Answer :
Sure! Let's solve this step by step.
### Part (a): Drawing the cumulative frequency diagram
To draw a cumulative frequency diagram:
1. Plot the points: Use the given cumulative frequency table to plot points on a graph. The x-axis represents the distance (km) and the y-axis represents the cumulative frequency.
[tex]\( (5, 10) \)[/tex]
[tex]\( (10, 30) \)[/tex]
[tex]\( (15, 50) \)[/tex]
[tex]\( (20, 80) \)[/tex]
2. Draw the axes: Label the x-axis as 'Distance (km)' and the y-axis as 'Cumulative Frequency'.
3. Join the points: Connect the points with straight lines to form the cumulative frequency curve.
### Part (b): Estimating the median and the interquartile range
From the cumulative frequency table, we know the total number of deliveries is [tex]\( 80 \)[/tex].
#### Find the median position:
1. Median position: The median is the 50th percentile of the data. For a total of [tex]\( 80 \)[/tex] deliveries, the median position is:
[tex]\[ \text{Median position} = \frac{80}{2} = 40 \][/tex]
2. Locate the median on the graph: Find the y-value [tex]\( 40 \)[/tex] on your cumulative frequency curve. Draw a horizontal line from [tex]\( 40 \)[/tex] on the y-axis to where it intersects the curve, then draw a vertical line down to the x-axis to find the median distance.
In this case, estimating visually from the points provided:
- At [tex]\( 10 \)[/tex] km, the cumulative frequency is [tex]\( 30 \)[/tex].
- At [tex]\( 15 \)[/tex] km, the cumulative frequency is [tex]\( 50 \)[/tex].
Hence, the median distance is between [tex]\( 10 \)[/tex] km and [tex]\( 15 \)[/tex] km. Interpolating between these points:
[tex]\[ \text{Median distance} \approx 12.5 \, \text{km} \quad \text{(midpoint between 10 and 15 km)} \][/tex]
#### Find the interquartile range (IQR):
1. Q1 and Q3 positions: The interquartile range is between the first quartile (Q1) and the third quartile (Q3):
[tex]\[ \text{Q1 position} = \frac{80}{4} = 20 \][/tex]
[tex]\[ \text{Q3 position} = \frac{3 \times 80}{4} = 60 \][/tex]
2. Locate Q1 and Q3 on the graph:
- For [tex]\( Q1 \)[/tex], find the y-value [tex]\( 20 \)[/tex] on your cumulative frequency curve. Interpolating:
- At [tex]\( 5 \)[/tex] km, the cumulative frequency is [tex]\( 10 \)[/tex].
- At [tex]\( 10 \)[/tex] km, the cumulative frequency is [tex]\( 30 \)[/tex].
Hence, [tex]\( Q1 \)[/tex] is between [tex]\( 5 \)[/tex] km and [tex]\( 10 \)[/tex] km. Interpolating between these points, it would be closer to [tex]\( 5 \)[/tex] km since [tex]\( 20 \)[/tex] is closer to [tex]\( 10 \)[/tex]:
[tex]\[ Q1 \approx 7.5 \, \text{km} \][/tex]
- For [tex]\( Q3 \)[/tex], find the y-value [tex]\( 60 \)[/tex] on your cumulative frequency curve. Interpolating:
- At [tex]\( 15 \)[/tex] km, the cumulative frequency is [tex]\( 50 \)[/tex].
- At [tex]\( 20 \)[/tex] km, the cumulative frequency is [tex]\( 80 \)[/tex].
Hence, [tex]\( Q3 \)[/tex] is between [tex]\( 15 \)[/tex] km and [tex]\( 20 \)[/tex] km. Interpolating between these points:
[tex]\[ Q3 \approx 17.5 \, \text{km} \][/tex]
3. Calculate the IQR: Subtract [tex]\( Q1 \)[/tex] from [tex]\( Q3 \)[/tex]:
[tex]\[ \text{IQR} = Q3 - Q1 = 17.5 \, \text{km} - 7.5 \, \text{km} = 10 \, \text{km} \][/tex]
Thus, the median delivery distance is approximately [tex]\( 12.5 \)[/tex] km, and the interquartile range (IQR) is [tex]\( 10 \)[/tex] km.
### Part (a): Drawing the cumulative frequency diagram
To draw a cumulative frequency diagram:
1. Plot the points: Use the given cumulative frequency table to plot points on a graph. The x-axis represents the distance (km) and the y-axis represents the cumulative frequency.
[tex]\( (5, 10) \)[/tex]
[tex]\( (10, 30) \)[/tex]
[tex]\( (15, 50) \)[/tex]
[tex]\( (20, 80) \)[/tex]
2. Draw the axes: Label the x-axis as 'Distance (km)' and the y-axis as 'Cumulative Frequency'.
3. Join the points: Connect the points with straight lines to form the cumulative frequency curve.
### Part (b): Estimating the median and the interquartile range
From the cumulative frequency table, we know the total number of deliveries is [tex]\( 80 \)[/tex].
#### Find the median position:
1. Median position: The median is the 50th percentile of the data. For a total of [tex]\( 80 \)[/tex] deliveries, the median position is:
[tex]\[ \text{Median position} = \frac{80}{2} = 40 \][/tex]
2. Locate the median on the graph: Find the y-value [tex]\( 40 \)[/tex] on your cumulative frequency curve. Draw a horizontal line from [tex]\( 40 \)[/tex] on the y-axis to where it intersects the curve, then draw a vertical line down to the x-axis to find the median distance.
In this case, estimating visually from the points provided:
- At [tex]\( 10 \)[/tex] km, the cumulative frequency is [tex]\( 30 \)[/tex].
- At [tex]\( 15 \)[/tex] km, the cumulative frequency is [tex]\( 50 \)[/tex].
Hence, the median distance is between [tex]\( 10 \)[/tex] km and [tex]\( 15 \)[/tex] km. Interpolating between these points:
[tex]\[ \text{Median distance} \approx 12.5 \, \text{km} \quad \text{(midpoint between 10 and 15 km)} \][/tex]
#### Find the interquartile range (IQR):
1. Q1 and Q3 positions: The interquartile range is between the first quartile (Q1) and the third quartile (Q3):
[tex]\[ \text{Q1 position} = \frac{80}{4} = 20 \][/tex]
[tex]\[ \text{Q3 position} = \frac{3 \times 80}{4} = 60 \][/tex]
2. Locate Q1 and Q3 on the graph:
- For [tex]\( Q1 \)[/tex], find the y-value [tex]\( 20 \)[/tex] on your cumulative frequency curve. Interpolating:
- At [tex]\( 5 \)[/tex] km, the cumulative frequency is [tex]\( 10 \)[/tex].
- At [tex]\( 10 \)[/tex] km, the cumulative frequency is [tex]\( 30 \)[/tex].
Hence, [tex]\( Q1 \)[/tex] is between [tex]\( 5 \)[/tex] km and [tex]\( 10 \)[/tex] km. Interpolating between these points, it would be closer to [tex]\( 5 \)[/tex] km since [tex]\( 20 \)[/tex] is closer to [tex]\( 10 \)[/tex]:
[tex]\[ Q1 \approx 7.5 \, \text{km} \][/tex]
- For [tex]\( Q3 \)[/tex], find the y-value [tex]\( 60 \)[/tex] on your cumulative frequency curve. Interpolating:
- At [tex]\( 15 \)[/tex] km, the cumulative frequency is [tex]\( 50 \)[/tex].
- At [tex]\( 20 \)[/tex] km, the cumulative frequency is [tex]\( 80 \)[/tex].
Hence, [tex]\( Q3 \)[/tex] is between [tex]\( 15 \)[/tex] km and [tex]\( 20 \)[/tex] km. Interpolating between these points:
[tex]\[ Q3 \approx 17.5 \, \text{km} \][/tex]
3. Calculate the IQR: Subtract [tex]\( Q1 \)[/tex] from [tex]\( Q3 \)[/tex]:
[tex]\[ \text{IQR} = Q3 - Q1 = 17.5 \, \text{km} - 7.5 \, \text{km} = 10 \, \text{km} \][/tex]
Thus, the median delivery distance is approximately [tex]\( 12.5 \)[/tex] km, and the interquartile range (IQR) is [tex]\( 10 \)[/tex] km.
