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Sagot :
Sure, I will guide you step-by-step to find the mean age using the Direct Method.
### Step 1: Identify Age Groups and Corresponding Number of People
From the given table:
- Age Groups: [tex]\(10-20\)[/tex], [tex]\(20-30\)[/tex], [tex]\(30-40\)[/tex], [tex]\(40-50\)[/tex], [tex]\(50-60\)[/tex]
- Number of People (in thousands): [tex]\(30, 32, 15, 12, 9\)[/tex]
### Step 2: Calculate the Midpoints (Class Marks) of Each Age Group
The class mark (midpoint) for each age group is calculated as follows:
[tex]\[ \text{Class Mark} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \][/tex]
For each age group:
- [tex]\(10-20\)[/tex]: [tex]\(\frac{10 + 20}{2} = 15\)[/tex]
- [tex]\(20-30\)[/tex]: [tex]\(\frac{20 + 30}{2} = 25\)[/tex]
- [tex]\(30-40\)[/tex]: [tex]\(\frac{30 + 40}{2} = 35\)[/tex]
- [tex]\(40-50\)[/tex]: [tex]\(\frac{40 + 50}{2} = 45\)[/tex]
- [tex]\(50-60\)[/tex]: [tex]\(\frac{50 + 60}{2} = 55\)[/tex]
### Step 3: Convert the Number of People into Absolute Numbers
The given numbers are in thousands. Therefore, we need to convert them into actual numbers by multiplying by 1000:
- [tex]\(30 \times 1000 = 30000\)[/tex]
- [tex]\(32 \times 1000 = 32000\)[/tex]
- [tex]\(15 \times 1000 = 15000\)[/tex]
- [tex]\(12 \times 1000 = 12000\)[/tex]
- [tex]\(9 \times 1000 = 9000\)[/tex]
### Step 4: Calculate the Total Number of People
Sum the absolute number of people:
[tex]\[ 30000 + 32000 + 15000 + 12000 + 9000 = 98000 \][/tex]
### Step 5: Calculate the Sum of Products of Midpoints and Corresponding Counts
Now, we find the product of midpoints and the corresponding number of people in each age group and then sum them up:
[tex]\[ (15 \times 30000) + (25 \times 32000) + (35 \times 15000) + (45 \times 12000) + (55 \times 9000) \][/tex]
Calculating each product:
[tex]\[ 15 \times 30000 = 450000 \][/tex]
[tex]\[ 25 \times 32000 = 800000 \][/tex]
[tex]\[ 35 \times 15000 = 525000 \][/tex]
[tex]\[ 45 \times 12000 = 540000 \][/tex]
[tex]\[ 55 \times 9000 = 495000 \][/tex]
Summing these products:
[tex]\[ 450000 + 800000 + 525000 + 540000 + 495000 = 2810000 \][/tex]
### Step 6: Calculate the Mean Age
Finally, we use the formula for the mean age:
[tex]\[ \text{Mean Age} = \frac{\text{Sum of Midpoint Times People}}{\text{Total Number of People}} \][/tex]
Substituting the values we have:
[tex]\[ \text{Mean Age} = \frac{2810000}{98000} \approx 28.7 \text{ years} \][/tex]
### Conclusion
Hence, the mean age of the people in the given country is approximately [tex]\(28.7\)[/tex] years.
### Step 1: Identify Age Groups and Corresponding Number of People
From the given table:
- Age Groups: [tex]\(10-20\)[/tex], [tex]\(20-30\)[/tex], [tex]\(30-40\)[/tex], [tex]\(40-50\)[/tex], [tex]\(50-60\)[/tex]
- Number of People (in thousands): [tex]\(30, 32, 15, 12, 9\)[/tex]
### Step 2: Calculate the Midpoints (Class Marks) of Each Age Group
The class mark (midpoint) for each age group is calculated as follows:
[tex]\[ \text{Class Mark} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \][/tex]
For each age group:
- [tex]\(10-20\)[/tex]: [tex]\(\frac{10 + 20}{2} = 15\)[/tex]
- [tex]\(20-30\)[/tex]: [tex]\(\frac{20 + 30}{2} = 25\)[/tex]
- [tex]\(30-40\)[/tex]: [tex]\(\frac{30 + 40}{2} = 35\)[/tex]
- [tex]\(40-50\)[/tex]: [tex]\(\frac{40 + 50}{2} = 45\)[/tex]
- [tex]\(50-60\)[/tex]: [tex]\(\frac{50 + 60}{2} = 55\)[/tex]
### Step 3: Convert the Number of People into Absolute Numbers
The given numbers are in thousands. Therefore, we need to convert them into actual numbers by multiplying by 1000:
- [tex]\(30 \times 1000 = 30000\)[/tex]
- [tex]\(32 \times 1000 = 32000\)[/tex]
- [tex]\(15 \times 1000 = 15000\)[/tex]
- [tex]\(12 \times 1000 = 12000\)[/tex]
- [tex]\(9 \times 1000 = 9000\)[/tex]
### Step 4: Calculate the Total Number of People
Sum the absolute number of people:
[tex]\[ 30000 + 32000 + 15000 + 12000 + 9000 = 98000 \][/tex]
### Step 5: Calculate the Sum of Products of Midpoints and Corresponding Counts
Now, we find the product of midpoints and the corresponding number of people in each age group and then sum them up:
[tex]\[ (15 \times 30000) + (25 \times 32000) + (35 \times 15000) + (45 \times 12000) + (55 \times 9000) \][/tex]
Calculating each product:
[tex]\[ 15 \times 30000 = 450000 \][/tex]
[tex]\[ 25 \times 32000 = 800000 \][/tex]
[tex]\[ 35 \times 15000 = 525000 \][/tex]
[tex]\[ 45 \times 12000 = 540000 \][/tex]
[tex]\[ 55 \times 9000 = 495000 \][/tex]
Summing these products:
[tex]\[ 450000 + 800000 + 525000 + 540000 + 495000 = 2810000 \][/tex]
### Step 6: Calculate the Mean Age
Finally, we use the formula for the mean age:
[tex]\[ \text{Mean Age} = \frac{\text{Sum of Midpoint Times People}}{\text{Total Number of People}} \][/tex]
Substituting the values we have:
[tex]\[ \text{Mean Age} = \frac{2810000}{98000} \approx 28.7 \text{ years} \][/tex]
### Conclusion
Hence, the mean age of the people in the given country is approximately [tex]\(28.7\)[/tex] years.
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