Given the regression equation
$\hat{y}=14.8+17 x$
and assuming $x=5$ is in the interval of experimental data, find the predicted value when $x=5$.
$\hat{y}=[?]$
Answer (1)
Substitute x = 5 into the regression equation: y ^ = 14.8 + 17 ( 5 ) .
Calculate the product: 17 × 5 = 85 .
Add the result to 14.8: y ^ = 14.8 + 85 = 99.8 .
The predicted value is 99.8 .
Explanation
Understanding the Problem We are given the regression equation y ^ = 14.8 + 17 x and we are asked to find the predicted value y ^ when x = 5 .
Substituting the Value of x To find the predicted value, we substitute x = 5 into the regression equation: y ^ = 14.8 + 17 ( 5 )
Calculating the Predicted Value Now, we perform the calculation: y ^ = 14.8 + 85 y ^ = 99.8
Final Answer Therefore, the predicted value when x = 5 is 99.8 .
Examples
Regression equations are used to predict outcomes based on input variables. For example, a store manager might use a regression equation to predict daily sales based on the amount spent on advertising. If the regression equation is y ^ = 50 + 2.5 x , where y ^ is the predicted daily sales and x is the amount spent on advertising (in dollars), then spending 20 o na d v er t i s in g w o u l d p re d i c t d ai l ys a l eso f \hat{y} = 50 + 2.5(20) = 50 + 50 = 100$ dollars. This helps in making informed decisions about advertising budgets.
