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Sagot :
Sure, I will guide you through the detailed steps to calculate the combined mean for each group separately and then for both groups taken together.
### Group 1:
The first group of students consists of:
- 10 third divisioners with mean marks 36
- 15 second divisioners with mean marks 56
- 25 first divisioners with mean marks 64
Let's calculate the total number of students in Group 1 and the total marks they obtained.
1. Calculate total students in Group 1:
[tex]\[ \text{Total students in Group 1} = 10 + 15 + 25 = 50 \][/tex]
2. Calculate total marks for Group 1:
[tex]\[ \text{Total marks for Group 1} = (10 \times 36) + (15 \times 56) + (25 \times 64) \][/tex]
[tex]\[ \text{Total marks for Group 1} = 360 + 840 + 1600 = 2800 \][/tex]
3. Calculate the combined mean for Group 1:
[tex]\[ \text{Combined mean for Group 1} = \frac{\text{Total marks for Group 1}}{\text{Total students in Group 1}} \][/tex]
[tex]\[ \text{Combined mean for Group 1} = \frac{2800}{50} = 56.0 \][/tex]
### Group 2:
The second group of students consists of:
- 13 third divisioners with mean marks 45
- 17 second divisioners with mean marks 55
- 20 first divisioners with mean marks 74
Let's calculate the total number of students in Group 2 and the total marks they obtained.
1. Calculate total students in Group 2:
[tex]\[ \text{Total students in Group 2} = 13 + 17 + 20 = 50 \][/tex]
2. Calculate total marks for Group 2:
[tex]\[ \text{Total marks for Group 2} = (13 \times 45) + (17 \times 55) + (20 \times 74) \][/tex]
[tex]\[ \text{Total marks for Group 2} = 585 + 935 + 1480 = 3000 \][/tex]
3. Calculate the combined mean for Group 2:
[tex]\[ \text{Combined mean for Group 2} = \frac{\text{Total marks for Group 2}}{\text{Total students in Group 2}} \][/tex]
[tex]\[ \text{Combined mean for Group 2} = \frac{3000}{50} = 60.0 \][/tex]
### Combined Mean of Both Groups Taken Together:
Now, let's calculate the combined mean for both groups taken together.
1. Calculate total students in both groups:
[tex]\[ \text{Total students in both groups} = 50 + 50 = 100 \][/tex]
2. Calculate total marks for both groups:
[tex]\[ \text{Total marks for both groups} = 2800 + 3000 = 5800 \][/tex]
3. Calculate the combined mean for both groups:
[tex]\[ \text{Combined mean for both groups} = \frac{\text{Total marks for both groups}}{\text{Total students in both groups}} \][/tex]
[tex]\[ \text{Combined mean for both groups} = \frac{5800}{100} = 58.0 \][/tex]
### Summary of Results:
- Combined mean for Group 1: [tex]\(56.0\)[/tex]
- Combined mean for Group 2: [tex]\(60.0\)[/tex]
- Combined mean for both groups taken together: [tex]\(58.0\)[/tex]
This detailed breakdown shows how we can calculate the combined means for each group and for the groups together.
### Group 1:
The first group of students consists of:
- 10 third divisioners with mean marks 36
- 15 second divisioners with mean marks 56
- 25 first divisioners with mean marks 64
Let's calculate the total number of students in Group 1 and the total marks they obtained.
1. Calculate total students in Group 1:
[tex]\[ \text{Total students in Group 1} = 10 + 15 + 25 = 50 \][/tex]
2. Calculate total marks for Group 1:
[tex]\[ \text{Total marks for Group 1} = (10 \times 36) + (15 \times 56) + (25 \times 64) \][/tex]
[tex]\[ \text{Total marks for Group 1} = 360 + 840 + 1600 = 2800 \][/tex]
3. Calculate the combined mean for Group 1:
[tex]\[ \text{Combined mean for Group 1} = \frac{\text{Total marks for Group 1}}{\text{Total students in Group 1}} \][/tex]
[tex]\[ \text{Combined mean for Group 1} = \frac{2800}{50} = 56.0 \][/tex]
### Group 2:
The second group of students consists of:
- 13 third divisioners with mean marks 45
- 17 second divisioners with mean marks 55
- 20 first divisioners with mean marks 74
Let's calculate the total number of students in Group 2 and the total marks they obtained.
1. Calculate total students in Group 2:
[tex]\[ \text{Total students in Group 2} = 13 + 17 + 20 = 50 \][/tex]
2. Calculate total marks for Group 2:
[tex]\[ \text{Total marks for Group 2} = (13 \times 45) + (17 \times 55) + (20 \times 74) \][/tex]
[tex]\[ \text{Total marks for Group 2} = 585 + 935 + 1480 = 3000 \][/tex]
3. Calculate the combined mean for Group 2:
[tex]\[ \text{Combined mean for Group 2} = \frac{\text{Total marks for Group 2}}{\text{Total students in Group 2}} \][/tex]
[tex]\[ \text{Combined mean for Group 2} = \frac{3000}{50} = 60.0 \][/tex]
### Combined Mean of Both Groups Taken Together:
Now, let's calculate the combined mean for both groups taken together.
1. Calculate total students in both groups:
[tex]\[ \text{Total students in both groups} = 50 + 50 = 100 \][/tex]
2. Calculate total marks for both groups:
[tex]\[ \text{Total marks for both groups} = 2800 + 3000 = 5800 \][/tex]
3. Calculate the combined mean for both groups:
[tex]\[ \text{Combined mean for both groups} = \frac{\text{Total marks for both groups}}{\text{Total students in both groups}} \][/tex]
[tex]\[ \text{Combined mean for both groups} = \frac{5800}{100} = 58.0 \][/tex]
### Summary of Results:
- Combined mean for Group 1: [tex]\(56.0\)[/tex]
- Combined mean for Group 2: [tex]\(60.0\)[/tex]
- Combined mean for both groups taken together: [tex]\(58.0\)[/tex]
This detailed breakdown shows how we can calculate the combined means for each group and for the groups together.
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