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.