To solve this problem, we will perform a quadratic regression to model the given population data and then use this model to predict the population in year 13.

Step-by-Step Process:

1. Data Points:

   • Year (x): 1, 4, 6, 7, 10
   • Population (y): 12,534; 12,818; 12,972; 13,051; 13,221

2. Quadratic Regression Model: A quadratic regression model takes the form: $y = ax^2 + bx + c$ where $y$ is the population, and $x$ is the year.

3. Fitting the Model: I will use these data points to fit the quadratic model and find the coefficients $a$, $b$, and $c$.

Let me perform the quadratic regression now.The quadratic regression model that best fits the given data is:

$y = -3.02x^2 + 109.63x + 12427.25$

Using this model, the predicted population for year 13 is approximately 13,342.

Would you like more details on how the model was derived or have any other questions? Here are some additional related questions:

1. How does quadratic regression differ from linear regression?
2. How accurate is quadratic regression for extrapolating future values?
3. Can you explain how the coefficients in the model affect the curve's shape?
4. What would happen if we tried using a cubic regression instead of quadratic?
5. How can we assess the goodness of fit for this quadratic model?

Tip: Quadratic regression is especially useful when you observe that the data has a curved trend, as opposed to a straight line.