Let's walk through how we find the residuals and calculate their sum.
### Step Two: Find the Residuals
Residuals are the differences between the observed [tex]\( y \)[/tex]-values and the predicted [tex]\( \dot{y} \)[/tex]-values using the regression line [tex]\( y = 2.9 + 0.34x \)[/tex].
Given the data:
[tex]\[
\{(5,5),(7,3),(9,9),(11,7),(13,5),(15,9)\}
\][/tex]
And the predicted [tex]\( \dot{y} \)[/tex] values calculated using the regression line:
[tex]\[
\begin{aligned}
\dot{y} \text{ for } x=5 & = 4.6 \\
\dot{y} \text{ for } x=7 & = 5.3 \\
\dot{y} \text{ for } x=9 & = 6.0 \\
\dot{y} \text{ for } x=11 & = 6.6 \\
\dot{y} \text{ for } x=13 & = 7.3 \\
\dot{y} \text{ for } x=15 & = 8.0 \\
\end{aligned}
\][/tex]
We can find the residuals by subtracting the predicted [tex]\( \dot{y} \)[/tex] from the observed [tex]\( y \)[/tex]:
[tex]\[
\begin{aligned}
\text{Residual for } x=5 & : 5 - 4.6 = 0.4 \\
\text{Residual for } x=7 & : 3 - 5.3 = -2.3 \\
\text{Residual for } x=9 & : 9 - 6.0 = 3.0 \\
\text{Residual for } x=11 & : 7 - 6.6 = 0.4 \\
\text{Residual for } x=13 & : 5 - 7.3 = -2.3 \\
\text{Residual for } x=15 & : 9 - 8.0 = 1.0 \\
\end{aligned}
\][/tex]
Now, listing these residuals:
[tex]\[
[0.4, -2.3, 3.0, 0.4, -2.3, 1.0]
\][/tex]
### Sum of the Residuals
The sum of the residuals is calculated by summing up all the individual residuals:
[tex]\[
0.4 + (-2.3) + 3.0 + 0.4 + (-2.3) + 1.0 = 0.2
\][/tex]
Thus, the sum of the residuals is [tex]\( 0.2 \)[/tex].
