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Compute sample sizes and error control

This question evaluates proficiency in experimental design and applied inferential statistics—specifically sample size calculation for means and proportions, non-inferiority testing, multiple-comparison error control, cluster-randomized design effects, and sequential monitoring boundaries—within the Statistics & Math domain for a Data Scientist role. It is commonly asked to measure the ability to apply statistical formulas and error-control principles under practical constraints, balancing power and type I error while accounting for clustering and interim looks; the assessment emphasizes practical application grounded in conceptual understanding of inferential methods.

Using the Biker experiment context, compute required sample sizes and describe error control under practical constraints. Show formulas and numeric answers where possible.

Assumptions:

  • Two-armed test (control vs Biker exposure).
  • Two-sided alpha unless stated otherwise.
  1. Primary mean metric: Baseline mean delivery time = 42 min, SD = 15 min. Target relative improvement = −5%. Alpha = 0.05 (two-sided), power = 0.80. Compute per-arm sample size for a two-sample t-test on means.
  2. Guardrail proportion metric: Baseline cancellation rate = 6%. You require non-inferiority with margin +0.5 percentage points (i.e., Biker cancel rate ≤ 6.5%). One-sided alpha = 0.05, power = 0.80. Compute per-arm sample size for a non-inferiority test on proportions.
  3. Multiple metrics: You have 1 primary, 2 guardrails (cancellations, ETA accuracy), and 1 secondary (orders per courier-hour). Propose and justify an error-control approach (e.g., Bonferroni, Holm/Hochberg, gatekeeping/HMP). State the effective alpha for each family and how you’d report adjusted CIs.
  4. Cluster randomization: You randomize at the zone-day level with average m = 300 orders per cluster and ICC = 0.03 for the primary metric. Compute the design effect DE and the adjusted per-arm sample size. How many zone-days per arm are needed to reach that sample size?
  5. Sequential monitoring: You plan 4 equally spaced looks with O’Brien–Fleming spending. Explain qualitatively how early critical values differ from the final look and how this affects runtime/MDE. Provide the final-look alpha spending approximation and how you’d implement boundary checks in practice.

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