A data set is shown in the table. The line of best fit modeling the data is [tex]$y=2.69 x-7.95$[/tex]. What is the residual value when [tex]$x=3$[/tex]?
A. -0.88
B. -0.12
C. 0.12
D. 0.88
Answer (2)
Calculate the predicted y-value using the line of best fit: y p re d i c t e d = 2.69 × 3 − 7.95 = 0.12 .
Find the actual y-value from the table: y a c t u a l = 1.0 .
Calculate the residual: res i d u a l = y a c t u a l − y p re d i c t e d = 1.0 − 0.12 = 0.88 .
The residual value when x = 3 is 0.88 .
Explanation
Understanding the Problem We are given a data set and a line of best fit y = 2.69 x − 7.95 . We need to find the residual value when x = 3 . The residual is the difference between the actual y -value and the predicted y -value from the line of best fit.
Calculating Predicted y-value First, we need to find the predicted y -value when x = 3 using the line of best fit. We substitute x = 3 into the equation: y p re d i c t e d = 2.69 × 3 − 7.95
Performing the Calculation Calculating the predicted y-value: y p re d i c t e d = 8.07 − 7.95 = 0.12
Finding Actual y-value Next, we find the actual y -value from the table when x = 3 . From the table, we see that when x = 3 , y a c t u a l = 1.0 .
Calculating the Residual Now, we calculate the residual value, which is the difference between the actual and predicted y -values: res i d u a l = y a c t u a l − y p re d i c t e d = 1.0 − 0.12 = 0.88
Final Answer Therefore, the residual value when x = 3 is 0.88 .
Examples
Residuals are used in many real-world applications, such as analyzing the accuracy of weather forecasting models. For example, if a weather model predicts a temperature of 25 degrees Celsius for tomorrow, but the actual temperature turns out to be 28 degrees Celsius, the residual is 3 degrees Celsius. By analyzing residuals over time, meteorologists can identify biases in their models and improve their accuracy. This helps in making more reliable predictions, which are crucial for agriculture, transportation, and disaster preparedness.
The residual value when x = 3 is calculated by finding the predicted y-value using the line of best fit, which equals 0.12 . Given the actual y-value is 1.0 , the residual comes out to be 0.88 . Therefore, the answer is 0.88 .
;
