The problem requires us to use a cumulative frequency diagram to estimate the median, the lower quartile, the interquartile range, and the number of houses with a floor area greater than $120 m^2$. The cumulative frequency represents the number of houses, with a total of 80 houses in the sample.

1. Problem Description

The problem requires us to use a cumulative frequency diagram to estimate the median, the lower quartile, the interquartile range, and the number of houses with a floor area greater than $120 m^2$. The cumulative frequency represents the number of houses, with a total of 80 houses in the sample.

2. Solution Steps

(i) The median corresponds to the value at the 50th percentile, which is half of the total frequency. Since there are 80 houses, the median corresponds to a cumulative frequency of $80/2 = 40$. From the graph, the floor area corresponding to a cumulative frequency of 40 is approximately $90 m^2$.

(ii) The lower quartile corresponds to the value at the 25th percentile, which is one-quarter of the total frequency. This corresponds to a cumulative frequency of $80/4 = 20$. From the graph, the floor area corresponding to a cumulative frequency of 20 is approximately $74 m^2$.

(iii) The upper quartile corresponds to the value at the 75th percentile, which is three-quarters of the total frequency. This corresponds to a cumulative frequency of $(3/4) \times 80 = 60$. From the graph, the floor area corresponding to a cumulative frequency of 60 is approximately $110 m^2$. The interquartile range (IQR) is the difference between the upper quartile and the lower quartile:

$IQR = Upper Quartile - Lower Quartile$

$IQR = 110 - 74 = 36 m^2$

(iv) To find the number of houses with a floor area greater than $120 m^2$, we first find the number of houses with a floor area less than or equal to $120 m^2$. From the graph, the cumulative frequency corresponding to a floor area of $120 m^2$ is approximately

6

4. Therefore, the number of houses with a floor area greater than $120 m^2$ is:

$80 - 64 = 16$

3. Final Answer

(i) The median: $90 m^2$

(ii) The lower quartile: $74 m^2$

(iii) The interquartile range: $36 m^2$

(iv) The number of houses with a floor area greater than $120 m^2$: $16$