Implement four DS coding tasks

You are completing a CodeSignal-style assessment (Python or R). Implement solutions for the following four independent questions.

1) Two-sample z-test: required sample size

You are given:

• x: numeric array of historical observations for the metric (use it to estimate the metric standard deviation sigma)
• alpha: significance level (e.g., 0.05)
• power: desired power (e.g., 0.8)
• effect_size: the minimum detectable absolute difference in means, (\Delta)

Assumptions:

• Two-sided two-sample z-test for a difference in means.
• Treatment and control have equal sample size (n).
• Use (\hat\sigma = \text{std}(x)) as the population standard deviation estimate.

Task:

• Return the minimum integer per-group sample size n required to detect effect_size at level alpha with power.

2) Difference-in-Differences (DiD) + parallel-trend validation

You are given three equal-length arrays:

• period[i]: time indicator (contains at least a “pre” and a “post” period; may contain multiple pre periods)
• group[i]: 0 = control, 1 = treatment
• outcome[i]: numeric outcome

And a numeric threshold for trend validation.

Definitions:

• Let (\bar{Y}_{g,t}) be the mean outcome for group (g\in{0,1}) in period (t).
• The DiD estimate is: [ \text{DiD} = (\bar{Y}{1,post}-\bar{Y}{1,pre}) - (\bar{Y}{0,post}-\bar{Y}{0,pre}). ]

Parallel-trend / trend validation requirement:

• If there are multiple pre periods, compute the group difference (d_t = \bar{Y}{1,t} - \bar{Y}{0,t}) for each pre period (t), sort pre periods by time, and validate: [ \max_t |d_{t} - d_{t-1}| \le \text{threshold}. ]
• If there is only a single pre period, treat trend validation as passing.

Task:

• Return (a) the DiD estimate and (b) whether the pre-trend validation passes under the threshold.

3) Bayes’ rule posterior probability

You are given probabilities (as floats) describing an event (A) and evidence (B), such as:

• p_A = (P(A))
• p_B_given_A = (P(B\mid A))
• p_B_given_not_A = (P(B\mid\neg A))

Task:

• Compute and return the posterior probability (P(A\mid B)).

4) Logistic regression: top-3 features

You are given:

• X: a 2D array where each row corresponds to one feature and each column corresponds to one observation (shape: num_features × num_samples)
• y: binary outcome array of length num_samples (values in {0,1})
• feature_names: array of length num_features

Task:

• Fit a logistic regression model to predict y from X (include an intercept).
• Rank features by absolute value of their fitted coefficient (exclude the intercept).
• Return the names of the top 3 features in descending order of importance.

Notes:

• Handle ties deterministically (e.g., break ties by feature name ascending).
• Assume inputs are well-formed and numeric.