Exam 01 review

Prof. Maria Tackett

Feb 13, 2025

Announcements

  • HW 02 due TODAY at 11:59pm

  • Exam 01: Tuesday, February 18 (in-class + take-home)

    • Go directly to assigned room (emailed Wednesday evening)

  • Friday’s lab: Exam 01 review - Graded on attendance and participation

  • No office hours February 18 - 20

Exam 01

  • 50 points total

    • in-class: 35 points

    • take-home: 15 points

  • In-class: 75 minutes during February 18 lecture

  • Take-home: due February 20 at 9pm (no lecture on Thursday)

  • If you miss any part of the exam for an excused absence (with academic dean’s note), your Exam 02 score will be counted twice

Outline of in-class portion

  • Closed-book, closed-note.

  • Question types:

    • Short answer (show work / explain response)
    • True/ False.
      • If false, write 1 - 2 sentence justification about why it is false.
    • Derivations

  • Will be provided all relevant R output and a page of matrix calculus and probability rules

  • Can use any results from class or assignments without reproving them (e.g., \(\mathbf{H}\) is symmetric and idempotent)

  • Just need a pencil or pen. No calculator permitted on exam.

Outline of take-home portion

  • Released: Tuesday, February 18 right after class
  • Due: Thursday, February 20 at 9pm (no lecture February 20)
  • Similar in format to a lab/ HW
    • Will receive Exam questions in README of GitHub repo
    • Formatting + using a reproducible workflow will be part of grade
  • Submit a PDF of responses to GitHub

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
    • e.g., Why do we include “holding all else constant” in interpretations?
  • 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 assessment and comparison

  • ANOVA

  • Categorical + interaction terms

  • Inference for model coefficients

  • 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 \(\mathbf{y}\) ?

Find least-squares estimator \(\hat{\boldsymbol{\beta}}\)

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

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

SSR

Show

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