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

Answer 1

See explanation

Explanation:

Given:

400 students takes same final exam.

some students have 90 minutes to complete the exam.

other students have 120 minutes to complete the exam.

Each student is randomly assigned one of the examination times, based on the flip of a coin.

Y denotes the number of points scored on the exam by the i-th student (0 <Yi < 100) so it is an dependent variable

X denotes the amount of time that the student has to complete the exam (Xi = 90 or 120) so it is an independent variable

the regression model:
`Y_(i) = \beta _(0) +\beta_(1) X_(1) +`μ
`_(i)`

a)

The term μ
`_(i)` represents other factors or unobserved factors that have an effect on individual student's exam score. For example, studying time varies with students and varying aptitude for the subject, capability possessed by different students etc. ui is basically the error term.

b)

Due to random assignment of X
`_(i)` , μ
`_(i)` is independent of X
`_(i)` . Since μ
`_(i)` represents deviations from the average value, so E(μ
`_(i)`) = 0. Also since X
`_(i)` and μ
`_(i)` are independent hence E (μ
`_(i)` | X
`_(i)`) = 0

c)

The assumption that (X
`_(i)`,
`Y_(i)`), i = 1,2,...,...n are independent and identically distributed drawn from the joint distribution is true if this year's students are viewed as random draws from population of students that take this class.

The assumption holds because 0≤
`Y_(i)` ≤ 100 and Xi can take on only two values that are 90 and 120. So these restricted values make the assumption true.

d)

Given is the estimated model:

Y = 49 + 0.24 X
`_(i)`

The computation of prediction for the average score of students given 90 minutes to complete the exam is:

Y = 49 + 0.24 * 90

= 49 + 21.6

Y = 70.6

The computation of prediction for the average score of students given 120 minutes to complete the exam is:

Y = 49 + 0.24 * 120

= 49 + 28.8

Y = 77.8

Average score of students given 150 minutes to complete the exam is:

Y = 49 + 0.24 * 150

= 49 + 36

Y = 85

The average predicted gain in score for a student who is given an additional 10 minutes on the exam is:

Y = 0.24 * i

Y = 0.24 * 10

Y = 2.4