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. Study \(=\) number of hours spent studying for an exam, Grade \(=\) grade on the exam. \(\widehat{\text { Grade }}=41.0+3.8\) (Study); data point is a student who studied 10 hours and received an 81 on the exam.

Short answer

The predicted grade for a student who studied for 10 hours is given by the equation \(\widehat{Grade} = 41.0 + 3.8 \times 10\). The residual is the actual grade minus the predicted grade. The slope, 3.8, suggests that for every additional hour spent studying, the student's grade increases by approximately 3.8 points. The intercept, 41.0, represents the predicted grade if no time is spent studying, which is not a realistic scenario.

Step-by-step solution

Calculate the Predicted Grade

Use the regression equation to figure out the predicted grade for a student who studied for 10 hours. Substitute 10 for 'Study' in the equation: \(\widehat{Grade} = 41.0 + 3.8 \times 10 \).

Compute the Residual

The residual is the difference between the actual grade and the predicted grade. As the student's actual grade was 81, subtract the predicted grade from 81 to find the residual.

Interpret the Slope

The slope in the regression equation is 3.8. This indicates that for each additional hour spent studying, a student is predicted to increase their exam grade by approximately 3.8 points, all other things being equal.

Interpret the Intercept and its Relevance

The intercept in the regression equation is 41.0. This is the predicted exam grade for a student who spends zero hours studying. This interpretation doesn't make sense since it's generally understood that not studying at all should result in a lower grade and also because an exam without studying doesn't reflect the student's potential accurately.