This preview shows pages 1–3. Sign up to view the full content.
Chapter 12: Oneway Analysis of Variance (ANOVA)
Readings: Chapter 12
1
Introduction
•
The OneWay ANOVA is used when you have
one categorical
and
one quantitative
variable and you want to compare the means of the quantitative variable across diﬀerent
categories.
–
The lifetime of D batteries from two brands are compared for their lifetime in a
ﬂashlight. 50 ﬂashlights are loaded with Brand 1 batteries and 45 with Brand 2
batteries. The lifetime (in hours) of each battery is then recorded.
–
In order to examine whether the drug Paxil aﬀect serotonin levels in healthy young
men, a random sample of 15 men are split into 3 groups of 5. They receive 0, 20 and
40mg of the drug per day for a week and then their serotonin levels are measured.
•
If the categorical variable has two groups (Brand 1 and Brand 2 batteries), use the
two
sample
t
procedures
in Chapter 7.
•
If the categorical variable has more than 2 groups (Paxil doses=0, 20, and 40mg), then
use the
oneway ANOVA
.
•
ANOVA
(ANalysis Of VAriance): used for comparing several means. It is a technique
that generalizes the twosample
t
procedure which compares two means to a situation
with more than two sample means.
•
Assumptions of the ANOVA:
–
Each distribution have a normal distribution
–
Samples are independent and random
–
The population standard deviations are equal
2
Oneway ANOVA
•
Goal of ANOVA: Is there at least one mean that is statistically signiﬁcantly diﬀerent from
the others?
•
Notation:
–
I
= number of groups
–
n
i
= then number of observations in the
i
th
group
–
N
=
n
1
+
n
2
+
...
+
n
I
= total number of observations
–
μ
i
= the population mean of the
i
th
group
–
¯
x
i
= the sample mean of the
i
th
group
–
σ
= the common population standard deviation
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentStandard Deviations
•
We assume that in the ANOVA models, the population standard deviations are all equal.
The oﬃcial test is quite complicated, so we use the following rule of thumb:
If the largest standard deviation is less than twice the smallest standard devia
tion, we can use methods based on the assumption of equal standard deviations
•
If we assume all the standard deviations are equal, each
s
is an estimate of
σ
. We combine
these into a
Pooled Estimator
of
σ
:
s
p
=
s
(
n
1

1)
s
2
1
+ (
n
2

1)
s
2
2
+
...
+ (
n
I

1)
s
2
I
(
n
1

1) + (
n
2

1) +
...
+ (
n
I

1)
Example 1
: An experiment was run to compare four groups. The sample sizes were 20,
220, 18, and 15, and the corresponding sample standard deviations were 62, 40, 52, and
48.
a. Is it reasonable to use the assumption of equal standard deviations when we analyze
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 Staff

Click to edit the document details