Answer :
Alright, let's analyze the population trends for these animals in both Yellowstone and Everglades national parks. We will calculate the slopes of the population trends over time to understand whether the populations are increasing or decreasing.
### Data for Yellowstone:
- Years (x): 0, 1, 2, 3, 4, 5, 8, 10
- Grizzly Bears (y1): 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 8.0, 9.0
- Pocket Gophers (y2): 4.0, 4.5, 5.3, 6.4, 8.1, 10.6, 28.6, 60.7
- Osprey (y3): 4.0, 5.8, 7.2, 8.2, 8.8, 9.0, 7.2, 4.0
### Data for Everglades:
- Years (x): 0, 1, 2, 3, 4, 5, 8, 10
- American Alligator (y1): 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 10.0, 11.0
- Marsh Rabbit (y2): 6.0, 6.5, 7.3, 8.4, 10.1, 12.6, 30.6, 62.7
- Florida Panther (y3): 6.0, 7.8, 9.0, 9.8, 10.0, 9.8, 6.0, 1.0
### Calculation of Slopes:
The slope for each animal population can be found using the formula:
[tex]\[ \text{slope} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \][/tex]
#### Yellowstone Slopes:
1. Grizzly Bears:
- Slope: 0.5
2. Pocket Gophers:
- Slope: 5.14841628959276
3. Osprey:
- Slope: -0.011764705882353215
#### Everglades Slopes:
1. American Alligator:
- Slope: 0.5
2. Marsh Rabbit:
- Slope: 5.14841628959276
3. Florida Panther:
- Slope: -0.516742081447964
### Interpretation:
- Yellowstone:
- Grizzly Bears: The population is increasing at a rate of 0.5 bears per square mile per year.
- Pocket Gophers: The population is increasing rapidly at a rate of approximately 5.15 gophers per square mile per year.
- Osprey: The population is essentially stable with a very slight decrease of about -0.012 ospreys per square mile per year.
- Everglades:
- American Alligator: The population is increasing at a rate of 0.5 alligators per square mile per year.
- Marsh Rabbit: The population is increasing rapidly at a rate of approximately 5.15 rabbits per square mile per year.
- Florida Panther: The population is decreasing at a rate of approximately -0.52 panthers per square mile per year.
### Conclusion:
From the slopes calculated, we observe general trends in animal populations in both parks. The populations of Grizzly Bears and American Alligators are increasing moderately, while Pocket Gophers and Marsh Rabbits show a significant increase in their numbers. On the contrary, the Osprey population in Yellowstone is stable with a slight decrease, and the Florida Panther population in Everglades is decreasing notably.
These trends can help park management understand and possibly intervene to support animal populations effectively.
### Data for Yellowstone:
- Years (x): 0, 1, 2, 3, 4, 5, 8, 10
- Grizzly Bears (y1): 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 8.0, 9.0
- Pocket Gophers (y2): 4.0, 4.5, 5.3, 6.4, 8.1, 10.6, 28.6, 60.7
- Osprey (y3): 4.0, 5.8, 7.2, 8.2, 8.8, 9.0, 7.2, 4.0
### Data for Everglades:
- Years (x): 0, 1, 2, 3, 4, 5, 8, 10
- American Alligator (y1): 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 10.0, 11.0
- Marsh Rabbit (y2): 6.0, 6.5, 7.3, 8.4, 10.1, 12.6, 30.6, 62.7
- Florida Panther (y3): 6.0, 7.8, 9.0, 9.8, 10.0, 9.8, 6.0, 1.0
### Calculation of Slopes:
The slope for each animal population can be found using the formula:
[tex]\[ \text{slope} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \][/tex]
#### Yellowstone Slopes:
1. Grizzly Bears:
- Slope: 0.5
2. Pocket Gophers:
- Slope: 5.14841628959276
3. Osprey:
- Slope: -0.011764705882353215
#### Everglades Slopes:
1. American Alligator:
- Slope: 0.5
2. Marsh Rabbit:
- Slope: 5.14841628959276
3. Florida Panther:
- Slope: -0.516742081447964
### Interpretation:
- Yellowstone:
- Grizzly Bears: The population is increasing at a rate of 0.5 bears per square mile per year.
- Pocket Gophers: The population is increasing rapidly at a rate of approximately 5.15 gophers per square mile per year.
- Osprey: The population is essentially stable with a very slight decrease of about -0.012 ospreys per square mile per year.
- Everglades:
- American Alligator: The population is increasing at a rate of 0.5 alligators per square mile per year.
- Marsh Rabbit: The population is increasing rapidly at a rate of approximately 5.15 rabbits per square mile per year.
- Florida Panther: The population is decreasing at a rate of approximately -0.52 panthers per square mile per year.
### Conclusion:
From the slopes calculated, we observe general trends in animal populations in both parks. The populations of Grizzly Bears and American Alligators are increasing moderately, while Pocket Gophers and Marsh Rabbits show a significant increase in their numbers. On the contrary, the Osprey population in Yellowstone is stable with a slight decrease, and the Florida Panther population in Everglades is decreasing notably.
These trends can help park management understand and possibly intervene to support animal populations effectively.
