Question

consider the data set shown in the table.

x	0	1	2	3	4	5	6	7	8	9	10
y	21	25	28	27	26	28	29	31	31	33	32

use technology to model the data set with both a linear equation and a square root equation. then select the correct answer from each drop - down menu. for help, see this worked example.
when the data set is modeled by a linear equation, the value of $r^{2}$ is.
when the data set is modeled by a square root equation, the value of $r^{2}$ is.
the better model of the data set is the equation.

Explanation:

Step1: Use statistical software

Use software like Excel, Python (with libraries like numpy, scipy.stats, matplotlib), or a graph - ing calculator. Input the x and y values from the data set.

Step2: Fit a linear model

In Python, for example, using numpy.polyfit to find the coefficients of the linear equation $y = ax + b$. Then calculate the $r^{2}$ value. The formula for $r^{2}$ is $1-\frac{\sum_{i = 1}^{n}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}$, where $y_{i}$ are the observed values, $\hat{y}_{i}$ are the predicted values from the model, and $\bar{y}$ is the mean of the observed y - values.

Step3: Fit a square - root model

Let the model be $y=a\sqrt{x}+b$. Transform the x - values (take square roots) and use a similar regression method as in Step 2 to find the coefficients and calculate the $r^{2}$ value.

Step4: Compare $r^{2}$ values

The model with the higher $r^{2}$ value is the better - fitting model. A higher $r^{2}$ (closer to 1) indicates a better fit of the model to the data.

Since we don't have the actual software output, we can't give the exact numerical values. But the general process is as above.

Answer:

The values of $r^{2}$ for the linear and square - root models need to be calculated using software. The better model is the one with the higher $r^{2}$ value.