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

| Distance (km) | Cumulative Frequency |
|---------------|----------------------|
| [tex]\(0 \ \textless \ d \leq 5\)[/tex] | 10 |
| [tex]\(0 \ \textless \ d \leq 10\)[/tex] | 30 |
| [tex]\(0 \ \textless \ d \leq 15\)[/tex] | 50 |
| [tex]\(0 \ \textless \ d \leq 20\)[/tex] | 80 |

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

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

- Median: ________
- IQR: ________

Sagot :

Certainly! Let's walk through the steps to find the median and interquartile range (IQR) of the delivery distances using the cumulative frequency table provided.

### Step-by-Step Solution:

a) Drawing the cumulative graph:

1. Plot the Points:
- Plot cumulative frequency of 10 at the upper bound of the first interval, 5 km.
- Plot cumulative frequency of 30 at the upper bound of the second interval, 10 km.
- Plot cumulative frequency of 50 at the upper bound of the third interval, 15 km.
- Plot cumulative frequency of 80 at the upper bound of the fourth interval, 20 km.

2. Join the Points:
- Once the points are plotted on the graph paper, join the points with straight lines to form the cumulative frequency diagram.

You now have a cumulative frequency graph which you can use to estimate the median and quartiles.

b) Estimating the median and interquartile range from the graph:

1. Finding the Median (50th percentile):
- The total number of deliveries (N) is given by the last cumulative frequency: 80.
- The median is the value at the 50th percentile, which is at [tex]\( N/2 = 80/2 = 40 \)[/tex].
- On the cumulative frequency graph, draw a horizontal line from 40 on the cumulative frequency axis until it intersects the cumulative frequency curve.
- From this intersection point, draw a vertical line down to the distance axis to find the median delivery distance.

2. Finding the First Quartile (25th percentile):
- The first quartile (Q1) is the value at the 25th percentile, which is at [tex]\( N/4 = 80/4 = 20 \)[/tex].
- On the cumulative frequency graph, draw a horizontal line from 20 on the cumulative frequency axis until it intersects the cumulative frequency curve.
- From this intersection point, draw a vertical line down to the distance axis to find the first quartile delivery distance.

3. Finding the Third Quartile (75th percentile):
- The third quartile (Q3) is the value at the 75th percentile, which is at [tex]\( 3N/4 = 3 \times 80/4 = 60 \)[/tex].
- On the cumulative frequency graph, draw a horizontal line from 60 on the cumulative frequency axis until it intersects the cumulative frequency curve.
- From this intersection point, draw a vertical line down to the distance axis to find the third quartile delivery distance.

4. Calculating the Interquartile Range (IQR):
- IQR is calculated as [tex]\( Q3 - Q1 \)[/tex].

Results:
- Median Delivery Distance: The median delivery distance is found to be 7.5 km.
- Interquartile Range (IQR): The IQR is found to be 1.67 km.

So, the estimated median delivery distance from the cumulative frequency table is 7.5 km, and the interquartile range is 1.67 km.