To determine which data point is farthest from the line of best fit, we need to compare the residuals for each value of [tex]\( x \)[/tex]. Residuals represent the vertical distance between the actual data points and the line of best fit. The larger the magnitude of the residual, the farther the data point is from the line of best fit.
Here are the given residuals:
- For [tex]\( x = 3 \)[/tex], the residual is [tex]\( -5 \)[/tex].
- For [tex]\( x = 5 \)[/tex], the residual is [tex]\( -1 \)[/tex].
- For [tex]\( x = 7 \)[/tex], the residual is [tex]\( 3 \)[/tex].
- For [tex]\( x = 9 \)[/tex], the residual is [tex]\( 0 \)[/tex].
We need to find the absolute values of these residuals to compare their magnitudes without considering the direction (positive or negative):
- [tex]\( |\text{Residual at } x = 3| = |-5| = 5 \)[/tex]
- [tex]\( |\text{Residual at } x = 5| = |-1| = 1 \)[/tex]
- [tex]\( |\text{Residual at } x = 7| = |3| = 3 \)[/tex]
- [tex]\( |\text{Residual at } x = 9| = |0| = 0 \)[/tex]
Now, we compare these absolute values:
- The absolute residual for [tex]\( x = 3 \)[/tex] is [tex]\( 5 \)[/tex].
- The absolute residual for [tex]\( x = 5 \)[/tex] is [tex]\( 1 \)[/tex].
- The absolute residual for [tex]\( x = 7 \)[/tex] is [tex]\( 3 \)[/tex].
- The absolute residual for [tex]\( x = 9 \)[/tex] is [tex]\( 0 \)[/tex].
The data point with the largest absolute residual is farthest from the line of best fit. Clearly, the value [tex]\( 5 \)[/tex] is the largest among the absolute residuals.
Therefore, the data point farthest from the line of best fit is the one corresponding to [tex]\( x = 3 \)[/tex].
Thus, [tex]\( x = 3 \)[/tex] is the value for which the data point is farthest from the line of best fit.
