To determine a model for the given data using quadratic regression, we will follow these steps:

1. Label the years as $x$ and the population as $y$:

$\text{Year (x):} \quad 1, 4, 6, 7, 9$ $\text{Population (y):} \quad 18,376, 18,730, 18,945, 19,037, 19,202$

2. Use a quadratic regression model of the form:

$y = ax^2 + bx + c$

3. Perform quadratic regression to find the coefficients $a$, $b$, and $c$.

Let's perform the calculation using the given data to find the quadratic model and then use it to predict the population in year 12. numpy as np

Given data

years = np.array([1, 4, 6, 7, 9]) population = np.array([18376, 18730, 18945, 19037, 19202])

Perform quadratic regression (find coefficients a, b, c)

coefficients = np.polyfit(years, population, 2) # 2 for quadratic regression

Get the model equation: y = ax^2 + bx + c

a, b, c = coefficients quadratic_model = f"y = {a:.4f}x^2 + {b:.4f}x + {c:.4f}"

Predict population for year 12 (x = 12)

population_year_12 = np.polyval(coefficients, 12)

quadratic_model, population_year_12The quadratic regression model for the population data is:

$y = -3.2635x^2 + 136.2240x + 18241.8354$

Using this model, the predicted population for the year 12 is approximately 19,407.

Would you like to dive into the steps of quadratic regression or explore something else? Here are a few related questions:

1. How do we interpret the coefficients $a$, $b$, and $c$ in the context of population growth?
2. What would happen if we used linear regression instead of quadratic regression for this data?
3. How would the model change if we had more years of data?
4. What other types of regression models could we use to model population growth?
5. How accurate is the quadratic regression model in predicting future populations?

Tip: In quadratic regression, the $a$ coefficient determines the curvature of the graph—whether it's opening upwards or downwards. Negative values for $a$ mean the curve opens downward.