Answer :
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.
### 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.
