Compute sample size and Bayes posterior

Q: Compute sample size and Bayes posterior

This question evaluates understanding of statistical inference and probabilistic reasoning by combining a frequentist power/sample-size computation for a two-sided two-sample z-test with a basic Bayesian conditional probability calculation.

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Question

You are implementing two small statistical utilities.

Part A — Power / sample size (two-sample z-test)

Input:
- x : a 1D numeric array of historical observations of a metric (assume i.i.d.).
- alpha : significance level (e.g., 0.05).
- power : desired power (e.g., 0.8).
- effect_size : the minimum detectable absolute difference in means, Δ (same units as x ).
Task: Compute the required per-group sample size n for an equal-sized A/B test using a two-sided two-sample z-test .
Assumptions:
- The metric variance is the same in both groups.
- Population standard deviation is unknown; estimate it from x using the sample standard deviation s .
- Independent samples, normal approximation.
Output: return the smallest integer n that achieves the requested power (round up).

Part B — Bayes’ rule

Input: probabilities p_A = P(A) , p_B_given_A = P(B|A) , p_B_given_notA = P(B|¬A) .
Task: compute and return P(A|B) .
Assumptions: 0 < p_A < 1 and all conditional probabilities are valid.

Compute sample size and Bayes posterior

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