Let's solve each part of the problem systematically.
### Part i: Constructing the Frequency Table
The given data is:
```
28, 35, 61, 29, 36, 48, 59, 67, 69, 50,
48, 40, 49, 42, 41, 37, 51, 62, 63, 33,
31, 32, 35, 40, 38, 39, 60, 51, 54, 56,
69, 46, 42, 38, 61, 59, 58, 44, 39, 57,
38, 44, 45, 45, 47, 38, 44, 47, 47, 64
```
We need to consider class intervals:
- [tex]\( 25-34 \)[/tex]
- [tex]\( 35-44 \)[/tex]
- [tex]\( 45-54 \)[/tex]
- [tex]\( 55-64 \)[/tex]
- [tex]\( 65-74 \)[/tex]
Let's count the number of data points in each interval to form the frequency table.
| Class Interval | Frequency |
|----------------|-----------|
| 25 - 34 | 7 |
| 35 - 44 | 21 |
| 45 - 54 | 11 |
| 55 - 64 | 8 |
| 65 - 74 | 3 |
### Part ii: Number of firms having profit between Rs. 37 lakhs and Rs. 58 lakhs
To answer this, we need to count the number of firms whose profits fall in the range from 37 to 58 lakhs. These profits lie within the class intervals [tex]\(35-44\)[/tex], [tex]\(45-54\)[/tex], and partly in [tex]\(55-64\)[/tex] (excluding those above 58 in that range):
- From [tex]\( 35-44 \)[/tex]: 21
- From [tex]\( 45-54 \)[/tex]: 11
- From [tex]\( 55-64 \)[/tex]: 10 (only counting those from 55 to 58)
Total firms = 21 + 11 + 10 = 42
So, the number of firms having profit between Rs. 37 lakhs and Rs. 58 lakhs is [tex]\( 42 \)[/tex].
### Part iii: Profit above which 10\% of the firm will have their profits
To find the cutoff for the top 10%, we need to find the 90th percentile. Given 50 data points, the [tex]\(90^{th}\)[/tex] percentile is the value below which 90% of the data fall.
Sorted data gives us the 45th value (since [tex]\(90\%\)[/tex] of 50 is 45).
From the sorted list, the profit at the 45th position is Rs. 62.
So, the profit above which 10% of the firms will have their profits is approximately Rs. 62.
### Part iv: Middle 50% profit group
The middle 50% profit group lies between the 25th percentile and the 75th percentile.
- For the 25th percentile (which is the 12.5th firm), rounding should be used based on nearest data: Rs. 38.
- For the 75th percentile (which is the 37.5th firm), as per sorted data, it's approximately at Rs. 56.
So, the middle 50% profit group lies between Rs. 38 and Rs. 56.
### Final Answer
- i. The number of firms having profit between Rs. 37 lakhs and Rs. 58 lakhs: [tex]\(42\)[/tex]
- ii. Profit above which 10% of the firm will have their profits: Rs. 62
- iii. Middle 50% profit group: Between Rs. 38 and Rs. 56
