Fiona wrote the predicted and residual values for a data set using the line of best fit [tex]y=3.71 x-8.85[/tex].
| x | Given | Predicted | Residual |
|---|---|---|---|
| 1 | -5.1 | -5.14 | 0.04 |
| 2 | -1.3 | -1.43 | -0.13 |
| 3 | 1.9 | 2.28 | -0.38 |
| 4 | 6.2 | 5.99 | 0.21 |
Which statements are true about the table? Select three options.
A. The data point for [tex]x=1[/tex] is above the line of best fit.
B. The residual value for [tex]x=3[/tex] should be a positive number because the data point is above the line of best fit.
C. Fiona made a subtraction error when she computed the residual value for [tex]x=4[/tex].
D. The residual value for [tex]x=2[/tex] should be a positive number because the given point is above the line of best fit.
E. The residual value for [tex]x=3[/tex] is negative because the given point is below the line of best fit.
Answer (1)
The data point for x = 1 is above the line of best fit.
The residual value for x = 2 should be a positive number because the given point is above the line of best fit.
The residual value for x = 3 is negative because the given point is below the line of best fit.
The true statements are selected, based on the analysis of given and predicted values and the calculation of residuals: Statements 1, 4, and 5 are true.
Explanation
Understanding the Data We are given a data set with x values, corresponding 'Given' y values, 'Predicted' y values based on the line of best fit y = 3.71 x − 8.85 , and 'Residual' values. Our task is to verify the truthfulness of several statements about this data.
Analyzing Statement 1 The first statement is: 'The data point for x = 1 is above the line of best fit.'
For x = 1 , the given y value is − 5.1 and the predicted y value is − 5.14 . Since -5.14"> − 5.1 > − 5.14 , the data point is above the line of best fit. Thus, this statement is TRUE.
Analyzing Statement 2 The second statement is: 'The residual value for x = 3 should be a positive number because the data point is above the line of best fit.'
For x = 3 , the given y value is 1.9 and the predicted y value is 2.28 . Since 1.9 < 2.28 , the data point is below the line of best fit. Therefore, the residual should be negative. Thus, this statement is FALSE.
Analyzing Statement 3 The third statement is: 'Fiona made a subtraction error when she computed the residual value for x = 4 .'
The residual is calculated as Given - Predicted. For x = 4 , the given y value is 6.2 and the predicted y value is 5.99 . The residual should be 6.2 − 5.99 = 0.21 . The table shows the residual as 0.21 , so Fiona did NOT make a subtraction error. Thus, this statement is FALSE.
Analyzing Statement 4 The fourth statement is: 'The residual value for x = 2 should be a positive number because the given point is above the line of best fit.'
For x = 2 , the given y value is − 1.3 and the predicted y value is − 1.43 . Since -1.43"> − 1.3 > − 1.43 , the data point is above the line of best fit. The residual is calculated as Given - Predicted, so the residual should be − 1.3 − ( − 1.43 ) = − 1.3 + 1.43 = 0.13 . Thus, the residual should be positive, and this statement is TRUE.
Analyzing Statement 5 The fifth statement is: 'The residual value for x = 3 is negative because the given point is below the line of best fit.'
For x = 3 , the given y value is 1.9 and the predicted y value is 2.28 . Since 1.9 < 2.28 , the data point is below the line of best fit. The residual is calculated as Given - Predicted, so the residual should be 1.9 − 2.28 = − 0.38 . Thus, the residual is negative, and this statement is TRUE.
Final Answer In conclusion, the true statements are:
The data point for x = 1 is above the line of best fit.
The residual value for x = 2 should be a positive number because the given point is above the line of best fit.
The residual value for x = 3 is negative because the given point is below the line of best fit.
Examples
Understanding residuals is crucial in many real-world applications. For example, in weather forecasting, a model predicts temperature based on various factors. The residual is the difference between the actual temperature and the predicted temperature. Analyzing these residuals helps meteorologists refine their models and improve the accuracy of future forecasts. Similarly, in financial modeling, residuals can indicate how well a model predicts stock prices or economic trends, aiding in investment decisions and risk management.
