### Part a) Drawing the Cumulative Frequency Diagram
1. Create the Axes:
- Horizontal Axis (X-axis): Label it as "Distance (km)" and mark intervals from 0 to 20 km.
- Vertical Axis (Y-axis): Label it as "Cumulative Frequency" and mark intervals from 0 to 80.
2. Plot the Points on the Graph:
- Plot the point at (5 km, 6)
- Plot the point at (10 km, 30)
- Plot the point at (15 km, 60)
- Plot the point at (20 km, 80)
3. Join the Points with Straight Lines:
- Connect the points in sequence: (0,0) to (5,6), then (5,6) to (10,30), then (10,30) to (15,60), and finally (15,60) to (20,80).
### Part b) Estimating the Median and the Interquartile Range (IQR)
#### Finding the Median
1. Total Number of Observations:
- There are 80 total deliveries.
2. Median Position:
- The median is located at the 50th percentile, which is at the 0.5 80 = 40th observation.
3. Locate the 40th Observation on the Graph:
- Draw a horizontal line from the cumulative frequency value of 40 to intersect the cumulative frequency curve.
- Drop a vertical line from this intersection point down to the x-axis to read the corresponding distance.
- The distance corresponding to the 40th observation (from the graph) is approximately 11.67 km.
#### Finding Q1 and Q3
1. First Quartile (Q1) Position:
- Q1 is located at the 25th percentile, which is at the 0.25 80 = 20th observation.
2. Locate the 20th Observation on the Graph:
- Draw a horizontal line from the cumulative frequency value of 20 to intersect the cumulative frequency curve.
- Drop a vertical line from this intersection point down to the x-axis to read the corresponding distance.
- The distance corresponding to the 20th observation (from the graph) is approximately 5.75 km.
3. Third Quartile (Q3) Position:
- Q3 is located at the 75th percentile, which is at the 0.75 * 80 = 60th observation.
4. Locate the 60th Observation on the Graph:
- Draw a horizontal line from the cumulative frequency value of 60 to intersect the cumulative frequency curve.
- Drop a vertical line from this intersection point down to the x-axis to read the corresponding distance.
- The distance corresponding to the 60th observation (from the graph) is approximately 12.83 km.
5. Calculate the Interquartile Range (IQR):
- IQR = Q3 - Q1
- IQR ≈ 12.83 km - 5.75 km = 7.08 km
### Answers
- Median: 11.67 km
- IQR: 7.08 km
