A set of data points has a line of best fit of [tex]y=2.5 x-1.5[/tex]. What is the residual for the point (4,7)?
A. -1.5
B. 8.5
C. 1.5
D. 7
Answer (1)
First, find the predicted y-value using the line of best fit: y ^ = 2.5 × 4 − 1.5 = 8.5 .
Then, calculate the residual by subtracting the predicted y-value from the observed y-value: res i d u a l = 7 − 8.5 = − 1.5 .
The residual for the point (4, 7) is − 1.5 .
Explanation
Understanding the Problem We are given a line of best fit y = 2.5 x − 1.5 and a data point ( 4 , 7 ) . We need to find the residual for this point. The residual is the difference between the observed value and the predicted value.
Calculating the Predicted Value First, we need to find the predicted y -value ( y ^ ) using the line of best fit for x = 4 . We substitute x = 4 into the equation of the line: y ^ = 2.5 × 4 − 1.5
Finding the Predicted Value Now, we calculate the predicted y -value: y ^ = 10 − 1.5 = 8.5
Calculating the Residual Next, we calculate the residual, which is the difference between the observed y -value and the predicted y -value. The observed y -value is 7. So, the residual is: res i d u a l = y − y ^ = 7 − 8.5 = − 1.5
Final Answer Therefore, the residual for the point ( 4 , 7 ) is − 1.5 .
Examples
In data analysis, understanding residuals is crucial for evaluating the accuracy of a regression model. For example, if you're predicting house prices based on size, the residual tells you how far off your prediction is for a specific house. Consistently large residuals might indicate that your model needs improvement by including other factors like location or age of the house. By analyzing residuals, you can refine your model to make more accurate predictions.
