10.55 Consider an experiment with four groups, with eight
values in each. For the ANOVA summary table below, fill in
all the missing results:
Source
Degrees of
Freedom
Sum of
Squares
Mean
Square
(Variance) F
Among
groups
c - 1 = ? SSA = ? MSA = 80 FSTAT = ?
Within
groups
n - c = ? SSW = 560 MSW = ?
Total n - 1 = ? SST = ?
To fill in the missing results in the ANOVA summary table, you need to understand the concepts and formulas used in analysis of variance (ANOVA).
1. Degrees of Freedom (df):
- Among groups: c - 1, where c is the number of groups. In this case, there are four groups, so the degrees of freedom among groups is 4 - 1 = 3.
- Within groups: n - c, where n is the total number of observations and c is the number of groups. In this case, there are eight values in each group, so the total number of observations is 4 groups * 8 values = 32. Therefore, the degrees of freedom within groups is 32 - 4 = 28.
- Total: n - 1, where n is the total number of observations. In this case, the total number of observations is 32, so the degrees of freedom total is 32 - 1 = 31.
2. Sum of Squares (SS):
- Among groups (SSA): This represents the sum of squared deviations from the overall mean for each group. Since the mean square (MSA) is the variance, you can calculate SSA by multiplying MSA by the degrees of freedom among groups (SSA = MSA * df among groups). However, the MSA value is missing in the table, so we cannot calculate SSA.
- Within groups (SSW): This represents the sum of squared deviations within each group. It measures the variability within the groups. Similarly, SSW is the product of MSW and the degrees of freedom within groups (SSW = MSW * df within groups). However, the MSW value is also missing, so we cannot calculate SSW.
- Total (SST): This represents the total sum of squares and measures the overall variability. It is calculated as the sum of the sums of squares among groups and within groups (SST = SSA + SSW). Since both SSA and SSW are missing, we cannot calculate SST.
3. Mean Square (MS):
- Among groups (MSA): This represents the variance between the groups and is calculated by dividing the sum of squares among groups by the degrees of freedom among groups (MSA = SSA / df among groups). However, since the SSA value is missing, we cannot calculate MSA.
- Within groups (MSW): This represents the variance within the groups and is calculated by dividing the sum of squares within groups by the degrees of freedom within groups (MSW = SSW / df within groups). However, since the SSW value is missing, we cannot calculate MSW.
4. F statistic (FSTAT):
This is the test statistic for ANOVA, calculated by dividing the mean square among groups by the mean square within groups (FSTAT = MSA / MSW). However, since both MSA and MSW are missing, we cannot calculate FSTAT.
Unfortunately, without the missing values, we cannot fill in the missing results in the ANOVA summary table. These missing values need to be provided or calculated based on additional information.