1{2 "task_id": "F5-conf-03",3 "statement_true": false,4 "counterexample_provided": true,5 "counterexample_values": {6 "construction": "X ~ N(0,1); S ~ Rademacher (P(S=1)=P(S=-1)=1/2) independent of X; Y = S*X. Then Y ~ N(0,1) but X+Y = (1+S)X equals 0 w.p. 1/2 and 2X otherwise (mixture of a point mass at 0 and N(0,4)), which is not normal.",7 "n": 100000,8 "seed": 20261029,9 "mean_x": -0.0033187423835634258,10 "sd_x": 0.9982238690893596,11 "mean_y": -0.00024909222432809854,12 "sd_y": 0.9982293548745821,13 "mean_sum": -0.0035678346078915233,14 "sd_sum": 1.4058866469982523,15 "ks_sum_vs_normal": 0.2508475723268392,16 "ks_x_vs_std_normal": 0.0028297029959603037,17 "ks_y_vs_std_normal": 0.001648188194033784,18 "frac_sum_exactly_zero": 0.50047,19 "ks_critical_5pct_approx": 0.004294373062508659520 },21 "statement_b_true": true,22 "feasibility": "complete",23 "feasibility_reason": null,24 "data_issues": [],25 "n_input_rows": {26 "propositions.csv": 2,27 "design.csv": 228 },29 "n_analysed": 100000,30 "random_seed": 20261029,31 "baseline_independent_sum": {32 "construction": "X, Y iid N(0,1), drawn from the same generator after the counterexample draws",33 "mean_sum": -0.015452131370751422,34 "sd_sum": 1.412400304390764,35 "ks_sum_vs_normal": 0.002027253643752169336 }37}