Exam 01 review

Prof. Maria Tackett

Feb 12, 2026

Announcements

  • HW 02 due TODAY at 11:59pm

  • Exam 01: Tuesday, February 17 (in-class + take-home)

  • Friday’s lab: Exam 01 review - due Sunday at 11:59pm

Exam 01

  • In-class: 75 minutes during February 17 lecture

  • Take-home: due February 19 at 9am

    • Will have lecture on February 19

  • If you miss any part of the exam for an excused absence (with academic dean’s note or other official documentation), the final exam grade will be imputed for the exam 01 grade

Outline of in-class portion

  • Closed-book, closed-note.
  • Question types:
    • Short answer (show work / explain response)
    • Interpretations
    • Derivations (show work)
  • Will be provided all relevant R output and page of math rules
  • Can use any results from class or assignments without reproving them. State the results you’re using!
  • Just need a pencil or pen. No calculator permitted on exam.

Outline of take-home portion

  • Released: Tuesday, February 17 right after class
  • Due: Thursday, February 19 at 9am
  • Similar in format to lab / applied HW exercises
    • Will receive Exam questions in README of GitHub repo
  • Push work to GitHub and submit a PDF of responses to Gradescope

Tips for studying

  • Rework derivations from assignments and lecture notes
  • Review exercises in AEs and assignments, asking “why” as you review your process and reasoning
  • Focus on understanding not memorization
  • Explain concepts / process to others
  • Ask questions in office hours
  • Review lecture recordings as needed

Content: Weeks 1 - 6

  • Exploratory data analysis

  • Fitting and interpreting linear regression models

  • Model evaluation

  • Different types of predictors

  • Inference for regression

  • Matrix representation of regression

  • Hat matrix

  • Finding the least-squares estimator

  • Assumptions for least-squares regression

Population-level vs. sample-level models

Statistical model (population-level model)

\[ \mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}, \quad \epsilon \sim N(\mathbf{0}, \sigma^2_{\epsilon}\mathbf{I}) \]


Estimated regression model (sample-level model)

\[ \hat{\mathbf{y}} = \mathbf{X}\hat{\boldsymbol{\beta}}\quad \quad \mathbf{e} = \mathbf{y} - \hat{\mathbf{y}} \]

Model in matrix form

\[ \mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\epsilon} \]


  1. What are the dimensions of \(\mathbf{y}, \mathbf{X}, \boldsymbol{\beta}, \boldsymbol{\epsilon}\) ?
  2. What assumption do we make about the columns of \(\mathbf{X}\)? Why is that important?

Model in matrix form

\[ \mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\epsilon} \]


  1. What assumptions do we make about \(\boldsymbol{\epsilon}\) making given this model?
  2. What does this model tell us about the distribution of \(\mathbf{y}\) ?

Least-squares estimator \(\hat{\boldsymbol{\beta}}\)

Expected value of \(\hat{\boldsymbol{\beta}}\)

Variance of \(\hat{\boldsymbol{\beta}}\)

Find \(Var(\hat{\boldsymbol{\beta}})\) under the usual assumptions.


Assume \(Var(\boldsymbol{\epsilon}) = \mathbf{XV}\), such that \(\mathbf{V}\) has the appropriate dimensions. All other assumptions hold.

  • What are the dimensions of \(\mathbf{XV}\)?

  • Derive \(Var(\hat{\boldsymbol{\beta}})\). What are the dimensions of \(\mathbf{V}\)?

SSR

Show

\[ SSR = \mathbf{y}^\mathsf{T}\mathbf{y} - \hat{\boldsymbol{\beta}}^\mathsf{T}\mathbf{X}^\mathsf{T}\mathbf{y} \]