Introduction to Statistics

Exam 2 Review

Exam 2 Review — Chapters 7, 8.1, and 9

Exam window (from Sam’s FA2026 MATH141-99 syllabus): Sunday, Oct. 25 – Monday, Oct. 26, 2026. Proctored online or in person. Covers Chapters 7, 8.1, and 9. Worth 12% of the final grade.

Ready for a full run-through? Take the Practice Exam for Test 2 — 12 questions with an answer key.

Test 2 Study Guide Videos

Video walkthroughs covering the exam material:

Video 1 — Chapters 7 & 8.1

Video 2 — Confidence intervals

What the exam pulls from each chapter

Chapter Core skills StatCrunch tool
7 — Normal Probability Distribution (§§7.1–7.2) Probability as area under the curve; z = (x − μ)/σ; inverse problems (find x from a percentile / “top/bottom %”) Stat > Calculators > Normal
8 — Sampling Distributions (§8.1 only) Sampling distribution of x̄; Central Limit Theorem; standard error σ/√n; z = (x̄ − μ)/(σ/√n) Stat > Calculators > Normal, with SD = σ/√n
9 — Estimating a Parameter (§§9.1–9.3) Confidence interval for μ (z if σ known, t if σ unknown, df = n − 1); CI for a proportion; margin of error; sample-size calculations (round UP) Stat > Z Stats / T Stats / Proportion Stats > One Sample > Confidence interval

Formula shortlist (know what StatCrunch is computing behind the scenes)

  • z-score: z = (x − μ)/σ · Back-conversion: x = μ + zσ
  • Standard error of the mean: σ_x̄ = σ/√n · z for a sample mean: z = (x̄ − μ)/(σ/√n)
  • CI for μ, σ known: x̄ ± z_{α/2}·(σ/√n) · CI for μ, σ unknown: x̄ ± t_{α/2}·(s/√n)
  • CI for p: p̂ ± z_{α/2}·√[p̂(1 − p̂)/n], requires np̂(1 − p̂) ≥ 10
  • Sample size for p (no prior estimate, use p̂ = 0.5): n = 0.25 · [z_{α/2}/E]², round up
  • Critical values worth knowing cold: z_0.05 = 1.645, z_0.025 = 1.96, z_0.01 = 2.33

Practice problems — work these in StatCrunch (answers included so you can check)

Do each one in StatCrunch first, the way you will on the exam. The answers are here to check your work.

  1. Normal probability (Ch. 7). Test scores are normal with μ = 72, σ = 8.
  2. Stat > Calculators > Normal. Enter Mean 72, Std. dev. 8.
  3. (a) P(X > 84): choose the “greater than” probability, enter 84 → 0.0668.
  4. (b) P(64 < X < 80): choose “between”, enter 64 and 80 → ≈ 0.683.
  5. Inverse normal (Ch. 7). Bottle fills are normal with μ = 500 mL, σ = 10 mL. What is the cutoff for the lowest 5%?
  6. Same calculator, but switch to the inverse option: enter area 0.05 for the left tail → x ≈ 483.6 mL.
  7. Sampling distribution (Ch. 8.1). Population μ = 50, σ = 12; samples of n = 36. Find P(x̄ > 53).
  8. Stat > Calculators > Normal, but enter the standard error as the SD: Mean 50, Std. dev. 12/√36 = 2. Choose “greater than”, enter 53 → 0.0668. (For a single individual you would enter SD = 12 — that √n difference is the whole point of §8.1.)
  9. CI for a mean, σ unknown (Ch. 9). n = 16, x̄ = 28, s = 4, 95% confidence.
  10. Stat > T Stats > One Sample > With Summary. Enter n, sample mean, and sample SD, then choose Confidence interval at level 0.95 → (25.87, 30.13).
  11. CI for a proportion (Ch. 9). 180 of 400 students work part-time. 95% confidence interval?
  12. Stat > Proportion Stats > One Sample > With Summary. # of successes 180, # of observations 400, then Confidence interval at level 0.95 → (0.401, 0.499).
  13. Sample size (Ch. 9). Estimate a proportion within ±0.04 at 95% confidence, with no prior estimate.
  14. Formula: n = 0.25 × (1.96/0.04)² = 0.25 × 2401 = 600.25 → 601 (always round up).

Traps the chapter guides already flag — worth repeating to students before the exam

  • Ch. 7: tables give area to the left; “top 10%” means cumulative area 0.90.
  • Ch. 8: using σ instead of σ/√n for a sample mean is the single most common error.
  • Ch. 9: “95% confident” ≠ “95% probability μ is in this interval”; use t (not z) when σ is unknown; round sample sizes up, never down.