Year	1	3	6	7	10
Population	18,815	19,169	19,651	19,793	20,167

Determine a model for the given data by using quadratic regression.
What will be the population in year 12?

To determine a model for the given population data using quadratic regression, we need to fit a quadratic equation of the form:

\[
P(x) = ax^2 + bx + c
\]

where \( P(x) \) represents the population at year \( x \), and \( a \), \( b \), and \( c \) are constants to be determined.

Given data points:
- \( x = 1, P(x) = 18815 \)
- \( x = 3, P(x) = 19169 \)
- \( x = 6, P(x) = 19651 \)
- \( x = 7, P(x) = 19793 \)
- \( x = 10, P(x) = 20167 \)

Now, I'll calculate the quadratic regression model using these points and then use the model to estimate the population for year 12.

Let me compute this for you.The quadratic regression model for the given data is:

\[
P(x) = -4.19x^2 + 196.60x + 18620.72
\]

Using this model, the estimated population in year 12 is approximately:

\[
P(12) \approx 20,377
\]

Would you like further details or clarification on how this model was derived? Here are some related questions you might explore:

1. How does quadratic regression compare to linear regression for this data?
2. What does the negative coefficient of \(x^2\) indicate about population growth?
3. How can we calculate the population growth rate from this model?
4. What is the population prediction for year 15 using this model?
5. Can we determine when the population might start decreasing, based on the model?

**Tip:** Quadratic models are useful for data that initially grows but eventually slows down or even decreases.

Determine a model for the given data by using quadratic regression. What will be the population in year 12?

Learn how to apply quadratic regression to model population data over time. In this example, we predict the population in year 12 using the quadratic equation P(x) = -4.19x^2 + 196.60x + 18620.72, based on given data for years 1, 3, 6, 7, and 10. This problem covers concepts in quadratic regression and statistics.

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