A data set is shown in the table. The line of best fit modeling the data is [tex]$y=2.69 x-7.95$[/tex].
| x | y |
|---|---|
| 1 | -5.1 |
| 2 | -3.2 |
| 3 | 1.0 |
| 4 | 2.3 |
| 5 | 5.6 |
What is the residual value when [tex]$x=3$[/tex]?
A. -0.88
B. -0.12
C. 0.12
D. 0.88
Answer (2)
Calculate the predicted y value using the line of best fit: y = 2.69 ( 3 ) − 7.95 = 0.12 .
Find the actual y value from the table: y = 1.0 .
Calculate the residual value: residual = actual y - predicted y = 1.0 − 0.12 = 0.88 .
The residual value when x = 3 is 0.88 .
Explanation
Understanding the Problem We are given a data set and a line of best fit y = 2.69 x − 7.95 . We need to find the residual value when x = 3 . The residual is the difference between the actual y value from the data set and the predicted y value from the line of best fit.
Calculating Predicted y Value First, we need to find the predicted y value when x = 3 using the line of best fit. We substitute x = 3 into the equation: y = 2.69 ( 3 ) − 7.95
Predicted y Value Calculating the predicted y value: y = 8.07 − 7.95 = 0.12 So, the predicted y value is 0.12 .
Finding Actual y Value Next, we find the actual y value from the table when x = 3 . From the table, when x = 3 , y = 1.0 .
Calculating Residual Value Now, we calculate the residual value, which is the difference between the actual y value and the predicted y value: residual = actual y − predicted y residual = 1.0 − 0.12 = 0.88
Final Answer Therefore, the residual value when x = 3 is 0.88 .
Examples
In data analysis, understanding residuals helps assess the accuracy of a model. For instance, if you're predicting sales based on advertising spend, a large residual indicates the model isn't accurately capturing the relationship for that particular data point. By analyzing residuals, businesses can refine their models to make more accurate predictions, leading to better decision-making and resource allocation. This ensures that predictions align more closely with actual outcomes, improving the reliability of forecasting and strategic planning.
The residual value when x = 3 is calculated to be 0.88 . This is found by determining the difference between the actual y value from the table ( 1.0 ) and the predicted y value from the line of best fit ( 0.12 ). Therefore, the correct choice is D. 0.88.
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