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Answer :
Sure! Let's go through the steps to complete the grouped frequency distribution and draw the frequency polygon.
### (a) Complete the Grouped Frequency Distribution
We have temperatures from eighteen cities:
95, 103, 106, 116, 118, 105, 97, 98, 110, 117, 109, 111, 100, 104, 102, 115, 107, 115
We are dividing these into classes, each with a width of 6. The class intervals and their midpoints are as follows:
1. Class Interval 94.5 to 100.5:
- Includes temperatures: 95, 97, 98, 100
- Frequency: 4
2. Class Interval 100.5 to 106.5:
- Includes temperatures: 103, 105, 106, 104, 102
- Frequency: 5
3. Class Interval 106.5 to 112.5:
- Includes temperatures: 110, 109, 111, 107
- Frequency: 4
4. Class Interval 112.5 to 118.5:
- Includes temperatures: 116, 118, 117, 115, 115
- Frequency: 5
Thus, the completed frequency distribution is:
| Temperatures (in [tex]\(^{\circ}F\)[/tex]) | Frequency |
|---------------------------------|-----------|
| 94.5 to 100.5 | 4 |
| 100.5 to 106.5 | 5 |
| 106.5 to 112.5 | 4 |
| 112.5 to 118.5 | 5 |
### (b) Draw the Frequency Polygon
To draw the frequency polygon, we use the midpoints of each class interval. The midpoints are calculated by averaging the upper and lower bounds of each class:
1. Midpoint of 94.5 to 100.5: [tex]\((94.5 + 100.5) / 2 = 97.5\)[/tex]
2. Midpoint of 100.5 to 106.5: [tex]\((100.5 + 106.5) / 2 = 103.5\)[/tex]
3. Midpoint of 106.5 to 112.5: [tex]\((106.5 + 112.5) / 2 = 109.5\)[/tex]
4. Midpoint of 112.5 to 118.5: [tex]\((112.5 + 118.5) / 2 = 115.5\)[/tex]
Using these midpoints and their corresponding frequencies, you can plot a frequency polygon. On a graph, plot points at:
- (97.5, 4)
- (103.5, 5)
- (109.5, 4)
- (115.5, 5)
Connect the points with straight lines to form the frequency polygon. You can start a little before the first midpoint and end a little after the last one, usually to make the polygon complete at both ends. This will give you a visual representation of the distribution of temperatures.
### (a) Complete the Grouped Frequency Distribution
We have temperatures from eighteen cities:
95, 103, 106, 116, 118, 105, 97, 98, 110, 117, 109, 111, 100, 104, 102, 115, 107, 115
We are dividing these into classes, each with a width of 6. The class intervals and their midpoints are as follows:
1. Class Interval 94.5 to 100.5:
- Includes temperatures: 95, 97, 98, 100
- Frequency: 4
2. Class Interval 100.5 to 106.5:
- Includes temperatures: 103, 105, 106, 104, 102
- Frequency: 5
3. Class Interval 106.5 to 112.5:
- Includes temperatures: 110, 109, 111, 107
- Frequency: 4
4. Class Interval 112.5 to 118.5:
- Includes temperatures: 116, 118, 117, 115, 115
- Frequency: 5
Thus, the completed frequency distribution is:
| Temperatures (in [tex]\(^{\circ}F\)[/tex]) | Frequency |
|---------------------------------|-----------|
| 94.5 to 100.5 | 4 |
| 100.5 to 106.5 | 5 |
| 106.5 to 112.5 | 4 |
| 112.5 to 118.5 | 5 |
### (b) Draw the Frequency Polygon
To draw the frequency polygon, we use the midpoints of each class interval. The midpoints are calculated by averaging the upper and lower bounds of each class:
1. Midpoint of 94.5 to 100.5: [tex]\((94.5 + 100.5) / 2 = 97.5\)[/tex]
2. Midpoint of 100.5 to 106.5: [tex]\((100.5 + 106.5) / 2 = 103.5\)[/tex]
3. Midpoint of 106.5 to 112.5: [tex]\((106.5 + 112.5) / 2 = 109.5\)[/tex]
4. Midpoint of 112.5 to 118.5: [tex]\((112.5 + 118.5) / 2 = 115.5\)[/tex]
Using these midpoints and their corresponding frequencies, you can plot a frequency polygon. On a graph, plot points at:
- (97.5, 4)
- (103.5, 5)
- (109.5, 4)
- (115.5, 5)
Connect the points with straight lines to form the frequency polygon. You can start a little before the first midpoint and end a little after the last one, usually to make the polygon complete at both ends. This will give you a visual representation of the distribution of temperatures.
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