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]

| Year | [tex]y[/tex] 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 8 is about -11. Does this prediction make sense? Why or why not?

The problem provides a quadratic regression equation for the owl population: y = − 1.34 x 2 + 10.75 x − 11.3 .
We calculate the predicted owl population for year 8 by substituting x = 8 into the equation, resulting in y = − 11.06 .
We observe that the predicted number of owls is negative, which is not physically possible.
Therefore, the prediction does not make sense because the number of owls cannot be negative. No, the prediction does not make sense because the number of owls cannot be negative. ​

Explanation

Understanding the Problem We are given a quadratic regression equation that models the population of owls in a park: 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 model's prediction of approximately -11 owls in year 8 makes sense.

Calculating the Predicted Number of Owls First, let's calculate the predicted number of owls in year 8 using the given equation. We substitute x = 8 into the equation:


y = − 1.34 ( 8 ) 2 + 10.75 ( 8 ) − 11.3

Evaluating the Expression Calculating the terms, we have:

y = − 1.34 ( 64 ) + 86 − 11.3
y = − 85.76 + 86 − 11.3
y = 0.24 − 11.3
y = − 11.06

Determining if the Prediction Makes Sense The model predicts that the number of owls in year 8 is -11.06, which is approximately -11. Since the number of owls cannot be negative, this prediction does not make sense in the context of the problem. A negative number of owls is not a realistic or meaningful value.

Conclusion Therefore, the prediction does not make sense because the number of owls cannot be negative.


Examples
Quadratic regression models are used in various fields, such as predicting population growth, analyzing financial trends, and modeling physical phenomena. However, it's crucial to interpret the results within the context of the problem. For instance, if a quadratic model predicts a negative population size, it indicates that the model's applicability is limited to a certain range and may not be accurate for extreme values. Understanding the limitations of mathematical models is essential for making informed decisions and avoiding nonsensical predictions.

Answered by GinnyAnswer | 2025-07-08