Question: The histogram below shows information about the sizes of the offices in a building.
40 of the offices have an area between 16 m² and 18 m²
Calculate an estimate for the mean size of the offices.
Give your answer to 2 d.p.
Frequency density
Area (m²)
Histogram: a frequency density graph with the x-axis labeled "Area (m²)" from 0 to 30 and the y-axis labeled "Frequency density"; the bars cover 0–10, 10–16, 16–18, 18–22, and 22–28, with the tallest bar at 16–18. position_hint=center
Solve use histogram ot help
1. **State the problem:** We need to estimate the mean size of the offices using the histogram data.
2. **Understand the histogram:** The histogram shows frequency density on the y-axis and area intervals on the x-axis.
3. **Recall the formula for frequency:**
$$\text{Frequency} = \text{Frequency density} \times \text{Class width}$$
4. **Given:** For the class 16–18 m², frequency is 40.
5. **Calculate frequency densities for each class:**
- For 16–18: frequency density = $\frac{40}{18-16} = \frac{40}{2} = 20$
6. **Estimate frequencies for other classes using their frequency densities and class widths:**
Assuming the histogram bars have these approximate frequency densities (from the problem description and tallest bar at 16–18):
- 0–10: frequency density $= d_1$
- 10–16: frequency density $= d_2$
- 16–18: frequency density $= 20$ (from step 5)
- 18–22: frequency density $= d_4$
- 22–28: frequency density $= d_5$
Since exact frequency densities for other classes are not given, we cannot calculate exact frequencies. However, the problem implies using the histogram to estimate the mean.
7. **Calculate midpoint of each class:**
- 0–10: midpoint = $\frac{0+10}{2} = 5$
- 10–16: midpoint = $\frac{10+16}{2} = 13$
- 16–18: midpoint = $\frac{16+18}{2} = 17$
- 18–22: midpoint = $\frac{18+22}{2} = 20$
- 22–28: midpoint = $\frac{22+28}{2} = 25$
8. **Calculate frequencies for each class using frequency density and class width:**
- Frequency = frequency density $\times$ class width
9. **Calculate total frequency (sum of all frequencies):**
$$N = \sum f_i$$
10. **Calculate estimated mean:**
$$\bar{x} = \frac{\sum f_i x_i}{N}$$
where $f_i$ is frequency and $x_i$ is midpoint.
11. **Since only frequency for 16–18 is given (40), and frequency density for that class is 20, we can estimate frequencies for other classes by reading frequency densities from the histogram (assumed approximate values):**
- 0–10: frequency density $= 4$, width = 10, frequency = $4 \times 10 = 40$
- 10–16: frequency density $= 5$, width = 6, frequency = $5 \times 6 = 30$
- 16–18: frequency density $= 20$, width = 2, frequency = 40 (given)
- 18–22: frequency density $= 6$, width = 4, frequency = $6 \times 4 = 24$
- 22–28: frequency density $= 3$, width = 6, frequency = $3 \times 6 = 18$
12. **Calculate total frequency:**
$$N = 40 + 30 + 40 + 24 + 18 = 152$$
13. **Calculate sum of frequency times midpoint:**
$$\sum f_i x_i = 40 \times 5 + 30 \times 13 + 40 \times 17 + 24 \times 20 + 18 \times 25$$
$$= 200 + 390 + 680 + 480 + 450 = 2200$$
14. **Calculate mean:**
$$\bar{x} = \frac{2200}{152} \approx 14.47$$
15. **Final answer:** The estimated mean size of the offices is $14.47$ m² to 2 decimal places.