Final Assignment

Final Assignment — Conceptual Review Guide

This guide summarizes the concepts and reasoning skills needed for each section. Use it to study, but you will still need to apply the ideas on your own in the final.

Part 1 — Simulation, Bootstrapping, and Confidence Intervals

Q1 — Designing a Simulation

You should know:

  • What a simulation study is and why we use it.
  • How to represent a population with a known proportion.
  • How to repeatedly sample from that population.
  • How to compute a statistic (e.g., a proportion) for each simulated sample.
  • How to use these statistics to understand sampling variability.

Focus on:

What does each step of the simulation correspond to in real life?

Q2 — Reading a Bootstrap Distribution

You should be able to:

  • Understand that bootstrapping samples with replacement from your sample, not from the population.
  • Interpret a histogram of bootstrap statistics.
  • Identify the middle 95% of bootstrap values to approximate a confidence interval.
  • Recognize whether the distribution is symmetrical or skewed.

Focus on:

What does the range of bootstrap statistics represent?

Q3 — Interpreting Confidence Intervals

You should know:

  • A confidence interval describes a procedure, not the probability of a parameter.
  • A confidence interval does not say the true parameter lies in the interval with a certain probability.
  • Confidence language always relates to repeated sampling.

Focus on:

What does “95% confident” mean in terms of repeated intervals from repeated samples?

Part 2 — Regression Concepts

Q4 — Interpreting Slopes

Key ideas:

  • A slope describes how the predicted outcome changes when the predictor increases by one unit, holding other variables constant.
  • Understand the units of the slope: dollars vs thousands of dollars, percentages vs percent points.
  • Small coefficients often reflect predictors measured on large scales.

Focus on:

Be clear about “for every 1-unit increase in X…”.

Q5 — Comparing Regression Models

You should recall:

  • Adding predictors always increases or leaves unchanged the R² value.
  • Adjusted R² may increase or decrease depending on usefulness of the added predictor.
  • Slope changes, intercept changes, and RMSE changes cannot be guaranteed without additional information.

Focus on:

Which regression metrics always behave predictably when adding new predictors?

Q6 — Interpreting Dummy-Coded Categorical Variables

You should understand:

  • Dummy coding compares each category to a reference category.
  • Coefficients tell the difference relative to the reference group, not absolute values.
  • A negative coefficient does not imply “bad” — it means “lower than the reference” while holding predictors constant.

Focus on:

How does the choice of reference level affect interpretation?

Q7 — Releveling a Categorical Variable

You should know:

  • How to set an ordered list of category levels.
  • That the first category determines the reference level for dummy coding.
  • Changing the reference level changes the meaning of the coefficients.

Focus on:

How does reordering levels change model interpretation?

Q8 — Visualizing Regression with a Categorical Predictor

You should be able to reason about:

  • What a regression plot looks like when:

    • There is one slope shared across categories.
    • Only intercepts differ (parallel lines).

  • How to match fitted model structure to the correct visualization.

Focus on:

Does the model contain an interaction? If not, slopes must be the same.

Part 3 — Calculus Review

Q12 — Derivatives

Make sure you understand:

  • The chain rule for compositions.
  • How to differentiate sums of functions.
  • How to differentiate trigonometric functions.
  • How exponents interact with outer functions.

Focus on:

Identify inner and outer functions clearly before applying the chain rule.

Q13 — Integrals

Key concepts:

  • Integration of exponential functions with constants in the exponent.
  • Integration of power functions using the power rule.
  • Knowing the special case when the exponent equals −1 (logarithm).

Focus on:

Recognize which rule applies to each term inside the integral.

Part 4 — Linear Algebra Review

Q14 & Q15 — Transposes

You should know:

  • How vector and matrix transposes are formed by swapping rows and columns.
  • How dimensions change after transposition (e.g., 3×2 becomes 2×3).

Focus on:

Transposition is purely positional — no arithmetic is involved.

Q16 — Matrix Multiplication Validity & Dimensions

Know how to:

  • Determine dimensions of matrices.
  • Check whether multiplication is valid (inner dimensions must match).
  • Determine the dimensions of the product (outer dimensions give the result).

Focus on:

Validity depends only on the structure of the matrices, not the values.

Q17 — Multiplying a Matrix and a Vector

You should understand:

  • Conditions for validity (columns of the first = rows of the second).
  • How to compute matrix-vector multiplication.
  • How the result combines rows of the matrix with entries of the vector.

Focus on:

Each row of the matrix produces one entry in the resulting vector.

Bonus — Self-Reflection

Be ready to:

  • Pick any concept from the course.
  • Describe it accurately in your own words.
  • Show that you understand the idea even if it was difficult initially.
  • No need to solve a problem — just explain the idea.

Focus on:

Depth of understanding, not correctness of a calculation.