To determine the equation that relates foot length ([tex]\( x \)[/tex]) to shoe size ([tex]\( S \)[/tex]), we can perform linear regression to find the best-fit line for the given data points.
Given data:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Foot length (in.)} \; (x) & \text{Shoe size} \; (S) \\
\hline
8 & 6 \\
\hline
8.5 & 7 \\
\hline
9 & 8 \\
\hline
9.5 & 9 \\
\hline
10 & 10 \\
\hline
\end{array}
\][/tex]
With linear regression, we look for a linear equation of the form [tex]\( S = mx + c \)[/tex] where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
After performing the necessary calculations:
1. Calculate the Slope ([tex]\( m \)[/tex]):
- The slope [tex]\( m = 2 \)[/tex].
2. Calculate the Intercept ([tex]\( c \)[/tex]):
- The intercept [tex]\( c = -10 \)[/tex].
Thus, the equation derived from the linear regression analysis is:
[tex]\[
S = 2x - 10
\][/tex]
Therefore, the correct equation to determine the approximate shoe size ([tex]\( S \)[/tex]) for a foot length ([tex]\( x \)[/tex]) is:
[tex]\[
S = 2x - 10
\][/tex]
So the correct answer is:
[tex]\[ \boxed{S = 2x - 10} \][/tex]
