Say we want a model that will help explain the relationship between a student's exam grade and their attendance. Below are two defined variables, a regression equation and two example data points.Variables:Grd = Exam grade in %Abs = Number of absences during semesterRegression Equation:Grd^ = 85.3 - 5.4*AbsTwo example data points (observations):A student that was absent 5 times and got 70% on the examA student that was absent 9 times and got 42% on the exam(a) Find the predicted value of exam grade (Grd^) for the student that was absent 5 times to 1 decimal place.Predicted exam grade for the student that was absent 5 times = Answer 1 Question 5 % (1dp)(b) The student observation above, that was absent 9 times and got 42% on the exam, would have a predicted exam grade of 36.7%. What is the residual for this observation to 1 decimal place?Residual for student that was absent 9 times = Answer 2 Question 5% (1dp)(c) Interpret the slope in context. Answer 3 Question 5(d) Interpret the intercept in context. Answer 4 Question 5.(e) Is the interpretation of the intercept meaningful in context? Answer 5 Question 5
Question
Say we want a model that will help explain the relationship between a student's exam grade and their attendance. Below are two defined variables, a regression equation and two example data points.Variables:Grd = Exam grade in %Abs = Number of absences during semesterRegression Equation:Grd^ = 85.3 - 5.4*AbsTwo example data points (observations):A student that was absent 5 times and got 70% on the examA student that was absent 9 times and got 42% on the exam(a) Find the predicted value of exam grade (Grd^) for the student that was absent 5 times to 1 decimal place.Predicted exam grade for the student that was absent 5 times = Answer 1 Question 5 % (1dp)(b) The student observation above, that was absent 9 times and got 42% on the exam, would have a predicted exam grade of 36.7%. What is the residual for this observation to 1 decimal place?Residual for student that was absent 9 times = Answer 2 Question 5% (1dp)(c) Interpret the slope in context. Answer 3 Question 5(d) Interpret the intercept in context. Answer 4 Question 5.(e) Is the interpretation of the intercept meaningful in context? Answer 5 Question 5
Solution
(a) To find the predicted value of the exam grade (Grd^) for the student that was absent 5 times, we substitute Abs = 5 into the regression equation:
Grd^ = 85.3 - 5.4Abs Grd^ = 85.3 - 5.45 Grd^ = 85.3 - 27 Grd^ = 58.3
So, the predicted exam grade for the student that was absent 5 times is 58.3% (to 1 decimal place).
(b) The residual for an observation is the difference between the observed value and the predicted value. In this case, the observed grade is 42% and the predicted grade is 36.7%. So, the residual is:
Residual = Observed grade - Predicted grade Residual = 42 - 36.7 Residual = 5.3
So, the residual for the student that was absent 9 times is 5.3% (to 1 decimal place).
(c) The slope of the regression line (-5.4) represents the change in the predicted exam grade for each additional absence. In this context, it means that for each additional absence, the predicted exam grade decreases by 5.4%.
(d) The intercept of the regression line (85.3) represents the predicted exam grade when the number of absences is zero. In this context, it means that if a student has no absences, their predicted exam grade is 85.3%.
(e) The interpretation of the intercept is meaningful in this context, as it is possible for a student to have zero absences. However, it is important to note that this interpretation assumes that the relationship between absences and exam grade is linear and starts at the intercept, which may not be the case in reality.
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