Question

A professor decides to run an experiment to measure the effect of time pressure on final exam scores. He gives each of the 400 students in her course the same final exam, but some students have 90 minutes to complete the exam, while others have 120 minutes. Each student is randomly assigned one of the examination times, based on the flip of a coin. Let Y; denote the number of points scored on the exam by the ith student (0 (a) Explain what the term ui represents. Why will different students have different values of ui?

(b) Explain why E(ui|X;) = 0 for this regression model.
(c) Are the other assumptions among SLR.1-SLR.4 satisfied? Explain why.
(d) The estimated model is Y; = 49+0.24X;.
i. Based on the estimated model, predict the average score of students given 90 minutes. Repeat for 120 minutes and 150 minutes.
ii. Compute the average predicted gain in score for a student who is given an additional 10 minutes on the exam.

Answer 1

In this experiment, the term ui represents the individual error or deviation of each student's exam score from the regression line. The regression model assumes that on average, the deviation of the exam scores from the regression line is zero. The other assumptions of simple linear regression might or might not be satisfied. Based on the estimated model, the average scores for students given different times can be predicted, and the average predicted gain in score for an additional 10 minutes can be computed.

Step-by-step explanation:

(a) The term ui represents the individual error or deviation of the ith student's exam score from the regression line. Different students will have different values of ui because each student has their own unique abilities, preparation, and factors that can affect their exam performance.

(b) In this regression model, E(ui|X;) is equal to zero because it assumes that the average deviation of the exam score from the regression line is zero. This means that on average, the students' scores are predicted accurately by the regression model.

(c) The other assumptions among SLR.1-SLR.4 (Simple Linear Regression) may or may not be satisfied in this specific experiment setting. SLR.1 assumes a linear relationship between the dependent and independent variables, SLR.2 assumes independence of errors, SLR.3 assumes constant variance of errors, and SLR.4 assumes normality of errors. These assumptions need to be checked using statistical tests and diagnostic plots to draw any conclusions.

(d)
i. Based on the estimated model, the average score of students given 90 minutes would be 49 + 0.24(90) = 70.6. Similarly, for 120 minutes, the average score would be 49 + 0.24(120) = 76.8. For 150 minutes, the average score would be 49 + 0.24(150) = 82.6.
ii. To compute the average predicted gain in score for a student who is given an additional 10 minutes on the exam, we subtract the predicted score with 90 minutes from the predicted score with 100 minutes. This would be 49 + 0.24(100) - (49 + 0.24(90)) = 0.24.

Answer 2

Kindly check explanation

Step-by-step explanation:

The regression model :

Y; = Bo + BiX; + ui

ui in the regression model represents other underlying factors aside the model variables which may affect the final exam score of student. These factor will almost likely vary from student to student and may include factors such as ; rate of assimilation, natural brilliance, psychological factors and so on.

E(ui|X) = 0 ; because ui and Xi are independent.

The estimated model is Y; = 49+0.24X;.

i. Based on the estimated model, predict the average score of students given 90 minutes.

X = 90 minutes

Y; = 49+0.24(90)

Y = 70.6

Repeat for 120 minutes and 150 minutes.

X = 120 minutes

Y; = 49+0.24(120)

Y = 77.8

X = 150 minutes

Y; = 49+0.24(150)

Y = 85

ii. Compute the average predicted gain in score for a student who is given an additional 10 minutes on the exam.

Gain in score for student Given additional 10 minutes :

Gain in score for X = 10

0.24X

= 0.24(10)

= 2.4