Chapter 12 Outline
Introduction
In Chapter 11 we developed methods to determine whether there is a difference between two
population means. What if we wanted to compare more than two population means? The two
sample tests used in Chapter 11 require that the population means be compared two at a time.
This would be very time consuming, but more importantly there would be a buildup of Type I
error. That is, the total value of
α
would become quite large as the number of comparisons
increased. In this chapter, we will describe a technique that is efficient when simultaneously
comparing several sample means to determine if they come from the same or equal populations.
This technique is known as
Analysis of Variance (ANOVA).
Analysis of Variance (ANOVA
).
A statistical technique for testing whether
several populations have the same mean.
A second test compares two sample variances to determine if the populations are equal. This test
is particularly useful for validating a requirement of the twosample
t
test presented in Chapter
11.
This test assumed that the population standard deviations were equal but unknown (see text
formulas [115] and [116]).
The
F
Distribution
The
F
distribution
can be used to test whether two samples are from populations having equal
variances.
It can also be used to compare several population means simultaneously (
ANOVA
).
F Distribution
.
A continuous probability distribution where
F
is always 0 or
positive. The distribution is positively skewed. It is based on two parameters,
the number of degrees of freedom in the numerator and the number of degrees
of freedom in the denominator.
The major characteristics of the
F
distribution are:
1.
There is a family of
F
distributions
. A particular member of the family is determined
by two parameters: the degrees of freedom in the numerator and the degrees of
freedom in the denominator.
2.
The
F
distribution is continuous.
This means that it can assume an infinite number of
values between 0 and plus infinity.
3.
The
F
distribution cannot be negative.
The smallest value
F
can assume is 0.
4.
It is positively skewed.
The long tail of the distribution is to the righthand side. As the
number of degrees of freedom increases in both the numerator and denominator, the
distribution approaches a normal distribution.
5.
It is asymptotic
. As the values of
X
increase, the
F
curve approaches the
X
axis but never
touches it. This is similar to the behavior of the normal distribution described in
Chapter 7.
Comparing Two Populations Variances
The
F
distribution is used to test the hypothesis that the variance of one normal population equals
the variance of another normal population. The
F
distribution can also be used to validate
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assumptions with respect to certain statistical tests. Regardless of whether we want to determine
if one population has more variation than another population does or validate an assumption with
respect to a statistical test, we still use the usual fivestep hypothesis testing procedure. The value
of the test statistic is determined using text formula [121].
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 Spring '08
 Staff
 Normal Distribution, Standard Deviation, Variance, critical value

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