If the predicted value when [tex]x=5[/tex] is
[tex]\hat{y}=99.8[/tex]
and the actual value at [tex]x=5[/tex] is measured to be [tex]y=98.9[/tex], find the value of the residual at this point.
residual = [tex]\square[/tex] [?]
Answer (1)
The residual is the difference between the actual value and the predicted value.
The formula for the residual is: res i d u a l = y − y ^ .
Substitute the given values: res i d u a l = 98.9 − 99.8 .
Calculate the residual: res i d u a l = − 0.9 .
Explanation
Understanding the Problem We are given the predicted value y ^ = 99.8 and the actual value y = 98.9 at x = 5 . We need to find the residual at this point.
Residual Formula The residual is the difference between the actual value and the predicted value. The formula for the residual is: res i d u a l = y − y ^
Substituting Values Substitute the given values into the formula: res i d u a l = 98.9 − 99.8
Calculating the Residual Calculate the residual: res i d u a l = − 0.9
Final Answer The residual at x = 5 is − 0.9 .
Examples
In weather forecasting, the residual helps assess the accuracy of a model's predictions. For instance, if a model predicts a temperature of 25°C for tomorrow, but the actual temperature turns out to be 24°C, the residual is -1°C. This value indicates the model underestimated the temperature by 1 degree. By analyzing residuals over time, meteorologists can refine their models to improve future forecasts, making them more reliable for planning daily activities or agricultural decisions. Understanding residuals is crucial for enhancing the precision of predictive models across various fields.
