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Sagot :
Certainly! Let's break down the solution step-by-step for each part of the question.
### Given data:
| Ages (in years) | [tex]\(0-20\)[/tex] | [tex]\(20-40\)[/tex] | [tex]\(40-60\)[/tex] | [tex]\(60-80\)[/tex] | [tex]\(80-100\)[/tex] |
|-----------------|----------|----------|----------|----------|-----------|
| Number of people | 50 | 60 | 80 | 50 | 10 |
### (a) What does [tex]\( f_0 \)[/tex] represent in the formula [tex]\( M_0 = L + \frac{c}{f} \left( \frac{f_1 - f_0}{2 f_1 - f_0 - f_2} \right) \)[/tex] for the calculation of the mode of the continuous series?
[tex]\( f_0 \)[/tex] represents the frequency of the modal class, which is the class with the highest frequency in the given data.
### (b) Calculate the modal age.
1. Identify [tex]\( f_0 \)[/tex]: The highest frequency in the data is 80, which corresponds to the age group [tex]\(40-60\)[/tex]. Therefore, [tex]\( f_0 = 80 \)[/tex].
2. Find the modal class index: The modal class is [tex]\( 40-60 \)[/tex], so its index is 2.
3. Determine the lower boundary [tex]\( L \)[/tex]: The lower boundary of the modal class (40-60) is [tex]\( L = 40 \)[/tex].
4. Identify class interval [tex]\( c \)[/tex]: Assuming equal class intervals, [tex]\( c = 20 \)[/tex] (e.g., for [tex]\(20-40\)[/tex], the interval is [tex]\(40-20 = 20\)[/tex]).
5. Identify frequencies [tex]\( f_1 \)[/tex] and [tex]\( f_2 \)[/tex]:
- [tex]\( f_1 \)[/tex] is the frequency of the class preceding the modal class (age group [tex]\(20-40\)[/tex]), so [tex]\( f_1 = 60 \)[/tex].
- [tex]\( f_2 \)[/tex] is the frequency of the class succeeding the modal class (age group [tex]\(60-80\)[/tex]), so [tex]\( f_2 = 50 \)[/tex].
6. Calculate the mode using the formula:
[tex]\[ M_0 = L + \frac{c}{f_0}\left( \frac{f_1 - f_0}{2 f_0 - f_1 - f_2} \right) \][/tex]
[tex]\[ M_0 = 40 + \frac{20}{80}\left( \frac{60 - 80}{2 \cdot 80 - 60 - 50} \right) \][/tex]
[tex]\[ M_0 = 40 + \frac{20}{80} \left( \frac{-20}{160 - 110} \right) \][/tex]
[tex]\[ M_0 = 40 + \frac{20}{80} \left( \frac{-20}{50} \right) \][/tex]
[tex]\[ M_0 = 40 + \frac{20}{80} \cdot -0.4 \][/tex]
[tex]\[ M_0 = 40 - 0.4 \cdot 20 \][/tex]
[tex]\[ M_0 = 40 - 8 \][/tex]
[tex]\[ M_0 = 32 \][/tex]
So, the modal age [tex]\( M_0 \)[/tex] is 32 years.
### (c) Find the average age.
1. Calculate total people:
[tex]\[ \text{Total people} = 50 + 60 + 80 + 50 + 10 = 250 \][/tex]
2. Determine midpoints of the age groups:
- [tex]\(0-20\)[/tex] midpoint: [tex]\(10\)[/tex]
- [tex]\(20-40\)[/tex] midpoint: [tex]\(30\)[/tex]
- [tex]\(40-60\)[/tex] midpoint: [tex]\(50\)[/tex]
- [tex]\(60-80\)[/tex] midpoint: [tex]\(70\)[/tex]
- [tex]\(80-100\)[/tex] midpoint: [tex]\(90\)[/tex]
3. Calculate total ages:
[tex]\[ \text{Total age} = (50 \times 10) + (60 \times 30) + (80 \times 50) + (50 \times 70) + (10 \times 90) \][/tex]
[tex]\[ = 500 + 1800 + 4000 + 3500 + 900 \][/tex]
[tex]\[ = 10700 \][/tex]
4. Calculate average age:
[tex]\[ \text{Average age} = \frac{\text{Total age}}{\text{Total people}} = \frac{10700}{250} = 42.8 \][/tex]
So, the average age is 42.8 years.
### (d) How many people of age group (40-60) years make 44 years as the average age of entire data?
1. Target average age: [tex]\(44\)[/tex] years.
2. Current total age without age group [tex]\(40-60\)[/tex] frequency:
[tex]\[ \text{Current sum} = 10700 - (80 \times 50) = 10700 - 4000 = 6700 \][/tex]
3. Required total age for target average:
[tex]\[ \text{Needed sum} = 44 \times 250 = 11000 \][/tex]
4. Calculate the additional number of people (let's call it [tex]\( x \)[/tex]) needed in the [tex]\(40-60\)[/tex] age group:
[tex]\[ x \cdot 50 + 6700 = 11000 \][/tex]
[tex]\[ 50x = 11000 - 6700 \][/tex]
[tex]\[ 50x = 4300 \][/tex]
[tex]\[ x = \frac{4300}{50} = 86 \][/tex]
So, to achieve an average age of 44 years, 86 people in the age group [tex]\(40-60\)[/tex] years are needed.
### Given data:
| Ages (in years) | [tex]\(0-20\)[/tex] | [tex]\(20-40\)[/tex] | [tex]\(40-60\)[/tex] | [tex]\(60-80\)[/tex] | [tex]\(80-100\)[/tex] |
|-----------------|----------|----------|----------|----------|-----------|
| Number of people | 50 | 60 | 80 | 50 | 10 |
### (a) What does [tex]\( f_0 \)[/tex] represent in the formula [tex]\( M_0 = L + \frac{c}{f} \left( \frac{f_1 - f_0}{2 f_1 - f_0 - f_2} \right) \)[/tex] for the calculation of the mode of the continuous series?
[tex]\( f_0 \)[/tex] represents the frequency of the modal class, which is the class with the highest frequency in the given data.
### (b) Calculate the modal age.
1. Identify [tex]\( f_0 \)[/tex]: The highest frequency in the data is 80, which corresponds to the age group [tex]\(40-60\)[/tex]. Therefore, [tex]\( f_0 = 80 \)[/tex].
2. Find the modal class index: The modal class is [tex]\( 40-60 \)[/tex], so its index is 2.
3. Determine the lower boundary [tex]\( L \)[/tex]: The lower boundary of the modal class (40-60) is [tex]\( L = 40 \)[/tex].
4. Identify class interval [tex]\( c \)[/tex]: Assuming equal class intervals, [tex]\( c = 20 \)[/tex] (e.g., for [tex]\(20-40\)[/tex], the interval is [tex]\(40-20 = 20\)[/tex]).
5. Identify frequencies [tex]\( f_1 \)[/tex] and [tex]\( f_2 \)[/tex]:
- [tex]\( f_1 \)[/tex] is the frequency of the class preceding the modal class (age group [tex]\(20-40\)[/tex]), so [tex]\( f_1 = 60 \)[/tex].
- [tex]\( f_2 \)[/tex] is the frequency of the class succeeding the modal class (age group [tex]\(60-80\)[/tex]), so [tex]\( f_2 = 50 \)[/tex].
6. Calculate the mode using the formula:
[tex]\[ M_0 = L + \frac{c}{f_0}\left( \frac{f_1 - f_0}{2 f_0 - f_1 - f_2} \right) \][/tex]
[tex]\[ M_0 = 40 + \frac{20}{80}\left( \frac{60 - 80}{2 \cdot 80 - 60 - 50} \right) \][/tex]
[tex]\[ M_0 = 40 + \frac{20}{80} \left( \frac{-20}{160 - 110} \right) \][/tex]
[tex]\[ M_0 = 40 + \frac{20}{80} \left( \frac{-20}{50} \right) \][/tex]
[tex]\[ M_0 = 40 + \frac{20}{80} \cdot -0.4 \][/tex]
[tex]\[ M_0 = 40 - 0.4 \cdot 20 \][/tex]
[tex]\[ M_0 = 40 - 8 \][/tex]
[tex]\[ M_0 = 32 \][/tex]
So, the modal age [tex]\( M_0 \)[/tex] is 32 years.
### (c) Find the average age.
1. Calculate total people:
[tex]\[ \text{Total people} = 50 + 60 + 80 + 50 + 10 = 250 \][/tex]
2. Determine midpoints of the age groups:
- [tex]\(0-20\)[/tex] midpoint: [tex]\(10\)[/tex]
- [tex]\(20-40\)[/tex] midpoint: [tex]\(30\)[/tex]
- [tex]\(40-60\)[/tex] midpoint: [tex]\(50\)[/tex]
- [tex]\(60-80\)[/tex] midpoint: [tex]\(70\)[/tex]
- [tex]\(80-100\)[/tex] midpoint: [tex]\(90\)[/tex]
3. Calculate total ages:
[tex]\[ \text{Total age} = (50 \times 10) + (60 \times 30) + (80 \times 50) + (50 \times 70) + (10 \times 90) \][/tex]
[tex]\[ = 500 + 1800 + 4000 + 3500 + 900 \][/tex]
[tex]\[ = 10700 \][/tex]
4. Calculate average age:
[tex]\[ \text{Average age} = \frac{\text{Total age}}{\text{Total people}} = \frac{10700}{250} = 42.8 \][/tex]
So, the average age is 42.8 years.
### (d) How many people of age group (40-60) years make 44 years as the average age of entire data?
1. Target average age: [tex]\(44\)[/tex] years.
2. Current total age without age group [tex]\(40-60\)[/tex] frequency:
[tex]\[ \text{Current sum} = 10700 - (80 \times 50) = 10700 - 4000 = 6700 \][/tex]
3. Required total age for target average:
[tex]\[ \text{Needed sum} = 44 \times 250 = 11000 \][/tex]
4. Calculate the additional number of people (let's call it [tex]\( x \)[/tex]) needed in the [tex]\(40-60\)[/tex] age group:
[tex]\[ x \cdot 50 + 6700 = 11000 \][/tex]
[tex]\[ 50x = 11000 - 6700 \][/tex]
[tex]\[ 50x = 4300 \][/tex]
[tex]\[ x = \frac{4300}{50} = 86 \][/tex]
So, to achieve an average age of 44 years, 86 people in the age group [tex]\(40-60\)[/tex] years are needed.
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