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

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

For the intervals:

• \(0 \leq t < 5\): \(\frac{23}{5} = 4.6\)
• \(5 \leq t < 10\): \(\frac{102}{5} = 20.4\)
• \(10 \leq t < 20\): \(\frac{135}{10} = 13.5\)
• \(20 \leq t < 30\): \(\frac{76}{10} = 7.6\)
• \(30 \leq t < 50\): \(\frac{24}{20} = 1.2\)

Draw bars with these heights, ensuring axes are labeled correctly.

(b) To estimate the mean time, use the midpoints of each interval:

• \(0 \leq t < 5\): midpoint = 2.5
• \(5 \leq t < 10\): midpoint = 7.5
• \(10 \leq t < 20\): midpoint = 15
• \(20 \leq t < 30\): midpoint = 25
• \(30 \leq t < 50\): midpoint = 40

Calculate the mean:

\(\frac{(2.5 \times 23) + (7.5 \times 102) + (15 \times 135) + (25 \times 76) + (40 \times 24)}{360}\)

\(= \frac{5707.5}{360} \approx 15.9\) minutes

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

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

Class widths: 10, 10, 20, 20, 40

Frequency densities: \(\frac{16}{10} = 1.6\), \(\frac{54}{10} = 5.4\), \(\frac{78}{20} = 3.9\), \(\frac{32}{20} = 1.6\), \(\frac{20}{40} = 0.5\)

Draw bars with these heights on the histogram.

(b) To estimate the mean time, use the midpoints of each interval:

Midpoints: 5, 15, 30, 50, 80

Mean = \(\frac{16 \times 5 + 54 \times 15 + 78 \times 30 + 32 \times 50 + 20 \times 80}{200}\)

\(= \frac{80 + 810 + 2340 + 1600 + 1600}{200}\)

\(= \frac{6430}{200} = 32.15\)

Accept 32.2 as the answer.

(c) The greatest possible value of the interquartile range is calculated by taking a value in the upper quartile (40-60) and subtracting a value in the lower quartile (10-20):

\(Greatest possible value = 60 - 10 = 50\)

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

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

Class widths: 5, 5, 10, 20, 30

Frequency densities: \(\frac{10}{5} = 2\), \(\frac{5}{5} = 1\), \(\frac{26}{10} = 2.6\), \(\frac{32}{20} = 1.6\), \(\frac{18}{30} = 0.6\)

Draw bars with these heights for each class interval.

(b) The lower quartile (LQ) is in the 11 – 20 interval, and the upper quartile (UQ) is in the 21 – 40 interval. The greatest possible interquartile range (IQR) is:

\(40 - 11 = 29\)

(c) To estimate the mean, calculate the midpoint of each class interval:

Midpoints: 3, 8, 15.5, 30.5, 55.5

Calculate the mean using:

\(\text{Mean} = \frac{3 \times 10 + 8 \times 5 + 15.5 \times 26 + 30.5 \times 32 + 55.5 \times 18}{91}\)

\(= \frac{30 + 40 + 403 + 976 + 999}{91}\)

\(= \frac{2448}{91} \approx 26.9\)

(a) To draw the histogram, calculate the class widths: \( 10,\ 5,\ 15,\ 20,\ 10 \). Then calculate the frequency density for each class:

\( \text{Frequency density} = \dfrac{\text{frequency}}{\text{class width}} \)

\( 1.8,\ 4.8,\ 2,\ 1,\ 0.8 \)

Ensure all heights are correct on the diagram using a linear scale. The correct bar ends are: \( 10.5,\ 15.5,\ 30.5,\ 50.5,\ 60.5 \).

(b) The greatest possible value of the interquartile range (IQR) is calculated using the class intervals \( 11\text{–}15 \) and \( 31\text{–}50 \). Thus, the greatest IQR \( = 50 - 11 = 39 \).

(c) To calculate the mean:

\( \text{Mean} = \frac{ 18 \times 5.5 + 24 \times 13 + 30 \times 23 + {}\\ 20 \times 40.5 + 8 \times 55.5 }{100}\) \(= \frac{2355}{100} = 23.6 \)

To calculate the variance:

\( \text{Var} = \frac{ 18 \times 5.5^2 + 24 \times 13^2 + 30 \times 23^2 + {}\\ 20 \times 40.5^2 + 8 \times 55.5^2 }{100}\) \(- \text{mean}^2 \)

\( = \frac{77917.5}{100} - 23.6^2\) \(= 224.57 \)

Standard deviation \( = \sqrt{224.57} = 15.0 \).