{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261005-advance-20261006-r07-descriptive-statistics-0019", "pilot_number": "R0056-28944", "number": 28944, "family_id": "qf-statistics-observation-mean", "title": { "zh-Hant": "第 28944 題：保留重數的算術平均", "en": "Question 28944: Arithmetic Mean Retaining Multiplicities" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261005-advance-20261006-r07-descriptive-statistics-0019", "batch_id": "batch-20261005-advance-20261006-r07-descriptive-statistics-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-descriptive-statistics" ], "title_zh": "保留重數的算術平均", "stem_zh": "明示非空觀測列 D=[21, 21, 21, -1, -1, -20]，保留重數，所有值為原創合成整數資料。求算術平均，答案為最簡正分母 p/q，整數亦保留 /1。", "answer": { "kind": "expression", "value": "41/6", "explanation_zh": "均值=41/6。每筆觀測都參與總和，再除以觀測筆數；先去重會改變資料，不把摘要當成機率律或母體推論。" }, "difficulty": { "band": "upper_secondary", "level": 3, "skills": [ "保留重數的算術平均" ] }, "variant": { "family_id": "qf-statistics-observation-mean", "seed": 202610067019, "parameters": { "values": [ 21, 21, 21, -1, -1, -20 ], "operation": "observation_mean", "semantic_template": "qf-statistics-observation-mean/{\"inputs\":[[\"-20\",\"1\"],[\"-1\",\"2\"],[\"21\",\"3\"]],\"operation\":\"observation_mean\"}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-05", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261005-advance-20261006-r07-descriptive-statistics-0019", "batch_id": "batch-20261005-advance-20261006-r07-descriptive-statistics-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "450fa301ebcec184b93f871a635626a7a9fb7008c54e7eb559f4c9a492f03360", "family_id": "qf-statistics-observation-mean", "seed": 202610067019, "semantic_template": "qf-statistics-observation-mean/{\"inputs\":[[\"-20\",\"1\"],[\"-1\",\"2\"],[\"21\",\"3\"]],\"operation\":\"observation_mean\"}" }, "fields": { "title": "Arithmetic Mean Retaining Multiplicities", "stem": "The nonempty observation list is D=[21, 21, 21, -1, -1, -20]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers.", "answer": { "kind": "expression", "value": "41/6", "explanation": "Sum all 6 observations with their multiplicities and divide by that positive count, obtaining 41/6. Do not average a deduplicated set or infer a population probability law." }, "skills": [ "Arithmetic Mean Retaining Multiplicities" ], "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": "明示非空觀測列 D=[21, 21, 21, -1, -1, -20]，保留重數，所有值為原創合成整數資料。求算術平均，答案為最簡正分母 p/q，整數亦保留 /1。", "en": "The nonempty observation list is D=[21, 21, 21, -1, -1, -20]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-statistics-observation-mean, observation_mean]", "after": "derived = [exact_empirical_mean_fraction]", "relation": "implies", "rule_id": "R56_source_answer_role_and_EXACT_family_interface", "conditions": [], "explanation": { "zh-Hant": "本題輸出依原scalar／set／是非／fraction角色；有理數整數保留/1。不加未宣告domain，不改中英來源、公式或題目。", "en": "Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [[21, 21, 21, -1, -1, -20]]", "after": "derived = [[-20, -1, -1, 21, 21, 21], 6]", "relation": "implies", "rule_id": "Statistics_FULL_observations_MULTIPLICITY_and_numeric_order", "conditions": [], "explanation": { "zh-Hant": "每次觀測都保留，包括相同數值的重複。numeric排序只改順序，不變n或總和；不能用set先去重，也不能lexical排序。只陳述本樣本，不推母體規律。", "en": "Retain EVERY observation,including repeated values.Numeric sorting changes order only,not n or the sum;do not deduplicate as a set or sort lexically.These are sample summaries,not inferred population laws." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Statistics ORIGINAL ALL observations sum := sum(21, 21, 21, -1, -1, -20)", "after": "Statistics ORIGINAL ALL observations sum = 41", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Statistics ORIGINAL ALL observations sum，不以小數近似替代。", "en": "Recompute Statistics ORIGINAL ALL observations sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "21", "21", "21", "-1", "-1", "-20" ], "output": "41", "label": "Statistics ORIGINAL ALL observations sum" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Statistics ORIGINAL observation sum over count := div(41, 6)", "after": "Statistics ORIGINAL observation sum over count = 41/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Statistics ORIGINAL observation sum over count，不以小數近似替代。", "en": "Recompute Statistics ORIGINAL observation sum over count with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "41", "6" ], "output": "41/6", "label": "Statistics ORIGINAL observation sum over count" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [[21, 21, 21, -1, -1, -20], 41, 6]", "after": "derived = [41/6]", "relation": "implies", "rule_id": "Statistics_exact_empirical_mean_not_frequency_dedup_mean", "conditions": [], "explanation": { "zh-Hant": "均值為每次觀測之和除觀測次數，重複值各算一次。可為負或未被觀測過的值，分母正約分且整數保/1。", "en": "The mean is the sum of EVERY observation divided by its count.Repetitions each contribute.It may be negative or unattained;reduce with positive denominator and retain/1." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null } ], "math_model": { "result": "41/6" }, "conclusion": { "canonical_value": "41/6", "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-20261005-advance-20261006-r07-descriptive-statistics-foundations", "batch_path": "public/data/questions/batches/batch-20261005-advance-20261006-r07-descriptive-statistics-foundations/questions.jsonl", "batch_sha256": "747ab8e474f0b3e38444e9bf7afff58120ac1b324b06ceb99e7b56747bcb3a70", "original_line": 19, "original_line_sha256": "3df2576fb2a56d6612824c4e07aa4d63573ceafcfec82c1e8b727c41c86cc886", "question_compact_sha256": "450fa301ebcec184b93f871a635626a7a9fb7008c54e7eb559f4c9a492f03360", "english_row_compact_sha256": "13c301ca09b8763b4149bdcb35cdbe9d8ba03563d8c9c5cabadf4a4682c128e1", "english_manifest_path": "public/data/i18n/editions/batch-20261005-advance-20261006-r07-descriptive-statistics-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261005-advance-20261006-r07-descriptive-statistics-foundations.en.jsonl" }, "mko_bindings": { "mko-descriptive-statistics": { "sha256": "bb188b4a4f67bc4e07da170bd9ea1b92aa0a51719cac24e1570cb156d8f76204", "english_sha256": "54dad0cd96ba57e6ace6bc691c6895d71b99feca5b90a348806c619d275baffe", "original_path": "public/data/mko/descriptive-statistics.json", "english_path": "public/data/i18n/editions/mko-descriptive-statistics.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "e340023047850507ee3fe2ae1466d606cfcbd26849aa3bb168d53248f38097e5", "original_candidate_sha256": "2d8feeb29472afeb58b55918a80bbbd8232d8d39d829b23235893fab7ed34b3e", "original_candidate_line_sha256": "2d8feeb29472afeb58b55918a80bbbd8232d8d39d829b23235893fab7ed34b3e" }, "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-p57", "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": "8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8", "question_first_witness_sha256": "3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 5, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "66df414d8625307ec3d03f921e6d3357b33a24806d30839b96e6e7b4a38bed4f", "arithmetic_steps": 2 }, "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." } }