LAB SESSION 15
ANALYSIS OF VARIANCE
INTRODUCTION:
In earlier sessions you have examined and compared means from
two
samples. We will now practice a
technique that tests hypothesis about several
means. While we could compare the means in pairs as
we have done before, the process could become too unwieldy to be of any
use. Analysis of variance (ANOVA)
allows us
to
test all the means at the same time to see if there is any significant
difference between them.
The Logic Underlying The
Anova Technique
We will be forming a comparison between two
estimates of the population variance: one based on the variance within each set
of data and the other between the sets of data. We
will use the F distribution for this
comparison. If there is relatively little
difference
within
each group and a large difference between the sample means, we will reject the
null
hypothesis. (Remember we always word
the null hypothesis to say “there is no difference...”). If there is a lot of variance within a group
and little between groups, we cannot conclude that the population means are
different. We also need to know that
the groups under investigation are approximately normally distributed and
independent.
ANOVA
is presented as a table, and we need to define our terms in order to understand
what
the table is telling us. The Factor is
the variable whose means we are interested in studying. When we first set up our data charts in
Excel, each column will represent different Levels of the Factor we are
examining. Each row will be a data
value from
repeated
samplings, called a Replicate. The
ANOVA table will give a summary of the data with the different levels of the
Factor in the first column, followed by each levels’ count, sum average and
variance in subsequent columns across the rows. It then gives you a chart
describing the sources of variation both Between Groups and Within
Groups
PERFORMING AN ANOVA
ANALYSIS
This
sample problem will take you through the steps of entering the data and
generating
the
ANOVA table for Illustration 121 in your textbook. The FACTOR we are looking
at
is
temperature and whether it has any effect on production. We will examine production
at
three different temperature levels: 68°, 72°, 76°.
These levels form our columns.
The production amounts are the replicates and form the rows of the data
table. You can name
the
columns and enter the data directly into the worksheet.
If we did a Box Plot of the three columns
some interesting things are shown.
Note that the points within each level
are fairly close, but the three levels hardly overlap at all.
The commands for generating the ANOVA
table is as follows:
Choose: Tools
> Data Analysis > ANOVA: Single Factor > OK
Enter: Input
Range: A3:C8
Select: Output
Range:
Enter: A10 (or the upper left
corner where you want it)
The worksheet will look like this:
Illustration
12.1 













Sample from 68^{o} 
Sample from 72^{o} 
Sample from 76^{o} 




10 
7 
3 




12 
6 
3 




10 
7 
5 




9 
8 
4 





7 












Anova:
Single Factor 













SUMMARY 






Groups 
Count 
Sum 
Average 
Variance 


Sample
from 68o 
4 
41 
10.25 
1.583333 


Sample
from 72o 
5 
35 
7 
0.5 


Sample
from 76o 
4 
15 
3.75 
0.916667 
















ANOVA 






Source of Variation 
SS 
df 
MS 
F 
Pvalue 
F crit 
Between
Groups 
84.5 
2 
42.25 
44.47368 
1.05E05 
4.102816 
Within
Groups 
9.5 
10 
0.95 










Total 
94 
12 











Compare the output to the calculations in
Illustration 121 in the text.
Note in particular that the calculated
value for F* = 44.47.
To make our decision, we
need to compare this to the critical
value F(2,10,.05) = 4.10. We can therefore conclude
that at least one of the temperatures has
an effect on the production level. The pvalue
given in the chart can also be used to
determine the conclusion. How would you
interpret it?
Exercise 12.32 in the chapter compares
the stopping distances for four brands of tires.
Using the data given there, is there
sufficient evidence to conclude that there is a
difference in the mean stopping distances
at the a = .05 level?
This data may be found on the Student
Suite CD as ex1228
a) State your null and alternative
hypotheses.
b) Find your critical region and value
for F.
c) 1) Enter your data in columns 1  4,
naming them A, B, C, D respectively.
2) Do a box plot to get a feel for how the data interact.
3) Perform an ANOVA to calculate F*.
What does the p value tell you?
Explain.
d) Draw your conclusion about the null
hypothesis and explain what it means to you.
How would your conclusion change if a changed?
ASSIGNMENT:
Do Exercises 12.17, 12.18, and 12.22 in your text. The data for
Exercises 12.17, 12.18 and 11.22 may be
found on the Student Suite CD.