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 (2)
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 identified as: Statement 1, Statement 4, and Statement 5. Statements 1, 4, and 5
Explanation
Problem Analysis Let's analyze the given statements one by one using the provided data and the line of best fit y = 3.71 x − 8.85 . Remember that the residual is calculated as Given - Predicted.
Analyzing Statement 1 Statement 1: The data point for x = 1 is above the line of best fit. For x = 1 , the given value is -5.1 and the predicted value is -5.14. Since -5.1 > -5.14, the data point is indeed above the line of best fit. So, this statement is TRUE.
Analyzing Statement 2 Statement 2: 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 value is 1.9 and the predicted 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. So, this statement is FALSE.
Analyzing Statement 3 Statement 3: Fiona made a subtraction error when she computed the residual value for x = 4 .
For x = 4 , the given value is 6.2 and the predicted value is 5.99. The residual value should be 6.2 − 5.99 = 0.21 . The table shows the residual as 0.21, so there is no error. So, this statement is FALSE.
Analyzing Statement 4 Statement 4: 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 value is -1.3 and the predicted value is -1.43. Since -1.3 > -1.43, the data point is above the line of best fit. The residual value is Given - Predicted = − 1.3 − ( − 1.43 ) = − 1.3 + 1.43 = 0.13 . Since 0.13 is positive, this statement is TRUE.
Analyzing Statement 5 Statement 5: The residual value for x = 3 is negative because the given point is below the line of best fit. For x = 3 , the given value is 1.9 and the predicted value is 2.28. Since 1.9 < 2.28, the data point is below the line of best fit. The residual value is Given - Predicted = 1.9 − 2.28 = − 0.38 . Since -0.38 is negative, this statement is TRUE.
Final Answer Therefore, the three 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 instance, in weather forecasting, residuals help assess the accuracy of a predictive model by quantifying the difference between actual and predicted temperatures. Similarly, in financial modeling, residuals can indicate how well a stock's performance aligns with its predicted behavior based on a specific model. By analyzing these differences, we can refine our models and make more accurate predictions, whether it's planning for the weather or making investment decisions. This concept is also used in quality control to ensure products meet expected standards.
The true statements are A, D, and E, which indicate that the data point for x=1 is above the line of best fit, the residual for x=2 is positive because it is above the line, and the residual for x=3 is negative because it is below the line.
;
