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A local delivery company has a cumulative frequency table to show the distance it travels to deliver parcels.

\begin{tabular}{c|c}
\begin{tabular}{c}
Distance \\
[tex]$( km )$[/tex]
\end{tabular} & \begin{tabular}{c}
Cumulative \\
frequency
\end{tabular} \\
\hline
[tex]$0\ \textless \ d \leq 5$[/tex] & 6 \\
[tex]$0\ \textless \ d \leq 10$[/tex] & 30 \\
[tex]$0\ \textless \ d \leq 15$[/tex] & 60 \\
[tex]$0\ \textless \ d \leq 20$[/tex] & 80
\end{tabular}

a) On graph paper, draw a cumulative frequency diagram to show this information. Join your points with straight lines.

b) Use your diagram to estimate the median and the interquartile range (IQR) of the delivery distances.

Median: [tex]$\square$[/tex]

IQR: [tex]$\square$[/tex]

Sagot :

GinnyAnswer
### 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
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