To address the question of whether the predicted number of owls in year 8 makes sense, we need to follow a detailed analysis.
1. Understanding the Data and the Model:
- We have data points representing the number of owls in different years:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Year (x)} & \text{Number of Owls (y)} \\
\hline 1 & 0 \\
\hline 2 & 2 \\
\hline 3 & 7 \\
\hline 4 & 14 \\
\hline 5 & 9 \\
\hline 6 & 4 \\
\hline
\end{array}
\][/tex]
- We aim to predict the number of owls in year 8 using a polynomial model fitted to the data.
2. Analyzing the Polynomial Model:
- A quadratic (second-order polynomial) model has been fitted to the data points. This model can be expressed in the form:
[tex]\[
y = ax^2 + bx + c
\][/tex]
- The coefficients for this quadratic model are approximately:
[tex]\[
a \approx -1.34, \quad b \approx 10.75, \quad c \approx -11.30
\][/tex]
3. Prediction for Year 8:
- Using the quadratic model, we predict the number of owls for [tex]\( x = 8 \)[/tex]:
[tex]\[
y = -1.34(8)^2 + 10.75(8) - 11.30
\][/tex]
- Plugging in the value:
[tex]\[
y \approx -1.34 \cdot 64 + 10.75 \cdot 8 - 11.30
\][/tex]
[tex]\[
y \approx -85.76 + 86 - 11.30
\][/tex]
[tex]\[
y \approx -11.042857142857278
\][/tex]
4. Interpreting the Prediction:
- The prediction for the number of owls in year 8 is approximately -11.04.
- In the context of predicting a population, a negative number of owls does not make sense because you cannot have a negative quantity of owls.
- This suggests that while the quadratic model fits the given data points, its extrapolation beyond the data's range (in this case, to year 8) results in an unrealistic prediction.
5. Conclusion:
- The prediction of about -11 owls for year 8 does not make sense as it is not feasible to have a negative number of owls.
- This indicates that the quadratic model, although it fits the existing data, may not be appropriate for predicting future values beyond the provided data points. This unrealistic prediction suggests the need for caution when using the model for extrapolation, and potentially, consideration of different modeling techniques or additional data could be necessary for more accurate future predictions.
