To determine which equation best models the given data, let's assess each equation by calculating the sum of squared errors (SSE) for each, and compare them. The equation with the lowest SSE will best fit the data points.
Let's evaluate each equation step by step:
1. Equation A: [tex]\( y = 33\sqrt{x} + 32.7 \)[/tex]
Calculate the predicted [tex]\( y \)[/tex] values using this equation and then compute the SSE:
[tex]\[
\begin{aligned}
&\text{For } x = 0: & 33\sqrt{0} + 32.7 & = 32.7 \\
&\text{For } x = 1: & 33\sqrt{1} + 32.7 & = 65.7 \\
&\text{For } x = 2: & 33\sqrt{2} + 32.7 & \approx 79.405 \\
&\text{For } x = 3: & 33\sqrt{3} + 32.7 & \approx 90.926 \\
&\text{For } x = 4: & 33\sqrt{4} + 32.7 & = 98.7 \\
&\text{For } x = 5: & 33\sqrt{5} + 32.7 & \approx 106.432 \\
&\text{For } x = 6: & 33\sqrt{6} + 32.7 & \approx 113.69 \\
&\text{For } x = 7: & 33\sqrt{7} + 32.7 & \approx 120.437 \\
&\text{For } x = 8: & 33\sqrt{8} + 32.7 & \approx 126.9 \\
&\text{For } x = 9: & 33\sqrt{9} + 32.7 & = 132.7 \\
\end{aligned}
\][/tex]
Now compute the SSE:
[tex]\[
\begin{aligned}
\text{SSE}_A & = (32 - 32.7)^2 + (67 - 65.7)^2 + (79 - 79.405)^2 + (91 - 90.926)^2 + \\
& + (98 - 98.7)^2 + (106 - 106.432)^2 + (114 - 113.69)^2 + \\
& + (120 - 120.437)^2 + (126 - 126.9)^2 + (132 - 132.7)^2 \\
& = 0.49 + 1.69 + 0.164025 + 0.005476 + 0.49 + 0.186624 + 0.0961 + 0.190969 + 0.81 + 0.49 \\
& \approx 4.52
\end{aligned}
\][/tex]
2. Equation B: [tex]\( y = 33x + 32.7 \)[/tex]
Calculate the predicted [tex]\( y \)[/tex] values using this equation:
[tex]\[
\begin{aligned}
&\text{For } x = 0: & 33 \cdot 0 + 32.7 & = 32.7 \\
&\text{For } x = 1: & 33 \cdot 1 + 32.7 & = 65.7 \\
&\text{For } x = 2: & 33 \cdot 2 + 32.7 & = 98.7 \\
&\text{For } x = 3: & 33 \cdot 3 + 32.7 & = 131.7 \\
&\text{For } x = 4: & 33 \cdot 4 + 32.7 & = 164.7 \\
&\text{For } x = 5: & 33 \cdot 5 + 32.7 & = 197.7 \\
&\text{For } x = 6: & 33 \cdot 6 + 32.7 & = 230.7 \\
&\text{For } x = 7: & 33 \cdot 7 + 32.7 & = 263.7 \\
&\text{For } x = 8: & 33 \cdot 8 + 32.7 & = 296.7 \\
&\text{For } x = 9: & 33 \cdot 9 + 32.7 & = 329.7 \\
\end{aligned}
\][/tex]
Now compute the SSE:
[tex]\[
\begin{aligned}
\text{SSE}_B & = (32 - 32.7)^2 + (67 - 65.7)^2 + (79 - 98.7)^2 + (91 - 131.7)^2 + \\
& + (98 - 164.7)^2 + (106 - 197.7)^2 + (114 - 230.7)^2 + \\
& + (120 - 263.7)^2 + (126 - 296.7)^2 + (132 - 329.7)^2 \\
& \approx 0.49 + 1.69 + 392.04 + 1667.69 + 4425.69 + 8418.49 + \\
& + 13612.69 + 20585.69 + 29394.49 + 40189.69 \\
& \approx 100296.85
\end{aligned}
\][/tex]
3. Equation C: [tex]\( y = 33x - 32.7 \)[/tex]
Calculate the predicted [tex]\( y \)[/tex] values using this equation:
[tex]\[
\begin{aligned}
&\text{For } x = 0: & 33 \cdot 0 - 32.7 & = -32.7 \\
&\text{For } x = 1: & 33 \cdot 1 - 32.7 & = 0.3 \\
&\text{For } x = 2: & 33 \cdot 2 - 32.7 & = 33.3 \\
&\text{For } x = 3: & 33 \cdot 3 - 32.7 & = 66.3 \\
&\text{For } x = 4: & 33 \cdot 4 - 32.7 & = 99.3 \\
&\text{For } x = 5: & 33 \cdot 5 - 32.7 & = 132.3 \\
&\text{For } x = 6: & 33 \cdot 6 - 32.7 & = 165.3 \\
&\text{For } x = 7: & 33 \cdot 7 - 32.7 & = 198.3 \\
&\text{For } x = 8: & 33 \cdot 8 - 32.7 & = 231.3 \\
&\text{For } x = 9: & 33 \cdot 9 - 32.7 & = 264.3 \\
\end{aligned}
\][/tex]
Now compute the SSE:
[tex]\[
\begin{aligned}
\text{SSE}_C & = (32 - (-32.7))^2 + (67 - 0.3)^2 + (79 - 33.3)^2 + (91 - 66.3)^2 + \\
& + (98 - 99.3)^2 + (106 - 132.3)^2 + (114 - 165.3)^2 + \\
& + (120 - 198.3)^2 + (126 - 231.3)^2 + (132 - 264.3)^2 \\
& = 4227.29 + 4468.89 + 2119.29 + 611.29 + \\
& + 1.69 + 696.29 + 2627.29 + 6091.29 + 11102.89 + 17582.49 \\
& \approx 44528.9
\end{aligned}
\][/tex]
4. Equation D: [tex]\( y = 33\sqrt{x - 32.7} \)[/tex]
Note that for some [tex]\( x \)[/tex], the term [tex]\( x - 32.7 \)[/tex] will be negative, leading to undefined values for square root operations. Therefore, it’s invalid for use with the given data set in its entirety.
From the calculations:
- The SSE for Equation A is approximately 4.52.
- The SSE for Equation B is approximately 100296.85.
- The SSE for Equation C is approximately 44528.9.
- Equation D is invalid since [tex]\(\sqrt{x - 32.7}\)[/tex] is not defined for our given [tex]\( x \)[/tex] values.
Thus, the equation that best models the given data is:
A. [tex]\( y = 33\sqrt{x} + 32.7 \)[/tex].
