{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0087", "pilot_number": "R0053-27112", "number": 27112, "family_id": "qf-mod49-ci-interpretation-boundary", "title": { "zh-Hant": "第 27112 題：有限模型信賴區間覆蓋率", "en": "Question 27112: Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0087", "batch_id": "batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-finite-confidence-interval-coverage" ], "title_zh": "有限模型信賴區間覆蓋率", "stem_zh": "完整finite laws與total precommitted theta-independent interval procedure責任。操作=ci-interpretation-boundary；資料={\"mode\":\"invalid_procedure\",\"model\":{\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[12,14],[2,14]],[[2,14],[12,14]]],\"intervals\":[[12,11],[12,12]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(invalid_procedure, 否, interval lower endpoint exceeds upper endpoint, 否)", "explanation_zh": "Wave6第十二層核對=(invalid_procedure, 否, interval lower endpoint exceeds upper endpoint, 否)；依完整明示資料獨立重算；pointwise coverage與worst-case minimum逐theta重算；average不等於uniform，posterior不是coverage；selection納入procedure後須重新計算。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "有限模型信賴區間覆蓋率" ] }, "variant": { "family_id": "qf-mod49-ci-interpretation-boundary", "seed": 202611183087, "parameters": { "mode": "invalid_procedure", "model": { "theta_values": [ 0, 1 ], "samples": [ "a", "b" ], "probability_rows": [ [ [ 12, 14 ], [ 2, 14 ] ], [ [ 2, 14 ], [ 12, 14 ] ] ], "intervals": [ [ 12, 11 ], [ 12, 12 ] ] }, "domain": "Exact_finite_confidence_coverage", "operation": "ci-interpretation-boundary", "semantic_template": "ci-interpretation-boundary|{mode:\"invalid_procedure\",model:{intervals:[[[12,1],[11,1]],[[12,1],[12,1]]],probability_rows:[[[6,7],[1,7]],[[1,7],[6,7]]],samples:[\"a\",\"b\"],theta:[0,1]}}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0087", "batch_id": "batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "312836704f673a5f25f5d4ea208fd676c1794fb0c14262eabfb68bd293807f4a", "family_id": "qf-mod49-ci-interpretation-boundary", "seed": 202611183087, "semantic_template": "ci-interpretation-boundary|{mode:\"invalid_procedure\",model:{intervals:[[[12,1],[11,1]],[[12,1],[12,1]]],probability_rows:[[[6,7],[1,7]],[[1,7],[6,7]]],samples:[\"a\",\"b\"],theta:[0,1]}}" }, "fields": { "title": "Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities", "stem": "Use mode=invalid_procedure and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[12,14],[2,14]],[[2,14],[12,14]]],\"intervals\":[[12,11],[12,12]]}. Give (mode, procedure_valid, validation_error, procedure_qualification_claim).", "answer": { "kind": "expression", "value": "(invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)", "explanation": "average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)." }, "skills": [ "Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "完整finite laws與total precommitted theta-independent interval procedure責任。操作=ci-interpretation-boundary；資料={\"mode\":\"invalid_procedure\",\"model\":{\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[12,14],[2,14]],[[2,14],[12,14]]],\"intervals\":[[12,11],[12,12]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use mode=invalid_procedure and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[12,14],[2,14]],[[2,14],[12,14]]],\"intervals\":[[12,11],[12,12]]}. Give (mode, procedure_valid, validation_error, procedure_qualification_claim)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [ci-interpretation-boundary]", "after": "derived = [[mode, procedure_valid, validation_error, procedure_qualification_claim]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "此頁直接列完整來源欄序與mode特定角色，不靠JavaScript或由答案外形猜介面；No、undefined、infinity各保原意。", "en": "List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [{'theta_values': [0, 1], 'samples': ['a', 'b'], 'probability_rows': [[[12, 14], [2, 14]], [[2, 14], [12, 14]]], 'intervals': [[12, 11], [12, 12]]}]", "after": "derived = [否, interval lower endpoint exceeds upper endpoint, 否]", "relation": "implies", "rule_id": "CI_invalid_procedure_ACTUAL_first_source_validation_error", "conditions": [], "explanation": { "zh-Hant": "逐來源檢查順序取得實際首錯誤，不用任意False代替缺row、未正規化、錯誤區間等不同原因；此mode沒有計算未資格化的coverage。", "en": "Retain the actual FIRST validation error in source order,not one generic False for missing rows,unnormalized laws or bad intervals.This mode does not calculate coverage for an unqualified procedure." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null } ], "math_model": { "result": [ "invalid_procedure", false, "interval lower endpoint exceeds upper endpoint", false ] }, "conclusion": { "canonical_value": "(invalid_procedure, 否, interval lower endpoint exceeds upper endpoint, 否)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations/questions.jsonl", "batch_sha256": "e32eae55169a7001013c379f42d8fb53292340b088f540d145ae9717c637db21", "original_line": 87, "original_line_sha256": "3d345cf6726306169bf1642c7beb7d36197431bc4e12f924b9dfb641505aba2f", "question_compact_sha256": "312836704f673a5f25f5d4ea208fd676c1794fb0c14262eabfb68bd293807f4a", "english_row_compact_sha256": "b27652f1a066922782449f57f5a6c1fdfae4a2883e2f65312d8343801e8b977c", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations.en.jsonl" }, "mko_bindings": { "mko-finite-confidence-interval-coverage": { "sha256": "5ac0ccb03c88f011ec1edd7c258fe04dc72e982b042a8dcea53ac59deb386363", "english_sha256": "e2602185873b2f9b57fed1ac6efd6d390016ea0a0ff7efc9dc970097b8fca2cf", "original_path": "public/data/mko/finite-confidence-interval-coverage.json", "english_path": "public/data/i18n/editions/mko-finite-confidence-interval-coverage.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a54409c0bc3f68730e1072c6d3d772970462c1bc7ffc203b4030a6136a610283", "original_candidate_sha256": "f33e29bed9fee70f40e6728d11dd648516e938e7a9c7dcb93c3638045b8c042a", "original_candidate_line_sha256": "f33e29bed9fee70f40e6728d11dd648516e938e7a9c7dcb93c3638045b8c042a" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-010-p54", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62", "question_first_witness_sha256": "79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 2, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "98b11d2a24df69968d280dcde26b1cb55b3395676848ca6bc797b81ce31b4599", "arithmetic_steps": 0 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }