We measured the flat area of 30 apartments and we have measured the following values in $$ m^{2} $$ : $$ 82.6,57.3,70.4,65,48.4,103.8,73.6,43.5,66.1,93,52.6,70,84.2,55,81.3,61.5,75.1,34.8 $$, $$ 62.4,116,70.1,63.6,93,59.2,65.9,77.2,52.8,68.7,79.2,87.4 $$. a) Create a table of grouped frequency distribution for the number of classes $$ k=9 $$. b) Construct a histogram of relative frequencies of the flat areas. c) From the specified values estimate the sample mean and the variance. d) From the middle values of the intervals and from their frequencies estimate the sample mean and the variance.

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### Problem a)
**Create a table of grouped frequency distribution for the number of classes $$ k=9 $$.**

To create a grouped frequency distribution with 9 classes for the given data set, follow these steps:

1. **Determine the Range:**
   - **Minimum value ($$ 	ext{Min} $$)**: 34.8
   - **Maximum value ($$ 	ext{Max} $$)**: 116
   - **Range ($$ R $$)**: $$ R = 	ext{Max} - 	ext{Min} = 116 - 34.8 = 81.2 $$

2. **Calculate Class Width:**
   - **Number of classes ($$ k $$)**: 9
   - **Class Width ($$ C $$)**: $$ C = \frac{R}{k} = \frac{81.2}{9} \approx 9.022 $$
   - **Rounded Class Width**: Typically, it's practical to round up to a convenient number. Here, we'll use **10** for simplicity.

3. **Establish Class Intervals:**
   - Starting from 30 (a convenient number below the minimum) to 120 to accommodate the maximum value.
   - **Class Intervals**:
     - 30 – <40
     - 40 – <50
     - 50 – <60
     - 60 – <70
     - 70 – <80
     - 80 – <90
     - 90 – <100
     - 100 – <110
     - 110 – <120

4. **Tally the Frequencies:**
   - Assign each data point to its respective class interval.

5. **Frequency Distribution Table:**

| **Class Interval (m²)** | **Frequency** |
|-------------------------|---------------|
| 30 – <40                | 1             |
| 40 – <50                | 2             |
| 50 – <60                | 5             |
| 60 – <70                | 7             |
| 70 – <80                | 7             |
| 80 – <90                | 4             |
| 90 – <100               | 2             |
| 100 – <110              | 1             |
| 110 – <120              | 1             |
| **Total**               | **30**        |

6. **Frequency Assignment Details:**

- **30 – <40:** 34.8
- **40 – <50:** 43.5, 48.4
- **50 – <60:** 52.6, 52.8, 55, 57.3, 59.2
- **60 – <70:** 61.5, 62.4, 63.6, 65, 65.9, 66.1, 68.7
- **70 – <80:** 70, 70.1, 70.4, 73.6, 75.1, 77.2, 79.2
- **80 – <90:** 81.3, 82.6, 84.2, 87.4
- **90 – <100:** 93, 93
- **100 – <110:** 103.8
- **110 – <120:** 116

This table effectively groups the 30 apartment flat areas into 9 classes, making it easier to analyze the distribution of the data.
**Grouped Frequency Distribution:** | **Class Interval (m²)** | **Frequency** | |-------------------------|---------------| | 30 – <40 | 1 | | 40 – <50 | 2 | | 50 – <60 | 5 | | 60 – <70 | 7 | | 70 – <80 | 7 | | 80 – <90 | 4 | | 90 – <100 | 2 | | 100 – <110 | 1 | | 110 – <120 | 1 | | **Total** | **30** |

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