Answer :
Let's go through the steps in detail for each part of the question.
### Given Data:
- Wage intervals: 500-600, 600-700, 700-800, 800-900, 900-1000, 1000-1100
- Number of workers in each interval: 3, 5, 6, 2, 3, 1
From this data, we can derive the midpoints of the wage intervals and proceed with calculations.
### a) Modal Class:
The modal class is the class interval with the highest frequency (most number of workers).
#### Steps:
1. Observe the frequencies for each class interval: 3, 5, 6, 2, 3, 1.
2. Identify the interval with the highest frequency. Here, the highest frequency is 6.
Thus, the modal class is 700-800.
### b) Median Class:
The median class is the class interval where the median lies. The median divides the dataset into two equal halves.
#### Steps:
1. Calculate the cumulative frequencies:
- 500-600: 3
- 600-700: 3 + 5 = 8
- 700-800: 8 + 6 = 14
- 800-900: 14 + 2 = 16
- 900-1000: 16 + 3 = 19
- 1000-1100: 19 + 1 = 20
2. Find the total number of workers, which is 20.
3. Determine the median's position, which is at [tex]\( \frac{\text{Total number of workers}}{2} = \frac{20}{2} = 10 \)[/tex].
4. Identify the interval where the 10th worker falls:
- The cumulative frequency just before 10 is 8 (for interval 600-700), and the next cumulative frequency is 14 (for interval 700-800).
So, the 10th worker lies in the 700-800 interval, making this the median class.
### c) Average Daily Income:
To find the average daily income, use the formula for the weighted average of the midpoints of the wage intervals.
#### Steps:
1. Calculate the midpoints of the wage intervals:
- 500-600: midpoint = 550
- 600-700: midpoint = 650
- 700-800: midpoint = 750
- 800-900: midpoint = 850
- 900-1000: midpoint = 950
- 1000-1100: midpoint = 1050
2. Calculate the total income by summing the product of midpoints and frequencies:
[tex]\[ \text{Total income} = (550 \times 3) + (650 \times 5) + (750 \times 6) + (850 \times 2) + (950 \times 3) + (1050 \times 1) \][/tex]
[tex]\[ = 1650 + 3250 + 4500 + 1700 + 2850 + 1050 = 15000 \][/tex]
3. Calculate the average daily income:
[tex]\[ \text{Average daily income} = \frac{\text{Total income}}{\text{Total number of workers}} = \frac{15000}{20} = 750 \][/tex]
So, the average daily income per worker is Rs. 750.
### d) Median and Modal Class Comparison:
Do the median and modal class always lie in the same class?
Reasoning:
In this case, both the median class and the modal class are the same (700-800). However, this is not always true. The modal class is determined solely by the highest frequency, while the median class depends on the position that divides the dataset into two equal halves. They can differ in different datasets.
Conclusion:
- Modal class: 700-800
- Median class: 700-800
- Average daily income: Rs. 750
- Median and modal class same?: Yes (in this case), but not necessarily always. They can differ based on the dataset distribution.
### Given Data:
- Wage intervals: 500-600, 600-700, 700-800, 800-900, 900-1000, 1000-1100
- Number of workers in each interval: 3, 5, 6, 2, 3, 1
From this data, we can derive the midpoints of the wage intervals and proceed with calculations.
### a) Modal Class:
The modal class is the class interval with the highest frequency (most number of workers).
#### Steps:
1. Observe the frequencies for each class interval: 3, 5, 6, 2, 3, 1.
2. Identify the interval with the highest frequency. Here, the highest frequency is 6.
Thus, the modal class is 700-800.
### b) Median Class:
The median class is the class interval where the median lies. The median divides the dataset into two equal halves.
#### Steps:
1. Calculate the cumulative frequencies:
- 500-600: 3
- 600-700: 3 + 5 = 8
- 700-800: 8 + 6 = 14
- 800-900: 14 + 2 = 16
- 900-1000: 16 + 3 = 19
- 1000-1100: 19 + 1 = 20
2. Find the total number of workers, which is 20.
3. Determine the median's position, which is at [tex]\( \frac{\text{Total number of workers}}{2} = \frac{20}{2} = 10 \)[/tex].
4. Identify the interval where the 10th worker falls:
- The cumulative frequency just before 10 is 8 (for interval 600-700), and the next cumulative frequency is 14 (for interval 700-800).
So, the 10th worker lies in the 700-800 interval, making this the median class.
### c) Average Daily Income:
To find the average daily income, use the formula for the weighted average of the midpoints of the wage intervals.
#### Steps:
1. Calculate the midpoints of the wage intervals:
- 500-600: midpoint = 550
- 600-700: midpoint = 650
- 700-800: midpoint = 750
- 800-900: midpoint = 850
- 900-1000: midpoint = 950
- 1000-1100: midpoint = 1050
2. Calculate the total income by summing the product of midpoints and frequencies:
[tex]\[ \text{Total income} = (550 \times 3) + (650 \times 5) + (750 \times 6) + (850 \times 2) + (950 \times 3) + (1050 \times 1) \][/tex]
[tex]\[ = 1650 + 3250 + 4500 + 1700 + 2850 + 1050 = 15000 \][/tex]
3. Calculate the average daily income:
[tex]\[ \text{Average daily income} = \frac{\text{Total income}}{\text{Total number of workers}} = \frac{15000}{20} = 750 \][/tex]
So, the average daily income per worker is Rs. 750.
### d) Median and Modal Class Comparison:
Do the median and modal class always lie in the same class?
Reasoning:
In this case, both the median class and the modal class are the same (700-800). However, this is not always true. The modal class is determined solely by the highest frequency, while the median class depends on the position that divides the dataset into two equal halves. They can differ in different datasets.
Conclusion:
- Modal class: 700-800
- Median class: 700-800
- Average daily income: Rs. 750
- Median and modal class same?: Yes (in this case), but not necessarily always. They can differ based on the dataset distribution.
