Maria gathered the data in the table. She finds the line of best fit to be [tex]$y=2.78 x-4.4$[/tex].
What is the residual value when [tex]$x=4$[/tex]?
A. -6.72
B. -0.58
C. 0.58
D. 6.72
Answer (1)
Calculate the predicted y-value using the line of best fit: y = 2.78 ( 4 ) − 4.4 = 6.72 .
Find the actual y-value from the table: y = 7.3 .
Calculate the residual: 7.3 − 6.72 = 0.58 .
The residual value when x = 4 is 0.58 .
Explanation
Understanding the Problem We are given a data table and a line of best fit y = 2.78 x − 4.4 . We want to find the residual value when x = 4 . The residual is the difference between the actual y value from the table and the predicted y value from the line of best fit.
Calculating the Predicted Value First, we need to calculate the predicted y value when x = 4 using the line of best fit: y = 2.78 x − 4.4 Substituting x = 4 :
y = 2.78 ( 4 ) − 4.4 y = 11.12 − 4.4 y = 6.72
Finding the Actual Value Next, we find the actual y value from the table when x = 4 . From the table, we see that when x = 4 , y = 7.3 .
Calculating the 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 = 7.3 − 6.72 Residual = 0.58
Final Answer Therefore, the residual value when x = 4 is 0.58 .
Examples
In real-world scenarios, understanding residuals is crucial in various fields like finance and engineering. For instance, in stock market analysis, a line of best fit can model stock prices over time. The residual value helps determine how much the actual stock price deviates from the predicted price, indicating potential investment opportunities or risks. Similarly, in engineering, residuals can help assess the accuracy of a model predicting the performance of a structure or system, allowing engineers to identify areas needing improvement or further investigation.
