A set of data points has a line of best fit of [tex]y=-0.2 x+1.7[/tex]. What is the residual for the point [tex](5,1)[/tex]?
A. -0.3
B. 1.5
C. 0.3
D. 0.7
Answer (2)
Calculate the predicted y-value using the line of best fit: y ^ = − 0.2 ( 5 ) + 1.7 = 0.7 .
Calculate the residual: res i d u a l = a c t u a l y − p re d i c t e d y = 1 − 0.7 = 0.3 .
The residual for the point ( 5 , 1 ) is 0.3 .
The final answer is 0.3 .
Explanation
Understanding the Problem We are given a line of best fit y = − 0.2 x + 1.7 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.
Calculating the Predicted y-value First, we need to find the predicted y-value ( y ^ ) for x = 5 using the line of best fit equation: y ^ = − 0.2 x + 1.7 Substituting x = 5 into the equation, we get: y ^ = − 0.2 ( 5 ) + 1.7 y ^ = − 1 + 1.7 y ^ = 0.7
Calculating the Residual Next, we calculate the residual using the formula: res i d u a l = a c t u a l y − p re d i c t e d y In this case, the actual y-value is 1, and the predicted y-value is 0.7. So, res i d u a l = 1 − 0.7 res i d u a l = 0.3
Final Answer Therefore, the residual for the point ( 5 , 1 ) is 0.3 .
Examples
In data analysis, understanding residuals is crucial for evaluating the accuracy of a regression model. For instance, if you're predicting house prices based on size, the residual tells you how far off your prediction is for a specific house. A small residual indicates a good fit, meaning your model is quite accurate for that data point. Conversely, large residuals might highlight outliers or suggest the need for a more complex model.
The residual for the point (5, 1) based on the line of best fit y = − 0.2 x + 1.7 is calculated to be 0.3. Therefore, the correct answer is 0.3 .
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