The table below shows the population of a town (in thousands) at five-year intervals from the year 2000 to 2020.
| Year | 2000 | 2005 | 2010 | 2015 | 2020 |
|—|—|—|—|—|—|
| Population (thousands) | 50 | 68 | 85 | 130 | 170 |
(a) Input this data into your GDC and perform a quadratic regression to find a model of the form \(P(x) = ax^2 + bx + c\), where \(x\) is the number of years since 2000 and \(P(x)\) is the population in thousands. Report the values of \(a\), \(b\), and \(c\) to three significant figures.
(b) Perform a cubic regression on the same data. Compare the residuals from both the quadratic and cubic models. Based on the residual analysis, explain which model is a better fit for this population data and justify your choice.
Question
Input the given population data into your GDC. Perform a quadratic regression to find a model of the form \(P(x) = ax^2 + bx + c\), where \(x\) is the number of years since 2000 and \(P(x)\) is the population in thousands. Report the values of \(a\), \(b\), and \(c\) to three significant figures.
Hint
Hint
Hint
Define \(x = \text{Year} – 2000\) before entering data. Use the quadratic regression function on your GDC to find the coefficients.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
Using GDC quadratic regression, we find \(a \approx 0.53\), \(b \approx -2.62\), and \(c \approx 50.4\). The quadratic model is approximately \(P(x) = 0.53x^2 – 2.62x + 50.4\).
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Answer Explanation
Answer Explanation
Question
Perform a cubic regression on the same data. Compare the residuals from both the quadratic and cubic models. Based on the residual analysis, explain which model is a better fit for this population data and justify your choice.
Hint
Hint
Hint
Calculate and compare the residuals for both models. Consider the magnitude and pattern of residuals to assess the fit. A better fit typically shows smaller and more randomly distributed residuals.
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Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
The quadratic model shows residuals that are generally small and reasonably distributed around zero. The cubic model will likely have even smaller residuals, but with only five data points, a cubic model may overfit the data, potentially leading to less reliable predictions for future years. In this case, the quadratic model is likely a better balance between simplicity and fit, offering a good representation of the trend without overfitting.
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Answer Explanation
Answer Explanation
