This table shows the rainfall (in centimeters) for a city in different months. The quadratic regression equation that models these data is [tex]y=-0.77 x^2+6.06 x-5.9[/tex]
| x Month | y Rainfall (cm) |
|---|---|
| 1 | 0 |
| 2 | 2 |
| 3 | 5 |
| 4 | 8 |
| 5 | 4 |
| 6 | 3 |
Using this model, the predicted rainfall for month 10 is about -22.3 centimeters. Does this prediction make sense? Why or why not?
Answer (1)
The quadratic regression equation predicts a rainfall of -22.3 cm for month 10.
Rainfall cannot be negative.
The prediction does not make sense.
The model produces unrealistic predictions outside the range of the given data.
The prediction is No, because rainfall cannot be negative.
Explanation
Understanding the Problem We are given a quadratic regression equation that models the rainfall in a city for different months. The equation is y = − 0.77 x 2 + 6.06 x − 5.9 , where x represents the month and y represents the rainfall in centimeters. We are asked to determine if the model's prediction of -22.3 cm for month 10 makes sense.
Considering the Context Rainfall is a physical quantity, and in the real world, rainfall cannot be a negative value. It can only be zero or a positive value.
Evaluating the Prediction The model predicts a rainfall of -22.3 cm for month 10. Since rainfall cannot be negative, this prediction is not physically possible or realistic.
Conclusion Therefore, the prediction of -22.3 cm for month 10 does not make sense because rainfall cannot be a negative value. The quadratic model, while it may fit the given data points reasonably well, produces unrealistic predictions outside the range of the given data.
Examples
In environmental science, regression models are often used to predict various environmental factors, such as rainfall, temperature, or pollution levels. However, it's crucial to understand the limitations of these models. For instance, a model might accurately predict rainfall for months 1 through 6, but if used to predict rainfall for month 10, it could give a negative value, which is physically impossible. Therefore, scientists must always validate model predictions against real-world constraints to ensure they are meaningful and accurate.
