Exam 1

• Exam: Tuesday, October 20, 2026
  • Coverage: Chapter 2 (through 2.7), Chapter 3 (3.1–3.3), and Chapter 5 (5.1–5.3) — not dummy variables or analysis of covariance (see Class Notes stat modeling 15.10.26, which covers that material for reference only, not exam scope)
  • Format: 50% R output (given output, provide the interpretation — similar to homework), 20% multiple choice (conceptual), 30% theoretical (one part similar to homework-style problems, one part covering results proved in class, e.g. $R_{adj}^2\le R^2$)
  • Allowed: a formula sheet, any size — no size restriction given
  • Memorize: the $R^2$/adjusted $R^2$ relationships — $R^2=1-SSE/SST$, $R_{adj}^2=1-\frac{RSS/(n-p-1)}{SST/(n-1)}=1-\frac{n-1}{n-p-1}(1-R^2)$
  • Quick reference (notation, identities, assumptions): Formula and Identity Sheet

Queue

1. Fitting a Simple Linear Regression

   • hand-compute an SLR fit; derive the LSE / normal equations

2. Residual and Sum-of-Squares Proofs

   • $\sum {\hat{e}}_i=0$, $\sum x_i{\hat{e}}_i=0$, $\sum {\hat{y}}_i{\hat{e}}_i=0$; $SST=S{S}_{reg}+RSS$

3. Regression Estimator Properties

   • Cov/Var bilinearity: $E({\hat{\beta }}_1)$, $\mathrm{Var}({\hat{\beta }}_1)$, $\mathrm{Cov}({\hat{\beta }}_0,{\hat{\beta }}_1)$ from the $c_i$ representation

4. Regression Inference and ANOVA

   • $R^2$; $t$-tests/CIs for the slope and intercept, including a nonzero null like ${\beta }_1=1$; ANOVA table; $F=t^2$

5. Regression Diagnostics and Leverage

   • residual-pattern recognition; leverage $h_{ii}$, the $4/n$ cutoff; standardized residuals; Cook’s distance

6. Reading R Regression Output

   • translate summary(lm()) / anova() / confint() back into the math — not a standalone question type, a presentation format the above could all appear in

7. Fitting a Multiple Linear Regression

   • matrix LSE $\hat{\beta }=(X^{\prime }X{)}^{-1}{X}^{\prime }Y$; hat matrix $H$ and its properties; generalized leverage $h_{ii}>2p/n$; polynomial regression as MLR

8. MLR Inference, ANOVA, and Adjusted R-Squared

   • global $F$-test; deriving $F$ from $R^2$ alone; coefficient $t$-tests as partial tests; adjusted $R^2$ and the $R_{adj}^2\le R^2$ proof

9. Partial F-Tests for Subsets of Predictors

   • testing a subset of predictors jointly; Type I (sequential) vs. Type III (added-last) sums of squares; nested-model exam checklist