To determine if there is a significant difference between the mean ages of marriage across the three racial groups, we need to calculate the mean square between (MSB). Here is a detailed, step-by-step solution to arrive at the mean square between.
### Step 1: Gather Information
First, let's compile the data provided:
- Black
- Sample size ([tex]\( N_{\text{Black}} \)[/tex]): 113
- Mean age ([tex]\( \text{mean}_{\text{Black}} \)[/tex]): 25.39
- White
- Sample size ([tex]\( N_{\text{White}} \)[/tex]): 904
- Mean age ([tex]\( \text{mean}_{\text{White}} \)[/tex]): 22.99
- Other
- Sample size ([tex]\( N_{\text{Other}} \)[/tex]): 144
- Mean age ([tex]\( \text{mean}_{\text{Other}} \)[/tex]): 23.87
- All groups
- Total sample size ([tex]\( N_{\text{Total}} \)[/tex]): 1,161
- Overall mean age ([tex]\( \text{mean}_{\text{Total}} \)[/tex]): 23.33
- Number of groups: 3
### Step 2: Calculate the Between-Group Sum of Squares [tex]\( \text{SSB} \)[/tex]
The formula for calculating the sum of squares between groups (SSB) is as follows:
[tex]\[ \text{SSB} = \sum_{i} N_i (\text{mean}_i - \text{mean}_{\text{Total}})^2 \][/tex]
We calculate it for each group:
1. Black Group:
[tex]\[ \text{SSB}_{\text{Black}} = 113 \times (25.39 - 23.33)^2 = 113 \times 4.3241 = 479.5268 \][/tex]
2. White Group:
[tex]\[ \text{SSB}_{\text{White}} = 904 \times (22.99 - 23.33)^2 = 904 \times 0.114244 = 104.5024 \][/tex]
3. Other Group:
[tex]\[ \text{SSB}_{\text{Other}} = 144 \times (23.87 - 23.33)^2 = 144 \times 0.288369 = 41.9904 \][/tex]
Adding these contributions together gives the total between-group sum of squares:
[tex]\[ \text{SSB}_{\text{Total}} = 479.5268 + 104.5024 + 41.9904 = 626.0196 \][/tex]
### Step 3: Calculate Degrees of Freedom for Between-Group Sum of Squares
The degrees of freedom between groups ([tex]\( df_{\text{between}} \)[/tex]) are calculated as:
[tex]\[ df_{\text{between}} = \text{Number of groups} - 1 \][/tex]
Hence,
[tex]\[ df_{\text{between}} = 3 - 1 = 2 \][/tex]
### Step 4: Calculate the Mean Square Between (MSB)
Mean Square Between (MSB) is calculated by dividing the total between-group sum of squares (SSB) by the degrees of freedom between:
[tex]\[ \text{MSB} = \frac{\text{SSB}_{\text{Total}}}{df_{\text{between}}} = \frac{626.0196}{2} = 313.0098 \][/tex]
Therefore, the mean square between (MSB) for the given data is [tex]\( 313.0098 \)[/tex].
This step-by-step solution details how to generate the MSB using the provided sample sizes and means for the three racial groups.
