4. A least squares regression line to predict a student’s Stat145 test score (from 0-to-100) from the number of hours studied was determined from a class of 55 Stat145 students: ̂ = 48.2 + 2.21x. One student in the class studied for 18 hours and scored 82 on the exam.
(a) (5 pts.) What is the predicted value of this student’s Stat145 exam score?
(b) (5 pts.) What is the residual for this student?
(c) Explain what the slope of this least squares regression line tells us with regard to the explanatory and response variables. Be specific with regard to the value of the slope
(a) To find the predicted value of the student's Stat145 exam score, we substitute the value of the number of hours studied (x) into the equation. In this case, the student studied for 18 hours.
So, we substitute x = 18 into the equation:
ŷ = 48.2 + 2.21(18)
= 48.2 + 39.78
= 87.98
Therefore, the predicted value of this student's Stat145 exam score is 87.98.
(b) The residual for this student is the difference between the actual exam score and the predicted exam score. In this case, the student scored 82 on the exam.
Residual = Actual Exam Score - Predicted Exam Score
= 82 - 87.98
= -5.98
Therefore, the residual for this student is -5.98.
(c) The slope of the least squares regression line tells us the relationship between the number of hours studied (explanatory variable) and the Stat145 exam scores (response variable).
In this case, the slope of the regression line is 2.21. A positive slope indicates a positive relationship between the number of hours studied and the exam scores. More specifically, for every additional hour studied, the predicted exam score increases by 2.21 points, on average.
So, the slope of the regression line tells us that as the number of hours studied increases, we can expect an increase in the predicted Stat145 exam score, with an average increase of 2.21 points per hour studied.