A set of data points has a line of best fit of $y=-0.2 x+1.8$. What is the residual for the point $(5,1)$?
A. -0.2
B. 0.2
C. 1.6
D. 0.8
Answer (1)
Calculate the predicted y-value using the line of best fit: y ^ = − 0.2 ( 5 ) + 1.8 = 0.8 .
Calculate the residual by subtracting the predicted y-value from the actual y-value: 1 − 0.8 = 0.2 .
The residual for the point ( 5 , 1 ) is 0.2 .
The final answer is 0.2 .
Explanation
Understanding the Problem We are given a line of best fit y = − 0.2 x + 1.8 and a data point ( 5 , 1 ) . The residual is the difference between the actual y-value of the data point and the predicted y-value from the line of best fit. First, we need to find the predicted y-value when x = 5 .
Calculating the Predicted y-value To find the predicted y-value, we substitute x = 5 into the equation of the line of best fit: y ^ = − 0.2 ( 5 ) + 1.8 y ^ = − 1 + 1.8 y ^ = 0.8
Calculating the Residual Now that we have the predicted y-value, we can calculate the residual. The residual is the actual y-value minus the predicted y-value: residual = actual y − predicted y residual = 1 − 0.8 residual = 0.2
Final Answer Therefore, the residual for the point ( 5 , 1 ) is 0.2 .
Examples
In data analysis, understanding residuals helps us evaluate how well a line of best fit represents the data. For example, if you're tracking a company's sales over time and create a linear model, the residual for a particular month tells you how much the actual sales differed from what the model predicted. A small residual indicates a good fit, while a large residual might suggest external factors influenced sales that month. By analyzing residuals, you can refine your model and make more accurate predictions.
