Question

Two variables are defined, a regression equation is given, and one data point is given. (a) Find the predicted value for the data point and compute the residual. (b) Interpret the slope in context. (c) Interpret the intercept in context, and if the intercept makes no sense in this context, explain why. \(\mathrm{Hgt}=\) height in inches, Age \(=\) age in years of a child. \(\widehat{H g t}=24.3+2.74(\) Age \() ;\) data point is a child 12 years old who is 60 inches tall.

Short answer

The predicted height for a 12-year-old child is obtained by substituting the age into the regression equation. The residual, or deviation from the regression prediction is obtained by subtracting the predicted height from the observed height. The slope indicates that for each extra year of age, the model predicts, on average, an increase in height of 2.74 inches. The intercept of 24.3 inches should represent a child's height at birth, which is not consistent with typical real-world observations.

Step-by-step solution

Calculate the Predicted Value

To find the predicted value (also known as the predicted height), substitute the age of the child into the regression equation. Here, the age of the child is given as 12 years. With the given regression equation \(\widehat{H g t}=24.3+2.74(\) Age \()), we have \(\widehat{H g t}=24.3+2.74(12)\)

Calculate the Residual

The residual is calculated as the observed data value minus the predicted data value. The observed height of the child (our data point) is 60 inches. To calculate the residual, perform the operation \(60 - \widehat{H g t}\)

Interpret the Slope

The slope of the regression equation, given by 2.74, is the estimated change in height for each additional year of age. That is, for each increase in age by 1 year, the predicted height increases by 2.74 inches, on average.

Interpret the Intercept

The intercept of the regression equation, given by 24.3, is the estimated value for height when the age of the child is 0 years. However, a child's height at birth is usually more than 24.3 inches. This signifies that the intercept might not make sense in the context of predicting a child's height because interpretation assumes you could have a child of age 0 years, which does not match with real-world understanding.