Quiz 1 Review

DS701 Session 9 — Mon Oct 5, 2026 · quiz is Wed Oct 7

Quiz 1 — Wednesday

Logistics

  • Wed Oct 7, in class. targeted 60 minutes but you can use entire 105 minutes of class, written for ~40 minutes of work. You’re dismissed when you hand it in.
  • Paper and pencil. Closed notes, no devices.
  • You get a printed copy of the reference sheet — the same one posted last Wednesday: quiz1-reference.pdf. It is frozen; what you see is what you’ll have.
  • Read it before Wednesday. Knowing where a formula is on the sheet is worth more than remembering it.
  • Bring a pencil and your BU ID. No calculator needed — numbers are chosen to work by hand, and you may leave answers as \(\sqrt{7}\), \(e^{-2}\), fractions.

What’s in scope — and what isn’t

Lectures 01–07

  • 01 What is Data Science
  • 02 Distances and similarity
  • 03 Clustering I: \(k\)-means
  • 04 Clustering II: hierarchical
  • 05 Probabilistic modeling, MLE, CI
  • 06 Clustering III: GMM and EM
  • 07 Generalization

Not on Quiz 1

  • Today’s lecture (trees and forests)
  • Anything in a module (M1–M13)
  • Python syntax or sklearn calls
  • Derivations from the appendices (05A, 06A–06E)

Calibration: the problems are at the level of HW1–HW4 plus the bias–variance problem coming in HW5. Fresh scenarios, fresh numbers.

Shape of the quiz

Five problems, 20 points each:

  1. Mechanical — compute distances/similarities and interpret the result
  2. Mechanical — run one step of an algorithm by hand (\(k\)-means, agglomerative merge, or EM)
  3. Conceptual — explain or evaluate claims; short derivation (MLE)
  4. Conceptual — reason about generalization, validation, hard vs soft assignment
  5. Transfer — a new scenario: pick the method, justify it, state its assumptions and what could go wrong

Partial credit is real: a correct setup with an arithmetic slip earns most of the points. A bare number with no work earns few.

Quick-fire check

Distances (L02)

Q1. Why is \(1 - \cos(\mathbf x, \mathbf y)\) not a metric? Which axiom fails?

Show answer

→ Triangle inequality — and \(d(\mathbf x, 2\mathbf x) = 0\), so “\(d=0 \iff\) equal” fails too.

Q2. As \(p \to \infty\), which coordinate dominates \(\|\mathbf x - \mathbf y\|_p\)?

Show answer

→ The single largest \(|x_i - y_i|\) (Chebyshev).

Q3. When would you prefer Jaccard over Hamming for two bit vectors?

Show answer

→ When the sets differ a lot in size — Jaccard normalizes by the union.

Q4. One feature is in dollars, another in years. What do you do before computing Euclidean distance, and why?

Show answer

→ Standardize; otherwise the large-range feature dominates.

Clustering (L03, L04, L06)

Q5. Across Lloyd iterations, can the \(k\)-means objective ever go up?

Show answer

→ No. Each step (assign, update) can only lower WCSS — but it stops at a local minimum, hence \(k\)-means++ and multiple restarts.

Q6. Single vs complete linkage: which one chains, and which prefers compact clusters?

Show answer

→ Single chains (min distance); complete prefers compact (max distance).

Q7. In the E-step, what does \(\sum_k \gamma_{ik}\) equal, and why?

Show answer

→ 1 — responsibilities are posterior probabilities \(P(C_k \mid x_i)\) via Bayes’ rule.

Q8. Name the one thing EM guarantees and the one thing it doesn’t.

Show answer

→ Log-likelihood never decreases; no guarantee of the global maximum.

Probability and generalization (L05, L07)

Q9. The MLE of \(\sigma^2\) divides by \(N\) or \(N-1\)?

Show answer

→ \(N\). The sample variance \(s^2\) divides by \(N-1\). Know which one a problem asks for.

Q10. Doubling \(n\) does what to the width of a 95% CI for the mean?

Show answer

→ Shrinks it by \(1/\sqrt{2}\) — width \(\propto 1/\sqrt{n}\).

Q11. \(k\) in a degree-\(k\) polynomial fit: parameter or hyperparameter? Who sets it?

Show answer

→ Hyperparameter; chosen by cross-validation, never by training error.

Q12. Training error falls to zero as complexity grows. Is that bias or variance you’re seeing?

Show answer

→ Variance: a different training sample would give a very different fit.

How to prepare

Between now and Wednesday

  • Work the sheet, don’t read it. For each algorithm box (Lloyd’s, agglomerative, EM), run one iteration by hand on 5–6 made-up numbers. That is exactly what the mechanical problems ask.
  • Redo one problem from each of HW1–HW4 without looking at your solution. The quiz is calibrated to them.
  • Re-answer the knowledge-check questions for lectures 01–07 from memory, then check the recap decks.
  • For the transfer problem, practice the techniques map on the sheet: given a scenario, which question type is it (Leek & Peng), which method, what assumption would break it?
  • Office hours and Piazza are open through Tuesday — ask early.

On the day

  • Skim all five problems first; start with the one you’re surest of.
  • Show work. Write the formula from the sheet, substitute, then compute.
  • State assumptions you make; label units; box your final answers.
  • If a number looks wrong, say so and keep going — flagging it earns credit.
  • Stuck on arithmetic? Leave it symbolic and move on.

After the quiz: the team midterm project is released Wednesday and HW5 goes out Thursday. Enjoy the rest of today’s activity — trees are not on the quiz.

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