
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