To determine which equation best models the periodic behavior of the coyote population data given, we need to assess how closely each of the provided equations fits the observed data. The equations in question are:
1. [tex]\( f_1(x) = 25 \cos \left( \frac{\pi}{3} x \right) + 50 \)[/tex]
2. [tex]\( f_2(x) = -25 \cos \left( \frac{\pi}{3} x \right) + 50 \)[/tex]
3. [tex]\( f_3(x) = -50 \cos \left( \frac{\pi}{3} x \right) + 25 \)[/tex]
4. [tex]\( f_4(x) = 50 \cos \left( \frac{\pi}{3} x \right) + 25 \)[/tex]
To evaluate which model fits best, we calculate the sum of squared errors (SSE) for each equation using the given population data. The SSE evaluates the discrepancies between the observed data and the data predicted by the model; the model with the smallest SSE is the best fit.
Given the observed data points:
| Time (years) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|--------------|----|----|----|----|----|----|----|----|----|----|----|----|
| Population | 25 | 40 | 62 | 75 | 66 | 35 | 26 | 38 | 64 | 74 | 61 | 32 |
Calculations of the SSE for each model are as follows:
1. For [tex]\( f_1(x) = 25 \cos \left( \frac{\pi}{3} x \right) + 50 \)[/tex]:
[tex]\[
\text{SSE} = 15261.999999999998
\][/tex]
2. For [tex]\( f_2(x) = -25 \cos \left( \frac{\pi}{3} x \right) + 50 \)[/tex]:
[tex]\[
\text{SSE} = 62.000000000000384
\][/tex]
3. For [tex]\( f_3(x) = -50 \cos \left( \frac{\pi}{3} x \right) + 25 \)[/tex]:
[tex]\[
\text{SSE} = 11111.999999999996
\][/tex]
4. For [tex]\( f_4(x) = 50 \cos \left( \frac{\pi}{3} x \right) + 25 \)[/tex]:
[tex]\[
\text{SSE} = 41512.00000000001
\][/tex]
The model with the smallest sum of squared errors is:
[tex]\[
\text{SSE} = 62.000000000000384
\][/tex]
This corresponds to the equation:
[tex]\[
f_2(x) = -25 \cos \left( \frac{\pi}{3} x \right) + 50
\][/tex]
Thus, the equation [tex]\( f(x) = -25 \cos \left( \frac{\pi}{3} x \right) + 50 \)[/tex] best models the behavior of the given population data.
