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]?
Answer (2)
First, calculate the predicted y-value using the line of best fit: y ^ = − 0.7 ( 5 ) + 2.36 = − 1.14 .
Next, identify the observed y-value from the table when x = 5 , which is y = − 2 .
Then, calculate the residual value using the formula res i d u a l = o b ser v e d − p re d i c t e d = y − y ^ .
Finally, the residual value is res i d u a l = − 2 − ( − 1.14 ) = − 0.86 , so the answer is − 0.86 .
Explanation
Find the Predicted y-value First, we need to find the predicted y -value ( y ^ ) using the line of best fit when x = 5 . The equation for the line of best fit is given as y = − 0.7 x + 2.36 .
Substitute x=5 Substitute x = 5 into the equation: y ^ = − 0.7 ( 5 ) + 2.36
Calculate Predicted y Calculate the predicted y -value: y ^ = − 3.5 + 2.36 = − 1.14
Find Observed y-value Next, we find the observed y -value from the table when x = 5 . From the table, we see that when x = 5 , the observed y -value is − 2 .
Calculate the Residual Now, we calculate the residual value using the formula: res i d u a l = o b ser v e d − p re d i c t e d = y − y ^ .
In this case, res i d u a l = − 2 − ( − 1.14 ) .
Final Calculation Calculate the residual: res i d u a l = − 2 + 1.14 = − 0.86
Conclusion Therefore, the residual value when x = 5 is − 0.86 .
Examples
Understanding residuals is crucial in various real-world applications. For instance, in weather forecasting, a line of best fit might predict temperature trends based on historical data. The residual value helps assess the accuracy of the forecast by showing the difference between the predicted temperature and the actual temperature on a given day. Similarly, in financial analysis, residuals can indicate how well a stock's performance aligns with market trends, aiding investors in making informed decisions. By analyzing these differences, we can refine our models and make more accurate predictions.
The residual value when x = 5 is calculated as − 0.86 . This value indicates the difference between the observed and predicted y-values. It shows how much the prediction deviates from the actual data point.
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