To calculate the between-group sum of squares (SSB), we follow a structured process. Here’s a detailed step-by-step solution:
1. Identify the Sample Sizes and Means:
- For the Black group:
- Sample size ([tex]\(N_{black}\)[/tex]) = 113
- Mean age ([tex]\(\bar{X}_{black}\)[/tex]) = 25.39
- For the White group:
- Sample size ([tex]\(N_{white}\)[/tex]) = 904
- Mean age ([tex]\(\bar{X}_{white}\)[/tex]) = 22.99
- For the Other group:
- Sample size ([tex]\(N_{other}\)[/tex]) = 144
- Mean age ([tex]\(\bar{X}_{other}\)[/tex]) = 23.87
- Overall:
- Total sample size ([tex]\(N_{total}\)[/tex]) = 1,161
- Overall mean age ([tex]\(\bar{X}_{total}\)[/tex]) = 23.33
2. Compute the SSB for Each Group:
The formula for the between-group sum of squares for each group is:
[tex]\[
SSB_{group} = N_{group} \times (\bar{X}_{group} - \bar{X}_{total})^2
\][/tex]
- For the Black group:
[tex]\[
SSB_{black} = 113 \times (25.39 - 23.33)^2
\][/tex]
Calculating the terms inside:
[tex]\[
(25.39 - 23.33) = 2.06 \quad \Rightarrow \quad (2.06)^2 = 4.2436
\][/tex]
Thus:
[tex]\[
SSB_{black} = 113 \times 4.2436 = 479.5268
\][/tex]
- For the White group:
[tex]\[
SSB_{white} = 904 \times (22.99 - 23.33)^2
\][/tex]
Calculating the terms inside:
[tex]\[
(22.99 - 23.33) = -0.34 \quad \Rightarrow \quad (-0.34)^2 = 0.1156
\][/tex]
Thus:
[tex]\[
SSB_{white} = 904 \times 0.1156 = 104.5024
\][/tex]
- For the Other group:
[tex]\[
SSB_{other} = 144 \times (23.87 - 23.33)^2
\][/tex]
Calculating the terms inside:
[tex]\[
(23.87 - 23.33) = 0.54 \quad \Rightarrow \quad (0.54)^2 = 0.2916
\][/tex]
Thus:
[tex]\[
SSB_{other} = 144 \times 0.2916 = 41.9904
\][/tex]
3. Sum Up the Individual SSB Components to Get the Total SSB:
[tex]\[
SSB_{total} = SSB_{black} + SSB_{white} + SSB_{other}
\][/tex]
Therefore:
[tex]\[
SSB_{total} = 479.5268 + 104.5024 + 41.9904 = 626.0196
\][/tex]
So, the between-group sum of squares (SSB) is:
[tex]\[
\boxed{626.0196}
\][/tex]
