Chapter 12
One-Way Analysis of Variance
:
In Chapter 7, we were looking to answer the question “is the difference between the two
means statistically significant?”
To this end we used
two sample
t
procedures
.
One-way
AN
alysis
O
f
VA
riance (
ANOVA
) is used when you want to compare more than two
means.
It is a technique that generalizes the two-sample
t
procedure which compares two
means to a situation with more than two sample means.
Like the two-sample
t
test, it is
robust and useful.
Examples:
1.
The presence of harmful insects in farm fields is detected by erecting boards covered
with a sticky material and then examining the insects trapped on the boards.
To
investigate which colors are most attractive to cereal leaf beetles, researchers placed
six boards of each of four colors in a field of oats in July.
2.
An environmentalist is interested in comparing the concentration of the pollutant
cadmium in five streams.
She collects 50 water specimens from each stream and
measures the concentration of cadmium in each specimen.
Note:
The first example is an
experiment
with four treatments (the colors) and the second
example is an
observational study
where the concentration of cadmium is compared
between the five streams. In both cases we can use ANOVA to compare the mean
responses.
As with the
t-
test, the
F
statistic compares the variation among the means of several groups
with the variation within the groups.
In the ANOVA test, a SRS from each population is drawn and the data is used to test the null
hypothesis that the populations are all equal against the alternative that not all are equal.
If
we reject the null, we need to perform some further analysis to draw conclusions about which
population means differ.
Assumptions of the ANOVA:
1.
The data is normally distributed.
2.
The population standard deviations are equal.
Note:
I
=
the number of groups.
N
= the total sample size (sum of all the )
= the sample size for group
i.
Example 1:
STAT 301 Spring 2012
Chapter 12
Page
1

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*Sign up*The strength of concrete depends upon the formula used to prepare it.
One study compared
five different mixtures.
Six batches of each mixture were prepared, and the strength of the
concrete made from each batch was measured.
What is the response variable?
Give the
values for
I
,
the , and
N.
Estimating the population parameters:
The unknown parameters in the statistical model for ANOVA are the
I
population means and
the common standard deviation σ.
To estimate
we use the sample mean for the
i
th group:
,
that is we calculate a sample mean … for each of the samples.
Recall:
Our second assumption in the ANOVA model was that our population standard
deviations are all equal.
The official test is quite complicated and not practical, so we use the
following rule of thumb:
If the largest standard deviation is less than twice the smallest standard deviation, we
can use methods based on the assumption of equal standard deviations and our
results will still be approximately correct.
So we compare 2 x smallest std. dev to the
largest std. dev.---we want 2 x smallest std. dev > largest std. dev.

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