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 a 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 0.92 (dropdown with 0.85, 0.90, 0.92).
when the data set is modeled by a square root equation, the value of... (partially visible dropdown).
the better model of the data set is the square root (dropdown) equation.

Explanation:

Step1: Calculate linear model $r^2$

Using regression software, input $(x,y)$ pairs.
Linear model: $\hat{y} = 1.19x + 22.67$, $r^2 = 0.92$

Step2: Calculate square root model $r^2$

Transform $x$ to $\sqrt{x}$, run linear regression on $(\sqrt{x}, y)$.
Square root model: $\hat{y} = 6.42\sqrt{x} + 20.83$, $r^2 \approx 0.98$

Step3: Compare $r^2$ values

Higher $r^2$ means better fit. $0.98 > 0.92$

Answer:

When the data set is modeled by a linear equation, the value of $r^2$ is 0.92
When the data set is modeled by a square root equation, the value of $r^2$ is 0.98
The better model of the data set is the square root equation.