# Contract (P0)

Question. In a fixed-design small sample (n=30) with heteroskedastic errors, y_i = 1 + 0.5 x_i + sigma_i e_i, e_i iid N(0,1), x and sigma fixed from data/design.csv, do nominal 95% t(28) CIs for the OLS slope attain 95% coverage?

Estimands. (a) P(beta in betahat ± t_{.975,28} se_conv), se_conv = conventional OLS SE with s^2 = RSS/(n-2). (b) Same with se_HC1 = sqrt(n/(n-2) * HC0 sandwich). (c) Exact SD of betahat: sqrt(sum w_i^2 sigma_i^2), w_i = (x_i - xbar)/Sxx (analytical). (d) E[se_conv], by MC. (a),(b),(d) estimated by Monte Carlo with exactly R=2000, fixed seed. Meaningful under the stated DGP (normal errors, fixed design).

Outputs. out/results.json keys: task_id, coverage_conventional, coverage_hc1, sd_beta_exact, sd_beta_mc, mean_se_conventional, conventional_ci_undercovers (cov_conv < 0.95 - 3sqrt(0.950.05/2000)), n_replications, n_obs_per_replication, feasibility, feasibility_reason, data_issues, n_input_rows, random_seed. Also out/report.md (6 sections incl. MC SEs of coverages), out/analysis.py, out/claims.json (incl. claim on conventional nominal coverage).

Information rules. Only data/design.csv; never redraw x or sigma; constants given in TASK.md.

Invalidity conditions. Design file not 30 rows / missing or nonpositive sigma / constant x; R != 2000; non-reproducible seed.