Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Experience the convenience of getting reliable answers to your questions from a vast network of knowledgeable experts. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
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.
### 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.
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
