To determine which equation can best model the given set of data, we need to check the accuracy and fit of a linear model.
The data points are:
[tex]\[
\begin{array}{ccccccccccc}
\hline
x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
y & 32 & 67 & 79 & 91 & 98 & 106 & 114 & 120 & 126 & 132 \\
\hline
\end{array}
\][/tex]
Firstly, calculate the correlation coefficient between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- The correlation coefficient (r) is approximately 0.955, and the coefficient of determination ([tex]\( r^2 \)[/tex]) is about 0.91. This indicates a very strong linear relationship.
Next, fit a linear model of the form [tex]\( y = mx + b \)[/tex]:
- The slope ([tex]\( m \)[/tex]) is approximately 9.67.
- The intercept ([tex]\( b \)[/tex]) is approximately 53.
Thus, the linear equation that best models the data is:
[tex]\[ y = 9.67x + 53 \][/tex]
Now let's compare this model to the given answer choices:
A. [tex]\( y = 33x - 32.7 \)[/tex]
B. [tex]\( y = 33x + 32.7 \)[/tex]
C. [tex]\( y = 33 \sqrt{x - 32.7} \)[/tex]
D. [tex]\( y = 33 \sqrt{x} + 32.7 \)[/tex]
Our model [tex]\( y = 9.67x + 53 \)[/tex] clearly does not match any of the provided choices exactly. It is important to identify the closest matching model among the given options. Considering the form of the equation and checking the constants:
None of the provided choices precisely match the calculated best-fit model. The closest comparison would need to be done within the context of other constraints or rounded approximations, but based solely on our initial calculations, none of the provided options are a suitable match for the best model generated from the given data.
