The problem presents a distribution of housing based on the number of rooms $x$. The table provides the number of rooms and the corresponding frequencies (number of houses with that many rooms). The problem asks to: 1. Identify the characteristic being studied, its nature, and modalities.

1. Problem Description

The problem presents a distribution of housing based on the number of rooms $x$. The table provides the number of rooms and the corresponding frequencies (number of houses with that many rooms). The problem asks to:

1. Identify the characteristic being studied, its nature, and modalities.

2. Represent the data using a bar chart, frequency polygon, and cumulative distribution chart.

3. Determine the mode, median, and quartiles $Q_1$ and $Q_3$.

4. Calculate the arithmetic mean, variance, standard deviation, and coefficient of variation.

2. Solution Steps

1. (a) The characteristic studied is the number of rooms in a house.

(b) The nature of the characteristic is quantitative discrete.

(c) The modalities of the characteristic are 1, 2, 3, 4, 5, 6+.

2. The diagrams (bar chart, frequency polygon, and cumulative distribution chart) are not possible to be plotted here. The charts can be plotted by plotting the Number of rooms on x-axis and the corresponding frequencies on y-axis to generate bar chart and the frequency polygon. The cumulative frequencies can be calculated and used to plot the cumulative distribution.

3. (a) Mode: The mode is the value with the highest frequency. In this case, the highest frequency is 27, which corresponds to 3 rooms. So, the mode is

3. (b) Median: The total number of houses is $15 + 24 + 27 + 19 + 9 + 6 = 100$. The median is the value that divides the data into two halves. Since the total number of houses is 100, the median will be the average of the 50th and 51st values.

The cumulative frequencies are:

- 1 room: 15

- 2 rooms: $15 + 24 = 39$

- 3 rooms: $39 + 27 = 66$

Since the 50th and 51st values fall within the "3 rooms" category, the median is

3. (c) Quartiles:

- $Q_1$: The first quartile is the value at the 25th percentile. $25\%$ of $100 = 25$. The cumulative frequency reaches 25 in the "2 rooms" category. So, $Q_1 = 2$.

- $Q_3$: The third quartile is the value at the 75th percentile. $75\%$ of $100 = 75$. The cumulative frequencies are calculated earlier:

- 1 room: 15

- 2 rooms: $15 + 24 = 39$

- 3 rooms: $39 + 27 = 66$

- 4 rooms: $66 + 19 = 85$

Since the 75th value falls within the "4 rooms" category, $Q_3 = 4$.

4. (a) Arithmetic Mean ($\mu$):

To calculate the mean, we need to estimate a value for the "6+" category. Let's assume it is

6. $\mu = \frac{(1 \times 15) + (2 \times 24) + (3 \times 27) + (4 \times 19) + (5 \times 9) + (6 \times 6)}{100}$

$\mu = \frac{15 + 48 + 81 + 76 + 45 + 36}{100} = \frac{301}{100} = 3.01$

(b) Variance ($\sigma^2$):

$\sigma^2 = \frac{\sum (x_i - \mu)^2 \times f_i}{N}$

$\sigma^2 = \frac{[(1-3.01)^2 \times 15] + [(2-3.01)^2 \times 24] + [(3-3.01)^2 \times 27] + [(4-3.01)^2 \times 19] + [(5-3.01)^2 \times 9] + [(6-3.01)^2 \times 6]}{100}$

$\sigma^2 = \frac{[4.0401 \times 15] + [1.0201 \times 24] + [0.0001 \times 27] + [0.9801 \times 19] + [3.9601 \times 9] + [8.9401 \times 6]}{100}$

$\sigma^2 = \frac{60.6015 + 24.4824 + 0.0027 + 18.6219 + 35.6409 + 53.6406}{100} = \frac{192.99}{100} = 1.93$ (approx.)

(c) Standard Deviation ($\sigma$):

$\sigma = \sqrt{\sigma^2} = \sqrt{1.93} \approx 1.39$

(d) Coefficient of Variation (CV):

$CV = \frac{\sigma}{\mu} \times 100$

$CV = \frac{1.39}{3.01} \times 100 \approx 46.18\%$

3. Final Answer

1. (a) The characteristic studied is the number of rooms in a house.

(b) The nature of the characteristic is quantitative discrete.

(c) The modalities of the characteristic are 1, 2, 3, 4, 5, 6+.

2. Diagrams were not plotted.

3. Mode = 3, Median = 3, $Q_1 = 2$, $Q_3 = 4$.

4. Mean = 3.01, Variance = 1.93, Standard Deviation = 1.39, Coefficient of Variation = 46.18%.