This table shows the population of owls in a park in different years. The quadratic regression equation that models these data is [tex]y=-1.34 x^2+10.75 x-11.3[/tex]
| x Year | y Number of owls |
|---|---|
| 1 | 0 |
| 2 | 2 |
| 3 | 7 |
| 4 | 14 |
| 5 | 9 |
| 6 | 4 |
Using this model, the predicted number of owls in year 9 is about -23. Does this prediction make sense? Why or why not?
Answer (1)
The problem provides a quadratic regression equation for owl population.
The model predicts a negative number of owls in year 9.
Since owl population cannot be negative, the prediction is not realistic.
The model's accuracy is limited to the range of the original data. Therefore, the final answer is: N o , t h e p re d i c t i o n d oes n o t mak ese n se b ec a u se t h e n u mb ero f o wl sc ann o t b e n e g a t i v e .
Explanation
Understanding the Problem We are given a quadratic regression equation that models the population of owls in a park over different years. The equation is y = − 1.34 x 2 + 10.75 x − 11.3 , where x represents the year and y represents the number of owls. We are asked to determine if the predicted number of owls in year 9, which is approximately -23, makes sense.
Analyzing the Prediction To determine if the prediction makes sense, we need to consider the context of the problem. The number of owls in a population cannot be a negative number. A negative value would indicate an absence or a decrease below zero, which is not possible in this scenario.
Conclusion The predicted number of owls in year 9 is about -23. Since the number of owls cannot be negative, this prediction does not make sense. The quadratic model may not be accurate for values of x outside the range of the given data (years 1 to 6).
Examples
Quadratic regression models are useful for predicting trends, but they have limitations. For example, if you're modeling the height of a plant over time, the model might predict negative height after a certain period, which is not physically possible. Similarly, in business, a sales forecast might become unrealistic if extrapolated too far into the future. It's important to understand the context and limitations of the model to make reasonable predictions.
