553.413/613, Fall 2021: Midterm Exam 2
THE EXAM CONTAINS 4 QUESTIONS.
You will have 75 minutes to complete the exam.
You will need to have your CAMERA ON in Zoom for the whole duration of the exam. You will
need to have your MIC OFF for the whole duration of the exam.
During this Exam, you can review the Seber & Lee, Kutner, and Faraway texts, your course notes, the
lecture videos and lecture notes, and your homework and homework solutions. You are permitted to
use a calculator and, where appropriate, R. You are not allowed to consult with other people,
share or discuss Exam topics or Exam questions during the examination period, nor to
use any educational resources other the ones listed above.
If you have technical issues when submitting the exam to Gradescope, email me the screen-
shots/images of all your work before the deadline. Then submit the exam once you sort
out the technical issues.
If you have questions during the exam, you can send a private message to the TA monitoring you in
the Zoom chat. Do not unmute yourself to speak.
By submitting this Exam, you certify that 1) you understand the rules and agree to abide by them;
and 2) you understand that violating these rules constitutes academic dishonesty.
Problem Max Points Points
1 30
2 25
3 30
4 20
Total 105
1
Question 1 (30 points). THE QUESTION HAS 4 PARTS: (a)-(d)
Consider the linear model
Y = Xβ + ε,
where Y is a n-by-1 vector of response variables, X is an n-by-p design matrix, β is a p-by-1 vector of
coefficients, ε is a multivariate normal Nn(0, σ
2In).
(a) (9pts) Write down the distributions with the corresponding means and variance-covariance matrices of
the following random vectors. You do not need to justify what you wrote.
(i) Y
(ii) Y
(iii) e
(b) (12pts) Find Cov(e, Y ). Show all the steps and simplify as much as possible.
(c) (3pts) What are the dimensions of the matrix Cov(e, Y )
(d) (6pts) Does your answer to (b) change if only Gauss-Markov conditions are met If yes, state the
modified results, if no, explain why they do not change.
2
Question 2 (25 points). THE QUESTION HAS 2 PARTS: (a)-(b)
A beverage company is currently interested in finding the effect of milkshakes and other drinks on weight
gain. The company performs a designed study in which X is the amount of milkshakes (in pints) consumed
per week, Z is the amount of soda (in pints) consumed per week, and W is the amount of coffee (in pints)
consumed per week. The response variable Y is the weight gain in kilogram after one month. The number
of data points is n = 50. The company then runs a linear regression according to the model
Yi = β0 + β1Wi + β2Xi + β3Zi + i
where the error terms i are independent normal random variables with mean 0 and constant variance σ
2
(σ2 is unknown).
The design matrix
X =
