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:
Degrees of Sum of Mean Square Source Freedom Squares (Variance)
Among groups c - 1 = ? SSA = ? MSA = 80 FSTAT = ?
Within groupsn - c = ? SSW = 560 MSW = ?
Total n - 1 = ? SST = ?
Degrees of Sum of Mean Square Source Freedom Squares (Variance)
Among groups c - 1 = 3 SSA = 80 MSA = 80 FSTAT = 10
Within groupsn - c = 4 SSW = 560 MSW = 140
Total n - 1 = 31 SST = 640
To fill in the missing values in the ANOVA summary table, we need to use the formulas for each calculation.
1. Degrees of Freedom:
- Among groups: c - 1 (number of groups minus one), where c represents the number of groups. In this case, c = 4, so degrees of freedom among groups = 4 - 1 = 3.
- Within groups: n - c (total number of values minus the number of groups), where n represents the total number of values and c represents the number of groups. In this case, n = 8 * 4 = 32 and c = 4, so degrees of freedom within groups = 32 - 4 = 28.
- Total: n - 1, where n represents the total number of values. In this case, n = 8 * 4 = 32, so degrees of freedom total = 32 - 1 = 31.
2. Sum of Squares:
- SSA (Sum of Squares among groups) = MSA * (c - 1), where MSA represents the Mean Square among groups. But MSA is missing, so we cannot calculate SSA without this value.
- SSW (Sum of Squares within groups) = MSW * (n - c), where MSW represents the Mean Square within groups. But MSW is missing, so we cannot calculate SSW without this value.
- SST (Total Sum of Squares) = SSA + SSW, where SSA represents the Sum of Squares among groups and SSW represents the Sum of Squares within groups. But both SSA and SSW are missing, so we cannot calculate SST without these values.
3. Mean Square:
- MSA (Mean Square among groups) = SSA / (c - 1), where SSA represents the Sum of Squares among groups. But SSA is missing, so we cannot calculate MSA without this value.
- MSW (Mean Square within groups) = SSW / (n - c), where SSW represents the Sum of Squares within groups. But SSW is missing, so we cannot calculate MSW without this value.
4. Variance:
- FSTAT (F-value) = MSA / MSW, where MSA represents the Mean Square among groups and MSW represents the Mean Square within groups. But both MSA and MSW are missing, so we cannot calculate FSTAT without these values.
In conclusion, without the missing values, we cannot fill in all the results in the ANOVA summary table.
To fill in the missing values in the ANOVA summary table, you need to understand the formulas used to calculate these values.
1. Degrees of Freedom (df):
- Among groups: c - 1, where c is the number of groups. In this case, there are four groups, so c = 4. Therefore, df = 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 and four groups, so n = 8 x 4 = 32 and c = 4. Therefore, df = 32 - 4 = 28.
- Total: n - 1. In this case, n = 8 x 4 = 32, so df = 32 - 1 = 31.
2. Sum of Squares (SS):
- Among groups (SSA): This represents the variation between the group means. It is calculated by summing the squared deviations of each group mean from the overall mean, weighted by the sample size of each group. In this case, the mean square (MSA) is given as 80, so SSA = MSA x df = 80 x 3 = 240.
- Within groups (SSW): This represents the variation within each group. It is calculated by summing the squared deviations of each individual value from their group mean. In this case, SSW is given as 560.
3. Mean Square (MS):
- Among groups (MSA): This is calculated by dividing SSA by the corresponding degrees of freedom (df). In this case, MSA is given as 80.
- Within groups (MSW): This is calculated by dividing SSW by the corresponding degrees of freedom (df). In this case, MSW is missing.
4. F-statistic (FSTAT):
- This is calculated by dividing the mean square between groups (MSA) by the mean square within groups (MSW). In this case, FSTAT is missing.
Now, let's fill in the missing values based on the above calculations:
Degrees of Sum of Mean Square Source Freedom Squares (Variance)
Among groups 4 - 1 = 3 SSA = 240 MSA = 80 FSTAT = ?
Within groups 32 - 4 = 28 SSW = 560 MSW = ?
Total 32 - 1 = 31 SST = ?