To determine for which value of [tex]\( x \)[/tex] the data point is farthest from the line of best fit, let's proceed step-by-step:
1. Understand the residuals: Residuals represent the difference between observed values and the values predicted by the line of best fit. Positive residuals mean the observed value is above the predicted value, and negative residuals mean the observed value is below the predicted value.
2. Given residuals:
- For [tex]\( x = 3 \)[/tex], the residual is -5.
- For [tex]\( x = 5 \)[/tex], the residual is -1.
- For [tex]\( x = 7 \)[/tex], the residual is 3.
- For [tex]\( x = 9 \)[/tex], the residual is 0.
3. Calculate the absolute value of each residual to determine the magnitude of deviation from the line:
- For [tex]\( x = 3 \)[/tex], [tex]\( | -5 | = 5 \)[/tex].
- For [tex]\( x = 5 \)[/tex], [tex]\( | -1 | = 1 \)[/tex].
- For [tex]\( x = 7 \)[/tex], [tex]\( | 3 | = 3 \)[/tex].
- For [tex]\( x = 9 \)[/tex], [tex]\( | 0 | = 0 \)[/tex].
4. Compare the absolute values to find the largest one, as the largest absolute value indicates the data point farthest from the line of best fit:
- For [tex]\( x = 3 \)[/tex], the absolute residual is 5.
- For [tex]\( x = 5 \)[/tex], the absolute residual is 1.
- For [tex]\( x = 7 \)[/tex], the absolute residual is 3.
- For [tex]\( x = 9 \)[/tex], the absolute residual is 0.
5. Identify the maximum absolute residual: The maximum absolute residual is 5, which occurs at [tex]\( x = 3 \)[/tex].
Therefore, the data point farthest from the line of best fit is for [tex]\( x = 3 \)[/tex].
