To determine the equation of the line of best fit for the given data, we need to perform linear regression. The goal is to find the equation in the form:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Steps:
1. Identify the data points:
Week ([tex]\(x\)[/tex]) and corresponding Miles Run ([tex]\(y\)[/tex]):
[tex]\[
\begin{array}{|c|c|}
\hline \text{Week} & \text{Miles Run} \\
\hline 1 & 5 \\
\hline 2 & 8 \\
\hline 4 & 13 \\
\hline 6 & 15 \\
\hline 8 & 19 \\
\hline 10 & 20 \\
\hline
\end{array}
\][/tex]
2. Calculate the slope [tex]\( m \)[/tex] and y-intercept [tex]\( b \)[/tex]:
By performing linear regression on the given data points, the best fit line is found to have:
[tex]\[ m \approx 1.671 \][/tex]
[tex]\[ b \approx 4.699 \][/tex]
3. Write the equation of the line:
Substitute the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] into the linear equation [tex]\( y = mx + b \)[/tex]:
[tex]\[ y \approx 1.671x + 4.699 \][/tex]
In conclusion, the equation of the line of best fit for the given data, rounded to the thousandths place, is:
[tex]\[ y \approx 1.671x + 4.699 \][/tex]
