Math Problem Statement
Year 1 3 6 7 10 Population 12,811 13,068 13,421 13,516 13,799
Determine a model for the given data by using quadratic regression. What will be the population in year 14?
Solution
To determine a model using quadratic regression and predict the population for year 14, we follow these steps:
Step 1: Define the quadratic model
The general form of a quadratic model is:
where:
- represents the year,
- represents the population at year ,
- , , and are constants to be determined from the data.
Step 2: Translate the years
To simplify calculations, let's translate the years so that the first year corresponds to . Subtract 1 from all years in the data.
- Year 1 →
- Year 3 →
- Year 6 →
- Year 7 →
- Year 10 →
The data now becomes:
Step 3: Perform quadratic regression
We now apply quadratic regression to fit a model of the form . I'll compute this regression to find the values of , , and .
Let me calculate this for you.The quadratic regression model for the population data is:
where represents the years starting from year 1 (i.e., corresponds to year 1, corresponds to year 3, and so on).
Using this model, the predicted population for year 14 (which corresponds to ) is approximately:
Would you like more details on the regression process or have any other questions?
Here are 5 related questions to expand on this:
- What does each coefficient in the quadratic equation represent?
- How would the population change if we used linear regression instead of quadratic?
- Can we use this model to predict populations beyond 20 years?
- How does quadratic regression differ from exponential regression for population growth?
- What are potential sources of error in using this model for long-term predictions?
Tip: When using regression models for predictions far outside the data range, be cautious, as models may not capture long-term trends accurately.
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Math Problem Analysis
Mathematical Concepts
Quadratic Regression
Polynomial Fitting
Predictive Modeling
Formulas
Quadratic equation: P(t) = at^2 + bt + c
Quadratic regression model: P(t) = -2.83t^2 + 135.27t + 12810.49
Theorems
Least Squares Method
Suitable Grade Level
Grades 11-12