Let's evaluate how well each of the given equations models the data set. To do this, we will calculate the prediction error for each equation. The prediction error is found by computing the sum of the squared differences between the observed values ([tex]\(y\)[/tex]) and the predicted values ([tex]\(y_{\text{pred}}\)[/tex]) from each equation.
The given data is:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
x & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 \\
\hline
y & 11 & 19 & 27 & 35 & 37 & 40 & 44 & 45 & 48 & 52 \\
\hline
\end{array}
\][/tex]
We will evaluate the following options:
Option A: [tex]\( y = 11 \sqrt{x + 0.3} - 4.3 \)[/tex]
Option B: [tex]\( y = 2x - 17 \)[/tex]
Option C: [tex]\( y = 2x + 17 \)[/tex]
Option D: [tex]\( y = 11 \sqrt{x - 0.3} + 4.3 \)[/tex]
Using the observed data, we calculate the prediction error for each option as follows:
- Option A: The prediction error is approximately 813.38.
- Option B: The prediction error is approximately 12220.
- Option C: The prediction error is approximately 116.
- Option D: The prediction error results in a non-numeric value (NaN), likely due to the square root of a negative number for some [tex]\(x\)[/tex] values.
We need to select the equation that has the least prediction error. The prediction errors for the options are:
1. [tex]\( y = 11 \sqrt{x + 0.3} - 4.3 \)[/tex] has an error of 813.38.
2. [tex]\( y = 2x - 17 \)[/tex] has an error of 12220.
3. [tex]\( y = 2x + 17 \)[/tex] has an error of 116.
4. [tex]\( y = 11 \sqrt{x - 0.3} + 4.3 \)[/tex] is not valid because it produces NaN errors.
The smallest prediction error among the valid options is for:
Option C: [tex]\( y = 2x + 17 \)[/tex]
Thus, the correct answer is:
C. [tex]\( y = 2x + 17 \)[/tex]
