Jace gathered the data in the table. He found the approximate line of best fit to be [tex]y=-0.7 x+2.36[/tex].
| x | y |
|---|---|
| 0 | 3 |
| 1 | 1 |
| 4 | 0 |
| 5 | -2 |
| 7 | -2 |
What is the residual value when [tex]x=5[/tex]?
A. -3.14
B. -0.86
C. 0.86
D. 3.14
Answer (2)
Calculate the predicted y-value using the line of best fit: y ^ = − 0.7 ( 5 ) + 2.36 = − 1.14 .
Find the actual y-value from the table: y = − 2 .
Calculate the residual: residual = actual y − predicted y = − 2 − ( − 1.14 ) = − 0.86 .
The residual value when x = 5 is − 0.86 .
Explanation
Understanding the Problem We are given a data table and a line of best fit, y = − 0.7 x + 2.36 . We need to find the residual value when x = 5 . The residual is the difference between the actual y-value and the predicted y-value from the line of best fit.
Calculating the Predicted y-value First, we need to calculate the predicted y-value, y ^ , using the line of best fit when x = 5 . We substitute x = 5 into the equation: y ^ = − 0.7 ( 5 ) + 2.36 y ^ = − 3.5 + 2.36 y ^ = − 1.14
Finding the Actual y-value Next, we find the actual y-value from the table when x = 5 . From the table, we see that when x = 5 , y = − 2 .
Calculating the Residual Value Now, we calculate the residual value using the formula: residual = actual y - predicted y. In this case, the residual is: residual = − 2 − ( − 1.14 ) residual = − 2 + 1.14 residual = − 0.86
Final Answer Therefore, the residual value when x = 5 is − 0.86 .
Examples
In data analysis, understanding residuals is crucial for evaluating the accuracy of a model. For example, if you're predicting house prices based on size, the residual tells you how far off your prediction is for a specific house. A large residual indicates the model didn't accurately predict the price for that house, which could be due to unique features not captured by the model. By analyzing residuals, you can refine your model to make more accurate predictions.
The residual value when x = 5 is calculated to be − 0.86 . This is found by determining the predicted y-value using the line of best fit and then subtracting this from the actual y-value from the data table.
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