{"abstract":"z-scores are roughly halved, so real effects look insignificant.","category":"Experiment statistics","checks":8,"contract":"Pooled two-proportion z-test: pool = (conv_a + conv_b) / (n_a + n_b), se = sqrt(pool(1 - pool)(1/n_a + 1/n_b)), z = (p_b - p_a) / se, two-sided p = erfc(|z| / sqrt 2). Nonpositive n -> None; se = 0 -> [0.0, 1.0]. Return [round(z, 6), round(p, 6)].","contract_signature":"conv_a, n_a, conv_b, n_b","evaluation_group":"w2-experiment-statistics-two-proportion-z","failed_approach":"Doubling 1/(n_a + n_b) matches only when the arms are equal in size.","family":"w2-experiment-statistics-two-proportion-z-standard-error-terms","id":"FA-74371","implementations":{"attempt":{"sha256":"843cfd653b99fac81d8ed5e20dbc1de0881be26caa0fb6066b7e93f6558e8295","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(conv_a, n_a, conv_b, n_b):\n    if n_a <= 0 or n_b <= 0:\n        return None\n    pa, pb = conv_a / n_a, conv_b / n_b\n    pool = (conv_a + conv_b) / (n_a + n_b)\n    se = math.sqrt(pool * (1 - pool) * (2 / (n_a + n_b)))\n    if se == 0:\n        return [0.0, 1.0]\n    z = (pb - pa) / se\n    p = math.erfc(abs(z) / math.sqrt(2))\n    return [round(z, 6), round(p, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[('unequal arm sizes weight the pooled rate', [50, 1000, 90, 3000], [-2.98032, 0.002879]),\n  ('treatment better gives positive z', [100, 1000, 130, 1000], [2.102741, 0.035488]),\n  ('treatment worse gives negative z', [130, 1000, 100, 1000], [-2.102741, 0.035488]),\n  ('no conversions anywhere', [0, 500, 0, 500], [0.0, 1.0]),\n  ('all conversions everywhere', [200, 200, 300, 300], [0.0, 1.0]),\n  ('conversion sample 1', [88, 1000, 75, 500], [3.637163, 0.000276]),\n  ('conversion sample 2', [446, 4000, 43, 2500], [-14.022988, 0.0]),\n  ('conversion sample 3', [86, 500, 18, 100], [0.192927, 0.847016])],\n [('treatment better gives positive z', [100, 1000, 130, 1000], [2.102741, 0.035488]),\n  ('treatment worse gives negative z', [130, 1000, 100, 1000], [-2.102741, 0.035488]),\n  ('no conversions anywhere', [0, 500, 0, 500], [0.0, 1.0]),\n  ('all conversions everywhere', [200, 200, 300, 300], [0.0, 1.0]),\n  ('identical rates', [40, 400, 80, 800], [0.0, 1.0]),\n  ('conversion sample 5', [14, 100, 17, 100], [0.586154, 0.557772]),\n  ('conversion sample 6', [455, 4000, 121, 1000], [0.642295, 0.520682]),\n  ('conversion sample 7', [16, 500, 457, 2500], [8.446636, 0.0])],\n [('treatment worse gives negative z', [130, 1000, 100, 1000], [-2.102741, 0.035488]),\n  ('no conversions anywhere', [0, 500, 0, 500], [0.0, 1.0]),\n  ('all conversions everywhere', [200, 200, 300, 300], [0.0, 1.0]),\n  ('identical rates', [40, 400, 80, 800], [0.0, 1.0]),\n  ('empty arm', [0, 0, 5, 10], None),\n  ('conversion sample 11', [263, 4000, 78, 1000], [1.374446, 0.169303]),\n  ('conversion sample 12', [5, 100, 238, 1000], [4.320765, 1.6e-05]),\n  ('conversion sample 13', [83, 1000, 484, 2500], [8.022534, 0.0])],\n [('unequal arm sizes weight the pooled rate', [50, 1000, 90, 3000], [-2.98032, 0.002879]),\n  ('no conversions anywhere', [0, 500, 0, 500], [0.0, 1.0]),\n  ('all conversions everywhere', [200, 200, 300, 300], [0.0, 1.0]),\n  ('identical rates', [40, 400, 80, 800], [0.0, 1.0]),\n  ('empty arm', [0, 0, 5, 10], None),\n  ('conversion sample 16', [200, 1000, 1, 100], [-4.687832, 3e-06]),\n  ('conversion sample 17', [17, 100, 12, 100], [-1.004125, 0.315318]),\n  ('conversion sample 19', [211, 1000, 115, 1000], [-5.811653, 0.0])],\n [('unequal arm sizes weight the pooled rate', [50, 1000, 90, 3000], [-2.98032, 0.002879]),\n  ('treatment better gives positive z', [100, 1000, 130, 1000], [2.102741, 0.035488]),\n  ('all conversions everywhere', [200, 200, 300, 300], [0.0, 1.0]),\n  ('identical rates', [40, 400, 80, 800], [0.0, 1.0]),\n  ('empty arm', [0, 0, 5, 10], None),\n  ('conversion sample 21', [20, 100, 104, 500], [0.180358, 0.856871]),\n  ('conversion sample 22', [1, 100, 105, 500], [4.787119, 2e-06]),\n  ('conversion sample 26', [128, 4000, 22, 100], [9.890895, 0.0])]]\nfor label, args, expected in fixtures[N - 1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"3f1cb72bd7ca96acd308aa17e0f3edb4675dc103d28327f361c9b702f5135622","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(conv_a, n_a, conv_b, n_b):\n    if n_a <= 0 or n_b <= 0:\n        return None\n    pa, pb = conv_a / n_a, conv_b / n_b\n    pool = (conv_a + conv_b) / (n_a + n_b)\n    se = math.sqrt(pool * (1 - pool) * (1 / (n_a + n_b)))\n    if se == 0:\n        return [0.0, 1.0]\n    z = (pb - pa) / se\n    p = math.erfc(abs(z) / math.sqrt(2))\n    return [round(z, 6), round(p, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[('unequal arm sizes weight the pooled rate', [50, 1000, 90, 3000], [-2.98032, 0.002879]),\n  ('treatment better gives positive z', [100, 1000, 130, 1000], [2.102741, 0.035488]),\n  ('treatment worse gives negative z', [130, 1000, 100, 1000], [-2.102741, 0.035488]),\n  ('no conversions anywhere', [0, 500, 0, 500], [0.0, 1.0]),\n  ('all conversions everywhere', [200, 200, 300, 300], [0.0, 1.0]),\n  ('conversion sample 1', [88, 1000, 75, 500], [3.637163, 0.000276]),\n  ('conversion sample 2', [446, 4000, 43, 2500], [-14.022988, 0.0]),\n  ('conversion sample 3', [86, 500, 18, 100], [0.192927, 0.847016])],\n [('treatment better gives positive z', [100, 1000, 130, 1000], [2.102741, 0.035488]),\n  ('treatment worse gives negative z', [130, 1000, 100, 1000], [-2.102741, 0.035488]),\n  ('no conversions anywhere', [0, 500, 0, 500], [0.0, 1.0]),\n  ('all conversions everywhere', [200, 200, 300, 300], [0.0, 1.0]),\n  ('identical rates', [40, 400, 80, 800], [0.0, 1.0]),\n  ('conversion sample 5', [14, 100, 17, 100], [0.586154, 0.557772]),\n  ('conversion sample 6', [455, 4000, 121, 1000], [0.642295, 0.520682]),\n  ('conversion sample 7', [16, 500, 457, 2500], [8.446636, 0.0])],\n [('treatment worse gives negative z', [130, 1000, 100, 1000], [-2.102741, 0.035488]),\n  ('no conversions anywhere', [0, 500, 0, 500], [0.0, 1.0]),\n  ('all conversions everywhere', [200, 200, 300, 300], [0.0, 1.0]),\n  ('identical rates', [40, 400, 80, 800], [0.0, 1.0]),\n  ('empty arm', [0, 0, 5, 10], None),\n  ('conversion sample 11', [263, 4000, 78, 1000], [1.374446, 0.169303]),\n  ('conversion sample 12', [5, 100, 238, 1000], [4.320765, 1.6e-05]),\n  ('conversion sample 13', [83, 1000, 484, 2500], [8.022534, 0.0])],\n [('unequal arm sizes weight the pooled rate', [50, 1000, 90, 3000], [-2.98032, 0.002879]),\n  ('no conversions anywhere', [0, 500, 0, 500], [0.0, 1.0]),\n  ('all conversions everywhere', [200, 200, 300, 300], [0.0, 1.0]),\n  ('identical rates', [40, 400, 80, 800], [0.0, 1.0]),\n  ('empty arm', [0, 0, 5, 10], None),\n  ('conversion sample 16', [200, 1000, 1, 100], [-4.687832, 3e-06]),\n  ('conversion sample 17', [17, 100, 12, 100], [-1.004125, 0.315318]),\n  ('conversion sample 19', [211, 1000, 115, 1000], [-5.811653, 0.0])],\n [('unequal arm sizes weight the pooled rate', [50, 1000, 90, 3000], [-2.98032, 0.002879]),\n  ('treatment better gives positive z', [100, 1000, 130, 1000], [2.102741, 0.035488]),\n  ('all conversions everywhere', [200, 200, 300, 300], [0.0, 1.0]),\n  ('identical rates', [40, 400, 80, 800], [0.0, 1.0]),\n  ('empty arm', [0, 0, 5, 10], None),\n  ('conversion sample 21', [20, 100, 104, 500], [0.180358, 0.856871]),\n  ('conversion sample 22', [1, 100, 105, 500], [4.787119, 2e-06]),\n  ('conversion sample 26', [128, 4000, 22, 100], [9.890895, 0.0])]]\nfor label, args, expected in fixtures[N - 1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":"A deterministic toy experiment-analysis model with a stipulated contract; results are rounded and are not a substitute for a validated statistics package. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"w2-experiment-statistics-two-proportion-z-standard-error-terms","generated_at":"2026-09-29T14:48:56.211338+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Online experiment readouts drive launch decisions; a silent formula slip flips conclusions.","root_cause":"The variance factor is 1 / (n_a + n_b) instead of 1/n_a + 1/n_b.","sha256":"c5e89ba2138ac45e501939a543330fb653dec71bc5073f26ef66524262db0956","title":"Conversion rate z-test: Standard error uses the combined sample size once · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verified":true,"visibility":"public","verification":{"attempt":{"elapsed_ms":98.606,"exit_code":1,"observations":[{"actual":[-4.866843,1e-06],"check":"unequal arm sizes weight the pooled rate","expected":[-2.98032,0.002879],"passed":false},{"actual":[2.973724,0.002942],"check":"treatment better gives positive z","expected":[2.102741,0.035488],"passed":false},{"actual":[-2.973724,0.002942],"check":"treatment worse gives negative z","expected":[-2.102741,0.035488],"passed":false},{"actual":[0.0,1.0],"check":"no conversions anywhere","expected":[0.0,1.0],"passed":true},{"actual":[0.0,1.0],"check":"all conversions everywhere","expected":[0.0,1.0],"passed":true},{"actual":[5.455745,0.0],"check":"conversion sample 1","expected":[3.637163,0.000276],"passed":false},{"actual":[-20.381631,0.0],"check":"conversion sample 2","expected":[-14.022988,0.0],"passed":false},{"actual":[0.366053,0.714325],"check":"conversion sample 3","expected":[0.192927,0.847016],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"unequal arm sizes weight the pooled rate\", \"actual\": [-4.866843, 1e-06], \"expected\": [-2.98032, 0.002879], \"passed\": false}, {\"check\": \"treatment better gives positive z\", \"actual\": [2.973724, 0.002942], \"expected\": [2.102741, 0.035488], \"passed\": false}, {\"check\": \"treatment worse gives negative z\", \"actual\": [-2.973724, 0.002942], \"expected\": [-2.102741, 0.035488], \"passed\": false}, {\"check\": \"no conversions anywhere\", \"actual\": [0.0, 1.0], \"expected\": [0.0, 1.0], \"passed\": true}, {\"check\": \"all conversions everywhere\", \"actual\": [0.0, 1.0], \"expected\": [0.0, 1.0], \"passed\": true}, {\"check\": \"conversion sample 1\", \"actual\": [5.455745, 0.0], \"expected\": [3.637163, 0.000276], \"passed\": false}, {\"check\": \"conversion sample 2\", \"actual\": [-20.381631, 0.0], \"expected\": [-14.022988, 0.0], \"passed\": false}, {\"check\": \"conversion sample 3\", \"actual\": [0.366053, 0.714325], \"expected\": [0.192927, 0.847016], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":40.433,"exit_code":1,"observations":[{"actual":[-6.882755,0.0],"check":"unequal arm sizes weight the pooled rate","expected":[-2.98032,0.002879],"passed":false},{"actual":[4.205481,2.6e-05],"check":"treatment better gives positive z","expected":[2.102741,0.035488],"passed":false},{"actual":[-4.205481,2.6e-05],"check":"treatment worse gives negative z","expected":[-2.102741,0.035488],"passed":false},{"actual":[0.0,1.0],"check":"no conversions anywhere","expected":[0.0,1.0],"passed":true},{"actual":[0.0,1.0],"check":"all conversions everywhere","expected":[0.0,1.0],"passed":true},{"actual":[7.715589,0.0],"check":"conversion sample 1","expected":[3.637163,0.000276],"passed":false},{"actual":[-28.823979,0.0],"check":"conversion sample 2","expected":[-14.022988,0.0],"passed":false},{"actual":[0.517678,0.604683],"check":"conversion sample 3","expected":[0.192927,0.847016],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"unequal arm sizes weight the pooled rate\", \"actual\": [-6.882755, 0.0], \"expected\": [-2.98032, 0.002879], \"passed\": false}, {\"check\": \"treatment better gives positive z\", \"actual\": [4.205481, 2.6e-05], \"expected\": [2.102741, 0.035488], \"passed\": false}, {\"check\": \"treatment worse gives negative z\", \"actual\": [-4.205481, 2.6e-05], \"expected\": [-2.102741, 0.035488], \"passed\": false}, {\"check\": \"no conversions anywhere\", \"actual\": [0.0, 1.0], \"expected\": [0.0, 1.0], \"passed\": true}, {\"check\": \"all conversions everywhere\", \"actual\": [0.0, 1.0], \"expected\": [0.0, 1.0], \"passed\": true}, {\"check\": \"conversion sample 1\", \"actual\": [7.715589, 0.0], \"expected\": [3.637163, 0.000276], \"passed\": false}, {\"check\": \"conversion sample 2\", \"actual\": [-28.823979, 0.0], \"expected\": [-14.022988, 0.0], \"passed\": false}, {\"check\": \"conversion sample 3\", \"actual\": [0.517678, 0.604683], \"expected\": [0.192927, 0.847016], \"passed\": false}], \"passed\": false}\n"}},"member_only":{"stages":["fixed"],"fields":["implementations.fixed","verification.fixed","harness","repair"],"note":"The verified repair, its recorded checks, the repair description, and the scoring harness are available to members."}}