1{2 "task_id": "F5-conf-03",3 "statement_true": false,4 "counterexample_provided": true,5 "counterexample_values": {6 "construction": "X ~ N(0,1); W independent of X with P(W=1)=P(W=-1)=1/2; Y = W*X. Then Y ~ N(0,1), and X+Y = (1+W)X is 0 with probability 1/2 and 2X otherwise (atom of mass 1/2 at 0, non-constant), hence 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": 0.250847572326839216 },17 "statement_b_true": true,18 "feasibility": "complete",19 "feasibility_reason": null,20 "data_issues": [],21 "n_input_rows": {22 "propositions.csv": 2,23 "design.csv": 224 },25 "n_analysed": 100000,26 "random_seed": 20261029,27 "supplementary": {28 "ks_x_vs_normal": 0.0020917798156042666,29 "ks_y_vs_normal": 0.0019518220846299617,30 "share_sum_exactly_zero": 0.50047,31 "baseline_ks_independent_sum_vs_normal": 0.0017646088927930625,32 "corr_x_y": -0.008226889835527693,33 "note": "baseline_* uses Z ~ N(0,1) independent of X (proposition B setting), drawn after x and w from the same generator; KS statistics use parameters estimated from the sample."34 }35}