Answer :
To solve the given problem, we need to analyze the income distribution table and determine the required values based on the cumulative distribution of the number of persons in each income group.
Here is the table provided:
| Income (Rs 000) | [tex]$0-4$[/tex] | [tex]$5-9$[/tex] | [tex]$10-14$[/tex] | [tex]$15-19$[/tex] | [tex]$20-24$[/tex] |
|-----------------|-------|-------|---------|---------|---------|
| No. of persons | 160 | 200 | 430 | 140 | 70 |
Let's start with the cumulative frequency distribution of the number of persons:
1. Cumulative frequency distribution:
- For income group [tex]$0-4$[/tex]: 160
- For income group [tex]$5-9$[/tex]: 160 + 200 = 360
- For income group [tex]$10-14$[/tex]: 360 + 430 = 790
- For income group [tex]$15-19$[/tex]: 790 + 140 = 930
- For income group [tex]$20-24$[/tex]: 930 + 70 = 1000
The total number of people is 1000.
### Step-by-Step Solution:
(1) Lowest income of the richest 10% of the people:
The richest 10% of the people constitute the top 100 persons (10% of 1000 = 100).
To find the lowest income of these 100 persons, we need to find the income group where the cumulative frequency exceeds 900 (i.e., 1000 - 100 = 900).
From the cumulative frequency distribution:
- Cumulative frequency of 790 corresponds to the income group [tex]$10-14$[/tex].
- Cumulative frequency of 930 corresponds to the income group [tex]$15-19$[/tex].
Since we need to include at least 100 people from the top when moving from 790 to 930, we can conclude that the lowest income of the richest 10% falls in the income group [tex]$15-19$[/tex].
Therefore, the lowest income of the richest 10% of the people is 14 (the lower boundary of the income group [tex]$15-19$[/tex]).
(2) Highest income of the poorest 40% of the people:
The poorest 40% of the people constitute the bottom 400 persons (40% of 1000 = 400).
To find the highest income of these 400 persons, we look for the income group where the cumulative frequency just reaches or exceeds 400.
From the cumulative frequency distribution:
- Cumulative frequency of 360 corresponds to the income group [tex]$5-9$[/tex].
- Cumulative frequency of 790 exceeds 400 which includes part of the income group [tex]$10-14$[/tex].
Since 400 lies between 360 and 790, this falls within the income group [tex]$10-14$[/tex].
Therefore, the highest income of the poorest 40% of the people is 14 (the upper boundary of the income group [tex]$10-14$[/tex]).
(3) Range of income of the middle 60% of the people:
The middle 60% of the people is exactly the group between the 20th percentile and the 80th percentile.
- Position at the 20th percentile (0.2 1000 = 200 people)
- Position at the 80th percentile (0.8 1000 = 800 people)
We need to determine the income group corresponding to the 200th person and the 800th person.
From the cumulative frequency distribution:
- For position 200: This lies in the income group [tex]$5-9$[/tex], since cumulative frequency exceeds 200 after the first group [tex]$0-4$[/tex] (160 + 200).
- For position 800: This lies in the income group [tex]$10-14$[/tex], since cumulative frequency reaches 800 within this group (360 + 430).
Therefore, the range of income for the middle 60% of the people is from 9 (upper boundary of [tex]$5-9$[/tex]) to 14 (lower boundary of [tex]$15-19$[/tex]).
### Final Answers:
1. The lowest income of the richest 10% of the people is 14.
2. The highest income of the poorest 40% of the people is 14.
3. The range of income of the middle 60% of the people is (9, 14).
Here is the table provided:
| Income (Rs 000) | [tex]$0-4$[/tex] | [tex]$5-9$[/tex] | [tex]$10-14$[/tex] | [tex]$15-19$[/tex] | [tex]$20-24$[/tex] |
|-----------------|-------|-------|---------|---------|---------|
| No. of persons | 160 | 200 | 430 | 140 | 70 |
Let's start with the cumulative frequency distribution of the number of persons:
1. Cumulative frequency distribution:
- For income group [tex]$0-4$[/tex]: 160
- For income group [tex]$5-9$[/tex]: 160 + 200 = 360
- For income group [tex]$10-14$[/tex]: 360 + 430 = 790
- For income group [tex]$15-19$[/tex]: 790 + 140 = 930
- For income group [tex]$20-24$[/tex]: 930 + 70 = 1000
The total number of people is 1000.
### Step-by-Step Solution:
(1) Lowest income of the richest 10% of the people:
The richest 10% of the people constitute the top 100 persons (10% of 1000 = 100).
To find the lowest income of these 100 persons, we need to find the income group where the cumulative frequency exceeds 900 (i.e., 1000 - 100 = 900).
From the cumulative frequency distribution:
- Cumulative frequency of 790 corresponds to the income group [tex]$10-14$[/tex].
- Cumulative frequency of 930 corresponds to the income group [tex]$15-19$[/tex].
Since we need to include at least 100 people from the top when moving from 790 to 930, we can conclude that the lowest income of the richest 10% falls in the income group [tex]$15-19$[/tex].
Therefore, the lowest income of the richest 10% of the people is 14 (the lower boundary of the income group [tex]$15-19$[/tex]).
(2) Highest income of the poorest 40% of the people:
The poorest 40% of the people constitute the bottom 400 persons (40% of 1000 = 400).
To find the highest income of these 400 persons, we look for the income group where the cumulative frequency just reaches or exceeds 400.
From the cumulative frequency distribution:
- Cumulative frequency of 360 corresponds to the income group [tex]$5-9$[/tex].
- Cumulative frequency of 790 exceeds 400 which includes part of the income group [tex]$10-14$[/tex].
Since 400 lies between 360 and 790, this falls within the income group [tex]$10-14$[/tex].
Therefore, the highest income of the poorest 40% of the people is 14 (the upper boundary of the income group [tex]$10-14$[/tex]).
(3) Range of income of the middle 60% of the people:
The middle 60% of the people is exactly the group between the 20th percentile and the 80th percentile.
- Position at the 20th percentile (0.2 1000 = 200 people)
- Position at the 80th percentile (0.8 1000 = 800 people)
We need to determine the income group corresponding to the 200th person and the 800th person.
From the cumulative frequency distribution:
- For position 200: This lies in the income group [tex]$5-9$[/tex], since cumulative frequency exceeds 200 after the first group [tex]$0-4$[/tex] (160 + 200).
- For position 800: This lies in the income group [tex]$10-14$[/tex], since cumulative frequency reaches 800 within this group (360 + 430).
Therefore, the range of income for the middle 60% of the people is from 9 (upper boundary of [tex]$5-9$[/tex]) to 14 (lower boundary of [tex]$15-19$[/tex]).
### Final Answers:
1. The lowest income of the richest 10% of the people is 14.
2. The highest income of the poorest 40% of the people is 14.
3. The range of income of the middle 60% of the people is (9, 14).
