The data below shows the population of a small town (in 1000s) over a 9 year period. Using a graphing calculator, determine the linear & quadratic regression curves. Compare their $R^2$ values. Which model is a better fit for the data?

• A. Both Linear with $R^2=0.93$ and quadratic with $R^2=0.999$ are good fits for the data.
• B. Quadratic with $R^2=0.999$ is a better fit than linear with with $R^2=0.96$.
• C. Both Linear with $R^2=0.96$ and quadratic with $R^2=0.999$ are good fits for the data.
• D. Quadratic with $R^2=0.999$ is a better fit than linear with with $R^2=0.93$.

Answer: (D)

Verified step by step guidance

1. Step 1: Enter the data into your graphing calculator. Input the years (1 through 9) as the independent variable (X) and the population values (48, 48.6, 50.6, ..., 86.6) as the dependent variable (Y).
2. Step 2: Use the calculator's regression functions to find the linear regression equation. This will typically be in the form \(Y = a + bX\), where \(a\) is the intercept and \(b\) is the slope. The calculator will also provide the coefficient of determination, \(R^2\), which measures how well the linear model fits the data.
3. Step 3: Next, use the calculator to find the quadratic regression equation. This will be in the form \(Y = a + bX + cX^2\), where \(a\), \(b\), and \(c\) are coefficients determined by the calculator. Again, note the \(R^2\) value for this quadratic model.
4. Step 4: Compare the \(R^2\) values from both models. The \(R^2\) value ranges from 0 to 1, with values closer to 1 indicating a better fit to the data. A higher \(R^2\) means the model explains more of the variability in the population data.
5. Step 5: Decide which model is better based on the \(R^2\) values. The model with the higher \(R^2\) value is generally considered the better fit. In this case, since the quadratic model has a higher \(R^2\) value, it better represents the population change over the years.