Summary. We define the following sums of squares:

We define the following sums of squares:

Within-groups:

Between groups:

Total:

We define the mean squares as follows:

Within-groups:

Between groups:

The null hypothesis is

The decision rule is

Reject if

where is the number for which and follows an F distribution with numerator degrees of freedom (K -1) and denominator degrees of freedom . (Table 6 of Appendix).

For convenience, these calculations are often recorded in a table called a one-way analysis of variance table or ANOVA table, shown below (Table5.1):

Table 5.1

Source of Sum of Degrees of Mean variation squares freedom squares	F ratio
Between groups SSG MSG Within groups SSW MSW Total SST

Example:

A company buys thousands of light bulbs every year. The company is considering three brands of light bulbs to choose from. Before the company decides which light bulbs to buy, it wants to investigate if the mean life of the three types of light bulbs is the same. The research department selects randomly a few bulbs of each type and tested them. Table lists number of hours (in thousands) that each of the bulbs in each brand survived before being burned out.

Brand I	Brand II	Brand III

At the 5% significance level, test the null hypothesis that the mean life of bulbs for each of these three brands is the same.

Solution:

There are 3 groups: Brand I, Brand II, and Brand III:

; ; ;

1) Let us calculate mean of each group

2) Overall mean is

3) In the first group, sum of squared deviations is

Similarly,

and

4)

5) Now, let us calculate between group variability

.

6)

Within-groups mean square is obtained as

Between-groups mean square is obtained as

The null hypothesis is

The decision rule is reject if

The value of test statistic is

.

Since 1.70 is not greater than 3.98 we fail to reject . We accept hypothesis that mean of all three populations are equal, in other words, there is no difference for company which brand to choose.

In the end, substituting the values of various quantities in Table 5.1, we write an ANOVA table for our example as

Source of Sum of Degrees of Mean variation squares freedom squares	F ratio
Between groups 20.36 2 10.18 Within groups 6611 6 Total 86.36 13

Remark1: An alternative formula for SSB and SSW are

where

the sum of the values in sample i

the sum of the values in all samples

the sum of the squares of the values in all samples.

Example:

Consider the following data obtained for two samples selected from two populations

Sample I	Sample II

Set out the analysis of variance table for these data.

Solution:

; ;

Substituting all the values in the formula for SSG and SSW, we obtain

Hence, the variance between samples MSG and the variance within samples MSW are

We write an ANOVA table for our example as

Source of Sum of Degrees of Mean variation squares freedom squares	F ratio
Between groups 16.81 1 16.81 Within groups 58.757 8.39 Total 75.56 8	2.00

Remark2:

To use MINITAB menu follow the following instructions:

1. Select Stat>ANOVA>One-way (Unstacked)

2. Select data columns

3. Click OK

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