Wolves were reintroduced into Yellowstone park and their population over the next 10 years. Which model type do you think best fit the data based on the table values which are "years since introduction" and "population".
You have linear regression, exponential regression, and quadratic regression.
I think it would be a linear expression. Am I right?
Here is the table provided:
Years since introduction ---- Population:
2 ---- 9
4 ---- 24
6 ---- 53
8 ---- 119
10 ---- 267
We wish to see whether this table can be represented by a linear, quadratic, or exponential function.
If this table could be represented by a linear function, then we would have that
(week 6 population) - (week 4 population) = (week 4 population) - (week 2 population)
because if it was linear the differences between two consecutive weeks would remain constant.(We can just say that the weeks are consecutive because they always increase by 2, and treat "consecutive" as "seperated by 2 weeks")
We can plug in for the populations to get:
$53-24=24-9$, which simplifies to
$29=13$, which is clearly false. So, it cannot be represented by a linear function.
Here is a neat trick for determining if a table can be represented by a quadratic function(or any higher power).
We can do this here! From here, we can just calculate the differences(We only need 3, can you see why? But I will calculate all of them to make sure)
$24-9=13$
$53-24=29$
$119-53=66$
$267-119=148$
By looking at these values, we can calculate the differences of the differences and evaluate them to see if they are linear.
$66-29=29-13$, which gives
$37=16$, which is obviously false. So, the table cannot be represented by a quadratic function.
The only other option is for it to be represented by an exponential function, so the answer is $\boxed{\text{exponential}}$