Points and their residual values are shown in the table.
| x | y | Residual |
|---|---|----------|
| 1 | 2 | -0.4 |
| 2 | 3.5 | 0.7 |
| 3 | 5 | -0.2 |
| 4 | 6.1 | 0.19 |
| 5 | 8 | -0.6 |
Which residual value is the farthest from the line of best fit?
A. 0.19
B. 0.7
C. 2
D. 8
Answer (1)
Calculate the absolute value of each residual.
Identify the maximum absolute value among the residuals.
The residual value corresponding to the maximum absolute value is the farthest from the line of best fit.
The residual value farthest from the line of best fit is 0.7 .
Explanation
Understanding the Problem We are given a set of points and their corresponding residual values. The residual value represents the difference between the observed value and the value predicted by the line of best fit. We want to find the residual value that is farthest from the line of best fit. This corresponds to the residual with the largest absolute value.
Identifying the Residuals To find the residual value farthest from the line of best fit, we need to find the residual with the largest absolute value. The given residual values are -0.4, 0.7, -0.2, 0.19, and -0.6.
Calculating Absolute Values Now, we calculate the absolute value of each residual:
∣ − 0.4∣ = 0.4
∣0.7∣ = 0.7
∣ − 0.2∣ = 0.2
∣0.19∣ = 0.19
∣ − 0.6∣ = 0.6
Finding the Maximum Absolute Value Comparing the absolute values, we have 0.4, 0.7, 0.2, 0.19, and 0.6. The largest of these values is 0.7.
Determining the Farthest Residual The residual value corresponding to the largest absolute value (0.7) is 0.7. Therefore, the residual value farthest from the line of best fit is 0.7.
Examples
In data analysis, residuals help us understand how well a model fits the data. For example, if you're predicting house prices based on size, the residual is the difference between the actual price and the predicted price. Identifying the largest residual helps you find the data point that your model predicts most poorly, which could indicate an unusual house or a problem with your model. This helps improve the accuracy and reliability of predictions in various fields, from finance to environmental science.
