(i) To draw the histogram, calculate the frequency density for each class using the formula:

\(\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}\)

Class widths: 5, 5, 10, 20, 40 (in thousands)

Frequency densities: 8, 6, 1.8, 1.7, 0.2

Plot these on the histogram with the horizontal axis labeled 'Capacity (1000s)' and the vertical axis labeled 'Frequency Density'.

(ii) Calculate the mean capacity using the midpoints of each class:

\(\text{Mean} = \frac{(5 \times 40) + (10 \times 30) + (17.5 \times 18) + (32.5 \times 34) + (62.5 \times 8)}{130}\)

\(= \frac{2420}{130} = 18.6 \text{ thousand}\)

(iii) To find the median and lower quartile groups:

Median position: \(\frac{130 + 1}{2} = 65.5\)

Lower quartile position: \(\frac{130 + 1}{4} = 32.75\)

The median group is 8000–12000 and the lower quartile group is 3000–7000.

(i) To find the number of journey times between 25 and 45 minutes, subtract the cumulative frequency at 25 minutes from the cumulative frequency at 45 minutes: \(50 - 18 = 32\).

(ii) To draw the histogram, calculate the frequency for each interval: \([0, 10]: 0\), \((10, 25]: 18\), \((25, 45]: 32\), \((45, 60]: 9\), \((60, 80]: 4\). Then, calculate the frequency density by dividing the frequency by the class width: \(\frac{18}{15} = 1.2\), \(\frac{32}{20} = 1.6\), \(\frac{9}{15} = 0.6\), \(\frac{4}{20} = 0.2\). Draw bars with these heights.

(iii) To estimate the mean journey time, use the midpoints of each interval: \((17.5 \times 18) + (35 \times 32) + (52.5 \times 9) + (70 \times 4)\). Sum these products and divide by the total frequency: \(\frac{2187.5}{63} = 34.7\) minutes.

(i) To draw the histogram, calculate the frequency density for each class interval. Frequency density is given by:

\(\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}\)

Class widths: 20, 5, 5, 10, 20

Frequency densities: \(\frac{18}{20} = 0.9\), \(\frac{15}{5} = 3\), \(\frac{21}{5} = 4.2\), \(\frac{52}{10} = 5.2\), \(\frac{28}{20} = 1.4\)

Draw bars with these heights for each class interval.

(ii) Calculate the mean:

Midpoints: 30.5, 43, 48, 55.5, 70.5

Mean: \(\frac{(30.5 \times 18) + (43 \times 15) + (48 \times 21) + (55.5 \times 52) + (70.5 \times 28)}{134} = \frac{7062}{134} = 52.701 \approx 52.7\)

Calculate the variance:

\(\text{Var} = \frac{(30.5^2 \times 18) + (43^2 \times 15) + (48^2 \times 21) + (55.5^2 \times 52) + (70.5^2 \times 28)}{134} - 52.701^2\)

\(= \frac{392203.5}{134} - 52.701^2 = 149.496\)

Standard deviation: \(\sqrt{149.496} = 12.2\)

To find \(a\) and \(b\), we use the formula for frequency density: \(\text{fd} = \frac{f}{\text{cw}}\), where \(f\) is the frequency and \(\text{cw}\) is the class width.

For the class 63 - 64, \(\text{cw} = 2\) and \(f = 9\). Therefore, \(a = \frac{9}{2} = 4.5\).

For the class 68 - 71, \(\text{cw} = 4\) and \(\text{fd} = 1.5\). Therefore, \(1.5 = \frac{b}{4}\), which gives \(b = 6\).

(i) To find the class containing the upper quartile, calculate the position of the upper quartile in the ordered data set. The upper quartile position is given by \( \frac{3}{4} \times 104 = 78 \). Cumulative frequencies are: 8, 33, 61, 92, 104. The 78th value falls in the class 5.5–7.0 cm.

(ii) To draw the histogram, calculate the frequency density for each class using the formula \( \text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}} \).

• Class 2.0–3.5: \( \text{Width} = 1.5,\ \text{Frequency} = 8,\ \text{Density} = \frac{8}{1.5} = 5.33 \)
• Class 3.5–4.5: \( \text{Width} = 1.0,\ \text{Frequency} = 25,\ \text{Density} = 25 \)
• Class 4.5–5.5: \( \text{Width} = 1.0,\ \text{Frequency} = 28,\ \text{Density} = 28 \)
• Class 5.5–7.0: \( \text{Width} = 1.5,\ \text{Frequency} = 31,\ \text{Density} = \frac{31}{1.5} = 20.7 \)
• Class 7.0–9.0: \( \text{Width} = 2.0,\ \text{Frequency} = 12,\ \text{Density} = \frac{12}{2.0} = 6 \)

Draw bars with these heights and ensure correct labels and bar widths with no gaps.