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

[tex]\[
\begin{tabular}{c|c}
\text{Distance} \, (km) & \text{Cumulative frequency} \\
\hline
0 \ \textless \ d \leq 5 & 8 \\
0 \ \textless \ d \leq 10 & 30 \\
0 \ \textless \ d \leq 15 & 70 \\
0 \ \textless \ d \leq 20 & 80 \\
\end{tabular}
\][/tex]

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

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

Sagot :

GinnyAnswer
Alright, let's tackle the problem step-by-step:

### Part a) Drawing a Cumulative Frequency Diagram

To draw a cumulative frequency diagram, follow these steps:

1. Plot the Cumulative Frequency Points:
- For [tex]\(0 < d \leq 5\)[/tex], the cumulative frequency is 8.
- For [tex]\(0 < d \leq 10\)[/tex], the cumulative frequency is 30.
- For [tex]\(0 < d \leq 15\)[/tex], the cumulative frequency is 70.
- For [tex]\(0 < d \leq 20\)[/tex], the cumulative frequency is 80.

2. Mark these points on a graph paper:
- On the x-axis (Distance in km): 5, 10, 15, 20
- On the y-axis (Cumulative frequency): 8, 30, 70, 80

3. Plot the points:
- (5, 8)
- (10, 30)
- (15, 70)
- (20, 80)

4. Join the points with straight lines:
- Draw straight lines between the points in the order they appear, connecting (5, 8) to (10, 30), from (10, 30) to (15, 70), and from (15, 70) to (20, 80).

### Part b) Estimating the Median and the Interquartile Range

#### 1. Find the Median

To estimate the median from the cumulative frequency diagram:

- Median corresponds to the [tex]\(n/2\)[/tex]th value in the ordered dataset, where [tex]\(n\)[/tex] is the total number of deliveries.
- In this case, the total cumulative frequency is 80 (sum total of deliveries).

Therefore, the Median is at the position:
[tex]\[ \text{Median position} = \frac{80}{2} = 40 \][/tex]

On the cumulative frequency diagram, locate 40 on the y-axis and draw a horizontal line to intersect the curve. Then drop down a vertical line to the x-axis. The distance at this point is your median estimate.

From the diagram information and calculations, the estimate for the median distance is 11 km.

#### 2. Find the Interquartile Range (IQR)

To estimate the IQR:

1. First Quartile (Q1):
- Position of Q1: [tex]\( \frac{1}{4} \times 80 = 20 \)[/tex]
- Locate y = 20 on the y-axis and find the corresponding x value. This gives an approximate delivery distance at the first quartile.

2. Third Quartile (Q3):
- Position of Q3: [tex]\( \frac{3}{4} \times 80 = 60 \)[/tex]
- Locate y = 60 on the y-axis and find the corresponding x value. This gives an approximate delivery distance at the third quartile.

From the diagram:

- Q1 is approximately 8 km.
- Q3 is approximately 13 km.

Finally, calculate the IQR:
[tex]\[ \text{IQR} = Q3 - Q1 = 13 - 8 = 5 \][/tex]

### Summary

- The estimated median distance is 11 km.
- The Interquartile Range (IQR) is 5 km.
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